AI-driven actigraphy data reconstruction

Aluno: Rodrigo Santos da Silva Orientador: Prof. Dr. Thiago Damasceno Cordeiro / Prof. Dr. Tiago Gomes de Andrade

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Master’s Thesis

AI-Driven Actigraphy Data
Reconstruction
by Rodrigo Santos da Silva
advised by

Prof. Ph.D. Thiago Damasceno Cordeiro
Prof. Ph.D. Tiago Gomes de Andrade

Federal University of Alagoas
Computing Institute
Maceió, Alagoas
December 12, 2024

FEDERAL UNIVERSITY OF ALAGOAS
Computing Institute

AI-DRIVEN ACTIGRAPHY DATA RECONSTRUCTION

Master’s thesis submitted to the Computing Institute from Federal University of Alagoas as
a partial requirement to obtain the master’s degree in Informatics.

Rodrigo Santos da Silva
Advisor: Prof. Ph.D. Thiago Damasceno Cordeiro
Advisor: Prof. Ph.D. Tiago Gomes de Andrade

Examining Board:
Bruno Alemida Pimentel
Glauber Rodrigues Leite
Allan de Medeiros Martins

Ph.D., UFAL
Ph.D., UFAL
Ph.D., UFRN

Maceió, Alagoas
December 12, 2024

MINISTÉRIO DA EDUCAÇÃO
UNIVERSIDADE FEDERAL DE ALAGOAS
INSTITUTO DE COMPUTAÇÃO
Av. Lourival Melo Mota, S/N, Tabuleiro do Martins, Maceió - AL, 57.072-970
PRÓ-REITORIA DE PESQUISA E PÓS-GRADUAÇÃO (PROPEP)
PROGRAMA DE PÓS-GRADUAÇÃO EM INFORMÁTICA

Folha de Aprovação
RODRIGO SANTOS DA SILVA
RECONSTRUÇÃO DE DADOS DE ACTIGRAFIA ORIENTADOS POR IA
AI-DRIVEN ACTIGRAPHY DATA RECONSTRUCTION
Dissertação submetida ao corpo docente do
Programa de Pós-Graduação em Informática da
Universidade Federal de Alagoas e aprovada em
12 de dezembro de 2024.
Banca Examinadora:

________________________________________
Prof. Dr. THIAGO DAMASCENO CORDEIRO
UFAL – PPGI- Instituto de Computação
Orientador

________________________________________
Prof. Dr. BRUNO ALMEIDA PIMENTEL
UFAL – PPGI- Instituto de Computação
Examinador Interno

________________________________________
Prof. Dr. GLAUBER RODRIGUES LEITE
UFAL – Instituto de Computação
Examinador Externo

________________________________________
Prof. Dr. TIAGO GOMES DE ANDRADE
UFAL – Faculdade de Medicina
Coorientador

________________________________________
Prof. Dr. ALLAN DE MEDEIROS MARTINS
UFRN – Universidade Federal do Rio Grande do Norte
Examinador Externo

Catalogação na fonte
Universidade Federal de Alagoas
Biblioteca Central
Divisão de Tratamento Técnico
Bibliotecária: Helena Cristina Pimentel do Vale – CRB4 - 661
S586a

Silva, Rodrigo Santos da.
AI-driven actigraphy data reconstruction / Rodrigo Santos da Silva. – 2025.
68 f.: il.
Orientador: Thiago Damasceno Cordeiro, Tiago Gomes de Andrade.
Dissertação (mestrado em Informática) – Universidade Federal de Alagoas,
Instituto de Computação. Maceió, 2024.
Texto em Inglês.
Bibliografia: f. 58-63.
1. Actigrafia. 2. Aprendizado de maquina. 3. Aprendizado profundo. 4. M10.
5. PIM. I. Título.
CDU: 004.8

Acknowledgments
I would like to express my gratitude to Prof Thiago Cordeiro for all the support, not only
in this thesis but in all my graduation time, for allowing me to become a better professional, a
better researcher and showing me the kind of professor that I want to be. A Special thanks to
Prof. Tiago Andrade and Prof. Lı́via Goes Gitaı́, who helped me with guidance and knowledge.
Additionally, I would like to express my gratitude to Prof. Rafael Mello and Prof. Bruno
Pimentel for generously dedicating their time and participating in the chair.
I want to thank my family, in special my parents Ivone and Roberval, my stepmother Marlene, and my siblings, Vitória and Lorenzo, for always being there for me and supporting my
studies. Thank you for showing me the importance of education.
There are no words to describe how Fernanda was important in this whole process, as a
wonderful partner, friend, and coworker. Thank you for always being there for me. Meeting
you in college was the greatest gift.
Also, a special thanks to all my friends from IFAL that have been by my side, even though
we are not studying together anymore. A special thanks to Aguida, Clara, Gabriel, Haniel,
Juliane, Karen, Luı́sa, Henrique, Márcio Henrique, Michael, João Falcão, and Pedro Henrique.
Lastly, a special thanks to my friends Bruno, Vı́vian, Sophia, Arthur, José Neto, Maria
Eduarda, Lucas, Larissa, Eduarda Chagas, Gabriel Germano, Rafael, Laryssa, Álvaro, Vanessa,
and Kevin for the pleasure of working, having fun, and learning with you all.
And thank you, dear reader, for having an interest in this work.

One learns from books and example only that certain things can be done. Actual learning
requires that you do those things.
Frank Herbert

Resumo
A actigrafia é uma técnica não invasiva que utiliza um dispositivo de pulso para monitorar padrões de sono e ritmos circadianos, fornecendo dados valiosos para o diagnóstico de
distúrbios do sono, como insônia e hipersonia. No entanto, lacunas nos dados de actigrafia
frequentemente ocorrem quando o dispositivo está “fora do pulso”. Este estudo explora o uso
de técnicas avançadas de Machine Learning e Deep Learning para imputar dados ausentes,
visando melhorar a precisão e a confiabilidade das análises. O principal objetivo é desenvolver pipelines robustos de imputação de dados, utilizando uma variedade de algoritmos, incluindo modelos de regressão (Random Forests, XGBoost e Support Vector Regression), com
estratégias de otimização de hiperparâmetros, como Grid Search e Randomized Search, e autoencoders otimizados pelo Keras Tuner. As principais estratégias de filtragem incluem o Dynamic Time Warping, aproveitando a variável M10, que representa as 10 horas de maior atividade diária, além de considerar os dados em torno dos perı́odos ”fora do pulso”. A metodologia
pré-processa conjuntos de dados de actigrafia do dispositivo ActTrust (Condor), incorporando
variáveis como temperatura, exposição à luz e métricas de atividade do paciente. Resultados
preliminares demonstram que os modelos de autoencoder superam os métodos tradicionais na
imputação de dados ausentes, reduzindo significativamente o erro médio quadrático. Modelos
de ensemble, ajustados para padrões especı́ficos de atividade, melhoram ainda mais a precisão
preditiva, conforme validado por testes estatı́sticos, como o teste de Wilcoxon. Esses achados
destacam o potencial desses modelos para aprimorar a prática clı́nica, permitindo avaliações
personalizadas do sono e apoiando intervenções direcionadas.
Palavras-chave: Actigrafia; Aprendizado de Máquina; Aprendizado Profundo; M10; PIM.

vii

Abstract
Actigraphy is a non-invasive technique that uses a wrist-worn device to track sleep patterns
and circadian rhythms, providing valuable data for diagnosing sleep disorders such as insomnia
and hypersomnia. However, gaps in actigraphy data often occur when the device is ”off-wrist.”
This study explores the use of advanced Machine Learning and Deep Learning techniques to
impute missing data, aiming to improve the accuracy and reliability of the analyses. The main
objective is to develop robust data imputation pipelines utilizing a range of algorithms, including
regression models (Random Forests, XGBoost, and Support Vector Regression), with hyperparameter optimization strategies such as Grid Search and Randomized Search and autoencoders
optimized by Keras Tuner. Along with these regression strategies, we have vital data filtering
strategies, including Dynamic Time Warping, leveraging the M10 variable, which represents
the 10 hours of highest daily activity, and considering data surrounding off-wrist periods. In
this study, the methodology preprocesses actigraphy datasets from the ActTrust device, incorporating variables such as temperature, light exposure, and activity metrics. Preliminary results
demonstrate that autoencoder models outperform traditional methods in imputing missing data,
significantly reducing Mean Squared Error. Combined models tailored to specific activity patterns further enhance predictive accuracy, as validated by statistical tests such as the Wilcoxon
test. These findings highlight the potential of these models to improve clinical practice by enabling personalized sleep assessments and supporting targeted interventions.
Keywords: Actigraphy; Machine Learning; Deep Learning; M10; PIM.

viii

List of Figures
2.1

Autoencoder structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1
3.2
3.3
3.4
3.5

Correlation Matrix - Patient 04. . . . . . . . . . . . . . . . . . . . . . . . . . .
Temperature distribution - patient 04. . . . . . . . . . . . . . . . . . . . . . . .
PIM distribution - Patient 04. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boxplots of Temperature x PIM - patient 04. . . . . . . . . . . . . . . . . . . .
Flowchart showing the methodology steps. . . . . . . . . . . . . . . . . . . . .

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4.1 PIM original data value from Patient 7. . . . . . . . . . . . . . . . . . . . . . .
4.2 OW detected moments for Patient 7 using the RFC. . . . . . . . . . . . . . . .
4.3 Results for patient 4 using Autoencoder method. . . . . . . . . . . . . . . . . .
4.4 Results for patient 4 using TMAE method. . . . . . . . . . . . . . . . . . . . .
4.5 Results for patient 4 using M10 method. . . . . . . . . . . . . . . . . . . . . .
4.6 Results for patient 4 using Around + DTW method. . . . . . . . . . . . . . . .
4.7 Summary Plot for patient 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Decision Plot for patient 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Summary Plot for patient 21. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10 Decision Plot for patient 21. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.11 Summary Plot for patient 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.12 Decision Plot for patient 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.13 Summary Plot for patient 25. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.14 Decision Plot for patient 25. . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Tables
3.1

Models Hyperparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Autoencoder Models Structure. . . . . . . . . . . . . . . . . . . . . . . . . . .
Autoencoder Performance Metrics for Patients. This method was developed
using and trained using cross-validation, with five folds and hyperparameter
optimization techniques, using Keras-tuner as a library. . . . . . . . . . . . . .
4.3 Wilcoxon Test for Patients Using Autoencoder . . . . . . . . . . . . . . . . . .
4.4 nparACT Results for Autoencoder: the sub-indices O means Original Data and
sub-indices R means Reconstructed Data. White and gray columns better visualize the same metrics in original and imputed datasets. . . . . . . . . . . . . .
4.5 TMAE Performance Metrics for Patients. This method was developed using
and trained using cross-validation, with also five folds and hyperparameter optimization techniques, using Optuna as a library. . . . . . . . . . . . . . . . . .
4.6 TMAE Models Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Wilcoxon Test for Patients Using TMAE . . . . . . . . . . . . . . . . . . . . .
4.8 nparACT Results for TMAE . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 ML Models with M10 Performance Metrics for Patients . . . . . . . . . . . . .
4.10 Wilcoxon Test for Patients Using ML Models with M10 . . . . . . . . . . . . .
4.11 nparACT Results for ML model with M10: the sub-indices O means Original
Data and sub-indices R means Reconstructed Data. White and gray columns
better visualize the same metrics in original and imputed datasets. . . . . . . .
4.12 Model Performance Metrics for Patients . . . . . . . . . . . . . . . . . . . . .
4.13 Wilcoxon Test for Patients Using ML Models with DTW . . . . . . . . . . . .
4.14 nparACT Results for ML models with DTW and Around Gap: the sub-indices
O means Original Data and sub-indices R means Reconstructed Data. White
and gray columns better visualize the same metrics in original and imputed
datasets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1
4.2

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List of Acronyms
AI - Artificial Intelligence.
DL - Deep Learning.
DTW - Dynamic Time Warping.
IS - Interdaily Stability.
IV - Intradaily Variability.
KNN - K-Nearest Neighbor.
L5 - Least 5.
M10 - Most 10.
MAE - Mean Absolute Error.
MASE - Mean Absolute Scaled Error.
ML - Machine Learning.
MSE - Mean Squared Error.
NLP - Natural Language Processing.
ON - On-Wrist.
OW - Off-Wrist.
PIM - Proportional Integral Mode.
PIMn - Normalized Proportional Integral Mode.
PSG - Polysomnography.
RA - Relative Amplitude.
R2 - Coefficient of Determination.

RFC - Random Forest Classifier.
RMSE - Root Mean Squared Error.
TAT - Time Above Threshold.
TATn - Normalized Time Above Threshold.
ZCM - Zero Crossing Mode.
ZCMn - Normalized Zero Crossing Mode.

xii

Contents
1

Introduction
1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Specific Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3
4

Theoretical Foundation
2.1 Actigraphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Dynamic Time Warping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Normalization and MinMaxScaler . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Dummy Regressor . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Decision Tree Regressor . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Random Forest Regressor . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 XGBoost Regressor . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.5 Support Vector Regressor . . . . . . . . . . . . . . . . . . . . . . . .
2.4.6 K-Nearest Neighbors Regressor . . . . . . . . . . . . . . . . . . . . .
2.4.7 Linear Regressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Autoencoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 General Autoencoder Structure . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Advantages and Disadvantages of Autoencoders: . . . . . . . . . . . .
2.5.3 Transformed Masked Autoencoder . . . . . . . . . . . . . . . . . . . .
2.6 Hyperparameter Optimization . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Grid Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 Randomized Search . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Coefficient of Determination . . . . . . . . . . . . . . . . . . . . . . .
2.7.2 Mean Squared Error . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.7.3 Root Mean Squared Error . . . . . . . . . . . . . . . . . . . . . . . .
2.7.4 Mean Absolute Error . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.5 Mean Absolute Scaled Error . . . . . . . . . . . . . . . . . . . . . . .
SHapley Additive exPlanations (SHAP) . . . . . . . . . . . . . . . . . . . . .
2.8.1 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Final Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Methodology
3.1 Random Forest Classifier (RFC) . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Cleaning Data, Feature transform and scaling . . . . . . . . . . . . . . . . . .
3.3 Removing Off-wrist remaining rows . . . . . . . . . . . . . . . . . . . . . . .
3.4 Filtering Data Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Vigil periods and M10 approaches . . . . . . . . . . . . . . . . . . . .
3.4.2 Around Gap with DTW approaches . . . . . . . . . . . . . . . . . . .
3.5 Regression Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Autoencoder and TMAE . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Machine Learning with Hyperparameter Tunning . . . . . . . . . . . .
3.5.3 Final Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Results
4.1 Autencoder with Full Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 TMAE with Full Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 ML Models with M10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 ML Models with Around Gap with DTW . . . . . . . . . . . . . . . . . . . .
4.5 SHAP Evaluations of ML Models . . . . . . . . . . . . . . . . . . . . . . . .

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Conclusion
5.1 Autoencoder with Full Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 TMAE with Full Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 ML Models with M10 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 ML Models with DTW and Around Gap Approach . . . . . . . . . . . . . . .
5.5 SHAP Evaluations of ML Models . . . . . . . . . . . . . . . . . . . . . . . .
5.6 General Observations and Future Directions . . . . . . . . . . . . . . . . . . .
5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.8
2.9

Bibliography

xiv

Chapter 1
Introduction
Sleep constitutes a fundamental pillar of our holistic well-being, exerting significant influence across multiple dimensions of our lives beyond mere repose. A pivotal function of sleep
is the facilitation of restoration and recuperation, affording our physiological framework the
opportunity for tissue, muscular, organic repair, and revitalization (Anafi et al., 2013). Furthermore, sleep assumes a critical role in cognitive processes, augmenting memory consolidation, problem-solving acumen, creative faculties, and aptitude for decision-making. It also
profoundly impacts emotional equilibrium, regulating mood, mitigating irritability, and fostering emotional equilibrium (Scott et al., 2021). Additionally, sleep is indispensable for physical
vitality, contributing to energy preservation, modulation of hormones governing appetite and
metabolic functions, and bolstering immune defenses against disorders and infections (Rogers
et al., 2024).
Moreover, optimal sleep quality manages stress levels, bolstering resilience and overall psychological well-being. Finally, adequate sleep enhances physical performance, refining coordination, reaction time, and athletic capabilities. In sum, prioritizing adequate and rejuvenating
sleep is imperative for optimizing our physical, mental, and emotional welfare (Charest and
Grandner, 2020).
Considering all the elements presented in the previous paragraphs, many methods were developed to monitor sleep and analyze the quality of sleep. One widely used method for tracking
sleep and activity in real-world settings is actigraphy, a non-invasive technique to identify individual sleep and wake cycles over a predetermined timeframe. It relies on using a portable
apparatus known as an actigraph, typically worn on the non-dominant wrist of adults (and occasionally positioned on the hip or ankle for children), resembling a wristwatch. This apparatus
aids in the detection of circadian rhythm disruptions, encompassing conditions such as hypersomnia, insomnia, circadian rhythm disorders, excessive somnolence, as well as advanced or
delayed sleep phase syndromes (Grandner and Rosenberger, 2019).
Its fundamental constituents comprise an accelerometer for recording movement, a data
storage unit, and a programmable timing mechanism. Specific models may incorporate additional components, such as spectrophotometers for discerning various luminosity levels and
1

2
thermometers for monitoring patient and environmental temperatures (Zambotti et al., 2019).
These supplementary components provide specialists with valuable insights into the patient’s
condition at specific junctures, thereby aiding in identifying sleep disorders. Leveraging the
data from the device, specialists can ascertain the onset and cessation of sleep, the latency period preceding sleep onset, total sleep duration, and the duration of wakefulness after sleep
onset during nocturnal periods.
Actigraphy may be employed autonomously or in tandem with supplementary sleep evaluation methods, such as sleep logs or overnight sleep studies (Garefelt et al., 2022). While it does
not attain the benchmark status attributed to polysomnography (PSG), actigraphy presents a viable alternative in circumstances where PSG proves impractical, notably with pediatric patients
or individuals necessitating tailored sleep assessments. This viability is owing to its capacity to
emulate a familiar wearing pattern akin to a watch, enhancing patient compliance and comfort.
Nonetheless, patients need to wear the device for prolonged durations poses several challenges, encompassing potential device damage, improper utilization, and instances of device
removal from the wrist, commonly denoted as ”off-wrist” (OW) occurrences. While specialists
advise patients against removing the device until the data collection process concludes, specific
individuals may opt to do so under varying circumstances, such as engaging in sports activities
(due to concerns regarding potential damage to the device), during showering, and, perhaps
most significantly, during specific periods of sleep.
In days where OW intervals exceed 4 hours, excluding that day from the analysis and focusing on the remaining days with less than 4 hours of OW periods is recommended (Tonon et al.,
2022). Attempting to reconstruct days with more than 4 hours of off-wrist time is comparable to
trying to infer data patterns with a significant amount of missing information, compromising the
imputation’s accuracy. This extended off-wrist duration introduces substantial uncertainty, as
prolonged periods without data reduce the reliability of pattern recognition algorithms and make
it difficult to maintain the temporal continuity necessary for accurate imputations. Moreover,
including such days in the analysis can introduce noise into the model, leading to less robust
predictions and potentially skewing the overall results. By excluding days with excessive offwrist time, we prioritize the integrity of the dataset, ensuring that the imputation process can
rely on a sufficient amount of valid, continuous data to generate more accurate reconstructions.
Moreover, attention must be directed towards the identification process for OW intervals,
as not all devices incorporate a structured mechanism to discern such instances. Even when
present, these mechanisms may not provide explicit indication to specialists regarding the nature
of a specific interval (i.e., whether it constitutes an OW moment or not). For instance, with the
ActTrust (Condor actigraph) and ActStudio (its accompanying software), it is customary to
interpolate OW intervals with 0 values (Fekedulegn et al., 2020). However, it’s imperative to
note that values of 0 hold significance within the data analysis spectrum for certain variables,
as exemplified by the Proportional Integral Mode (PIM), which signifies the level of activity
exhibited by the patient during a given moment.

Objectives

In this context, developing a tool capable of performing data imputation for patients exhibiting OW moments within their actigraphy examination results could aid specialists in diagnosing
sleep disorders or other conditions associated with sleep quality. Such a tool must be able to
adapt to each patient’s unique realities and patterns, as individuals inherently exhibit distinct
activity patterns attributable to variations in their daily routines and lifestyles. By accommodating these individualized patterns, the tool can enhance its efficacy in accurately assessing sleep
parameters and facilitating tailored interventions to address specific patient needs.

1.1

Objectives

The primary objective of this study is to present and test different approaches to perform
data imputation in actigraphy data using artificial intelligence techniques (mainly regression
techniques, with Machine Learning and Deep Learning approaches), with the central goal of
accurately imputing missing data within applicable scenarios while adapting to the unique data
patterns of individual patients.

1.2

Specific Objectives
For the primary goal, the following specific objectives will be contemplated.

1. Examine the efficacy of employing machine learning regression models and autoencoders
in the data imputation process.
2. Perform a comprehensive analysis of the current variables within the datasets.
3. Conduct pre-processing and feature transformation procedures on the existing data.
4. Construct AI models using Machine Learning and Deep Learning approaches with hyperparameter tunning techniques to optimize predictive performance across tested models for
individual patients.

1.3

Applications

The imputation of actigraphy data holds the potential to aid specialists in diagnosing or
predicting certain illnesses related to the circadian cycle. Additionally, by utilizing distinct
groups with predefined diagnoses of specific diseases, ML and DL models can be tailored for
imputing data in targeted scenarios, where each model demonstrates superior performance for
the corresponding group.
In practical applications, such as web-based platforms integrated with distributed systems,
deploying these models within an adaptable AI pipeline facilitates seamless replication and

Structure

scalability across diverse healthcare contexts. This deployment enables healthcare providers
to efficiently utilize imputed actigraphy data for informed decision-making, personalized treatment planning, and monitoring of patient progress over time. By harnessing the potential of
targeted modeling in data imputation, healthcare professionals can unlock valuable insights
from actigraphy data, improving patient care and managing circadian-related illnesses.

1.4

Structure

The introductory chapter of the thesis will delineate the subject matter under investigation.
Subsequently, the second chapter will provide an in-depth exploration of the background informing the methodologies elucidated within this thesis. This exploration elucidates actigraphy
data, a spectrum of ML and DL algorithms, and the procedural framework to perform the imputation and some state of rate articles. In the third chapter, a comprehensive exposition of
the methodology will ensue, outlining the procedural steps to derive the outcomes. The fourth
chapter will present the findings obtained from the study conducted. Lastly, the conclusion
chapter presents critical insights and considerations from the research endeavor.

Chapter 2
Theoretical Foundation
This section aims to provide readers with a solid foundation for understanding each step
of data imputation and concepts related to actigraphy. Key topics include Machine Learning
models, particularly regressors, and statistical metrics for evaluating model performance on
our datasets. We will also discuss techniques for handling and comparing data vectors, such
as Dynamic Time Warping, and methods for transforming and normalizing data to compatible
ranges. We will cover strategies for selecting the optimal model within a specified range of hyperparameters. Additionally, we will explore autoencoders, a technique used in the imputation
process. Regarding actigraphy, we will discuss the variables in our dataset, their importance in
our context, and the standard actigraphy metrics used to evaluate the circadian cycle.

2.1

Actigraphy

The first actigraphy devices were developed by J.S. Szymanski in the early 1970s and
have been continuously improved over the years to reach the current state of the art. As mentioned, modern actigraphy devices, such as accelerometers, include movement detectors and
have enough memory to record data for several weeks. These devices sample movement multiple times per second and store the data for later analysis. Researchers use computer programs,
like ActStudio, which comes with the ActTrust device we used in this study, to analyze the data.
As showed by Ancoli-Israel et al. (2003), these programs calculate levels of activity/inactivity,
rhythm parameters (such as amplitude or acrophase), and sleep/wake parameters (including total sleep time, percentage of time spent asleep, total wake time, percentage of time spent awake,
and the number of awakenings).
Different actigraphy devices collect varying data types, including documentation, luminosity, temperature, and others (Jumabhoy et al., 2019). These variables directly impact how researchers evaluate sleep quality and assess the patient’s circadian cycle definition. Researchers
commonly use actigraphy along with other methods such as PSG (Kaplan et al., 2012), Sleep
Diaries (Mazza et al., 2020), and Questionnaires (Brink-Kjaer et al., 2023) since actigraphy by

Actigraphy

itself does not have the same amount of information such as PSG, which is the gold pattern for
this area. But, the fact that it is usually not used to perform diagnoses doesn’t subtract the impact
that actigraphy can have in the process of facilitating collecting patient data since PSG requires
the participant to stay overnight in a sleep lab with various sensors attached to their body, actigraphy allows the individual to maintain their routine while wearing a small, wristwatch-like
device, which is especially useful for studying sleep patterns in real-world settings.
Furthermore, Actigraphy is much more affordable than PSG, making it accessible for largescale population studies or long-term monitoring in clinical practice. It’s often used for initial
screenings for sleep disorders or to track the effectiveness of treatments over time (Lehrer et al.,
2022). Actigraphy is especially useful for populations where sleep patterns are highly variable
or complex to monitor in a lab setting, such as children (Schoch et al., 2021), older persons
(de Feijter et al., 2021), or individuals with neurodegenerative conditions (Yang et al., 2023)
like Parkinson’s or Alzheimer’s disease.
In addition, Actigraphy is extensively used to assess sleep and circadian disruptions in populations exposed to irregular schedules, such as shift workers (Baek et al., 2020) and frequent
travelers (Feliciano et al., 2019). It helps in studying shift work disorder and the impact of jet
lag, which affects circadian alignment. Tracking real-time sleep-wake patterns provides critical
insights into the severity of circadian disruptions and the adaptation periods required to recover.

2.1.1

Variables

Since we are working with data from the Condor company, ActTrust, where the reader can find
all the same references in this hyperlink: ActTrust References, we have the following variables
for each patient:
• DATE/TIME: Date and time of the log.
• MS: Milliseconds value of the log.
• EVENT: Value of the event. Mixed Mode: Indicates whether there was an event during
this period.
• TEMPERATURE: Value of the temperature (in °C).
• EXT. TEMPERATURE: Value of the external temperature (in °C).
• ORIENTATION: Value of the device orientation.
• PIM: Activity value calculated in Proportional Integral Mode.
• PIMn: Value of the PIM normalized per second.
• TAT: Activity value calculated in Time Above Threshold mode.

Actigraphy

• TATn: Value of the TAT normalized per second.
• ZCM: Activity value calculated in Zero Crossing Mode.
• ZCMn: Value of the ZCM normalized per second.
• LUX: Value of the light intensity in lux (Only hardware 2.x).
• AMB LIGHT: Value of the ambient light intensity in µw/m² (Only hardware 3.x).
• RED LIGHT: Value of red light intensity in µw/cm² (Only hardware 3.x).
• GREEN LIGHT: Value of green light intensity in µw/cm² (Only hardware 3.x).
• BLUE LIGHT: Value of blue light intensity in µw/cm² (Only hardware 3.x).
• IR LIGHT: Value of IR light intensity in µw/cm² (Only hardware 3.x).
• UVA LIGHT: Value of Ultraviolet A light intensity in µw/cm² (Only hardware 3.x).
• UVB LIGHT: Value of Ultraviolet B light intensity in µw/cm² (Only hardware 3.x).
• STATE: Value generated on the sleep score screen that indicates the patient’s state:
Awake, Sleeping, Resting, OW (device removed), or Editable states.

2.1.2

Metrics

Actigraphy provides several metrics to evaluate various contexts in the field, including M10, L5,
and others. These metrics allow researchers to analyze and interpret patterns of movement and
rest, offering valuable insights into different aspects of the subject’s activity levels. Actigraphy
enables a deeper understanding of behavioral and physiological states by measuring parameters
like the most active and least active periods. This tool is especially useful in studies on sleep
patterns, circadian rhythms, and overall activity monitoring, contributing significantly to the
analysis of human behavior and health (Mitchell et al., 2017).
• Most ten (M 10): represents the 10 hours of highest daily activity, especially for identifying patient activity patterns and cycles.
• Least five (L5): the five consecutive hours of lowest average activity.
• Relative Amplitude (RA): the difference between periods of rest and activity defined as
follows in Equation 2.1:
M 10 − L5
RA =
(2.1)
M 10 + L5

Dynamic Time Warping

• Interday Variability (IV ): quantifies the alternation between phases of rest and activity. N
represents the total number of observations (or measurements) recorded by the actigraph,
Xi represents the activity count at the i-th time interval, Xi−1 represents the activity count
at the previous time interval, and Xm represents the mean (average) activity count across
all observed time intervals. The value is calculated in Equation 2.2:
IV =

2
i=2 (Xi − Xi−1 )
PN
(N − 1) i=1 (Xm − Xi )2

(2.2)

• Interday Stability (IS): indicates how daylight synchronizes with the patient’s activity. N
represents the total number of observations (or measurements) recorded by the actigraph
over a while, p denotes the number of periods into which the data is divided (often the
number of days in the study), Xh represents the average activity count for a specific period
h, and Xm represents the overall mean (average) activity count across all observations.
The value is calculated in Equation 2.3:
IS =

2.2

2
h=1 (Xh − Xm )
PN
p i=1 (Xm − Xi )2

(2.3)

Dynamic Time Warping

Dynamic Time Warping (DTW) is a powerful technique for measuring similarity between
two time-dependent sequences, commonly called time series. When working with time series,
comparing sequences with temporal misalignments or non-linear distortions is often necessary.
Two major approaches for this are Euclidean distance and DTW. Euclidean distance provides
a straightforward comparison but assumes perfect synchronization between sequences, making
it suitable only when the timing between data points is aligned. DTW, however, offers more
flexibility by allowing non-linear adjustments to the time axis, making it particularly valuable
in applications involving time series data across various domains, such as time series analysis
(Salvador and Chan, 2007), speech recognition (Ismail et al., 2020), handwriting recognition,
and gesture recognition (Vikram et al., 2013).
The main objective of DTW is to find an optimal alignment between two temporal sequences, say X and Y , by allowing non-linear adjustments along the time axis to minimize
their overall distance. Unlike traditional methods, such as Euclidean distance, which assume
that the sequences are perfectly synchronized, DTW dynamically stretches and compresses segments of the sequences as needed to obtain the best possible match. This approach is beneficial
for data where timing can vary, like speech, where similar phonetic sounds can occur at different
speeds, or in motion analysis, where the same gestures may take different lengths to perform.
To achieve this, DTW computes a cumulative cost matrix, which records the cost of aligning each element in sequence X with each component in sequence Y . This cost matrix enables

Normalization and MinMaxScaler

the calculation of the minimum distance path between the sequences by summing up the minor alignment costs. The warping path, a sequence of index pairs, then maps elements from
sequence X to elements in sequence Y , indicating the best alignment between them.
DTW’s flexibility in handling sequences of varying lengths and timings makes it worthwhile
in fields where exact synchronization between time series elements is unrealistic or undesirable.
It is also robust to time shifts and distortions, meaning it can capture similarities in patterns that
occur at different rates or time frames. This property is essential in real-world applications,
where temporal variations often exist due to environmental or biological factors, as in our case.
However, DTW is computationally intensive, especially for long sequences, because it requires the calculation of the entire cumulative cost matrix, which has a complexity of O(n×m),
where n and m are the lengths of the two sequences. This can become a limitation when dealing
with large datasets, as DTW can be slower than other distance measures. Additionally, DTW
does not directly account for the amplitude differences between sequences, which can be problematic if the comparison requires both timing and amplitude matching. In situations where
these problems happen, we may have different techniques to deal with each case, such as fastDTW and SegmentedDTW to deal with lower computational complexity, Z-normalization, and
Derivative DTW to handle normalization problems.

2.3

Normalization and MinMaxScaler

Normalization is a preprocessing technique in ML designed to adjust feature scales so
that each feature contributes equally to distance-based calculations. This process is used with
algorithms sensitive to feature magnitudes, such as k-nearest neighbors and support vector machines, which rely on accurate distance measurements for optimal performance (Ali, 2022).
Generally, normalization has a more pronounced effect on models that leverage distance as
a core strategy. In contrast, its impact is less significant in tree-based models like Decision
Tree, Random Forest, and XGBoost, which use rule-based paths rather than distances to guide
decisions. This way, normalization promotes consistency and enhances learning by rescaling
features to a uniform range.
A commonly applied normalization technique transforms each feature individually to a specific range, often between 0 and 1. This scaling is achieved by subtracting the minimum value
of each feature and dividing by the range (i.e., the difference between the maximum and minimum values). This linear transformation preserves the relationships between data points while
maintaining the original distribution shape. Equation 2.4 defines this normalization formula.
x′i =

xi − min(x)
max(x) − min(x)

Where:
• xi is the original value of the feature for the i-th sample.

(2.4)

Models

• min(x) is the minimum value of the feature x in the dataset.
• max(x) is the maximum value of the feature x in the dataset.
• x′i is the normalized value of xi , scaled to the range [0, 1].
One key advantage of using MinMaxScaler is its simplicity and effectiveness in preparing
data for machine learning models that assume normalized input. By aligning feature scales,
the MinMaxScaler reduces bias introduced by features with more extensive ranges and facilitates faster convergence in optimization algorithms. This is particularly beneficial for neural
networks, where varying feature scales can lead to slower training and convergence issues due
to skewed gradient updates.
MinMaxScaler is straightforward to use and integrates seamlessly into a data processing
pipeline. Its impact is improving the accuracy and performance of machine learning models
across for specific algorithms a wide range of tasks. In summary, normalization via MinMaxScaler is a recommended technique in the data preprocessing arsenal, enabling more robust and
reliable model training and deployment across diverse domains.

2.4

Models

Artificial intelligence incorporates a variety of learning methodologies, including supervised, unsupervised, reinforcement learning, and others. Given that the dataset in question consists of time-series data collected over multiple days, with each record precisely timestamped
to specific patient events, employing supervised learning—specifically regression analysis—is
judiciously choseWe specific algorithms, we will provide illustrative examples to enhance understanding, as the mathematical formalism may not be immediately apparent. All examples
will be implemented in the Python programming language, utilizing libraries such as Scikitlearn for machine learning models.

2.4.1

Dummy Regressor

The Dummy method defines possible baselines for the process the user is trying to reach. It uses
essential approaches, such as the DummyRegressor, which builds a regressor to give a constant
value, the median, or the mean of the analyzed data. It is essential to inform that these methods
are not used as final predictors but only for comparison.
In addition to serving as a baseline, the DummyRegressor is valuable for assessing the performance of more complex models by providing a simple reference point. Using basic strategies
such as always predicting the mean, median, or constant value allows users to evaluate whether
a more sophisticated model adds real value over trivial approaches. The DummyRegressor can
also help identify potential data issues. For instance, if a complex model performs similarly to

Models

or worse than a DummyRegressor, this might indicate problems such as data noise, poor feature
selection, or model overfitting. A running example can be seen in 2.1.
Algorithm 2.1: Dummy Regressor example in Python.
from sklearn.dummy import DummyRegressor
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error
import numpy as np
import pandas as pd
# Generate a simple dataset
np.random.seed(42)
X = np.random.rand(100, 1) * 10 # Features
y = 2.5 * X.squeeze() + np.random.randn(100) * 2

# Target with some noise

# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2,
random_state=42)
# Initialize the Dummy Regressor with ’mean’ strategy
dummy_regressor = DummyRegressor(strategy="mean")
# Fit the Dummy Regressor to the training data
dummy_regressor.fit(X_train, y_train)
# Predict the target values for the test set
y_pred = dummy_regressor.predict(X_test)
# Evaluate the model
mse = mean_squared_error(y_test, y_pred)

2.4.2

Decision Tree Regressor

A Decision Tree algorithm is structured akin to a tree-like data framework, centralizing
on interconnected decision nodes via branches that enforce specific data rules to evaluate the
assessed statements’ veracity. The data is then transmitted to subsequent decision nodes or
terminal nodes, commonly referred to as leaves, which do not possess further branching and
represent the conclusion of the analysis. In the context of regression tasks, the Decision Tree
Regressor scrutinizes the attributes of the problem and, by utilizing predetermined hyperparameters such as maximum depth and number of leaves, forecasts continuous or discrete outcomes.
Decision trees are esteemed for their high interpretability, offering an explicit visual representation of the decision-making process. This attribute renders them particularly advantageous
for applications wherein comprehending the model’s underlying logic is imperative. Unlike linear models, Decision Trees adeptly manage intricate, non-linear relationships between features

Models

and the target variable. Consequently, this capability enhances their robustness across diverse
regression tasks. These characteristics highlight the application of decision tree regressors in
several areas, from economic problems (Sai et al., 2023) to health care issues (Oyeleye et al.,
2022).
Moreover, Decision Trees are scale-invariant, obviating the need for normalizing or scaling
features. This characteristic significantly simplifies the preprocessing stage of data analysis.
Furthermore, Decision Trees can handle numerical and categorical data, facilitating their application to various problems without requiring extensive data transformations.
Additionally, Decision Trees may undergo pruning during training to circumvent overfitting. This pruning process diminishes the tree’s size by excising segments that contribute
minimally to the prediction of the target variable. Lastly, by modifying hyperparameters such
as maximum depth, minimum samples split, and minimum samples leaf, users can regulate the
model’s complexity, balancing under-fitting and over-fitting. A running example can be seen in
2.2.
Algorithm 2.2: Decision Tree Regressor example in Python.
from sklearn.tree import DecisionTreeRegressor
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error
import numpy as np
# Generate a simple dataset
np.random.seed(42)
X = np.sort(np.random.rand(100, 1) * 10, axis=0) # Features (sorted for
visualization)
y = np.sin(X).ravel() + np.random.randn(100) * 0.1 # Target with some
noise
# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2,
random_state=42)
# Initialize the Decision Tree Regressor
tree_regressor = DecisionTreeRegressor(max_depth=4, random_state=42)
# Fit the regressor to the training data
tree_regressor.fit(X_train, y_train)
# Predict the target values for the test set
y_pred = tree_regressor.predict(X_test)
# Evaluate the model
mse = mean_squared_error(y_test, y_pred)
print(f"Mean Squared Error: {mse:.2f}")

Models

2.4.3

Random Forest Regressor

Random Forest is fundamentally an ensemble learning technique that employs the bagging
method. This approach entails constructing multiple Decision Trees, as previously described,
each developed from a bootstrap sample of the data, a random selection with replacement from
the entire dataset. This methodology aims to reduce the model’s variance without significantly
elevating bias. By aggregating the predictions of individually cultivated trees through averaging
or majority voting methods, Random Forest enhances predictive accuracy and mitigates overfitting, a prevalent issue in single Decision Tree models.
In conjunction with bagging, Random Forest incorporates randomness into selecting features considered for splitting at each node within the decision trees. The algorithm randomly
selects a subset of the total features at each node during tree construction. This strategy ensures
that the trees remain de-correlated and introduces an additional layer of variance reduction.
Typically, this leads to a more robust ensemble that outperforms individual trees when applied
to unseen data.
Furthermore, Random Forest features an intrinsic mechanism to estimate its prediction error
without requiring a separate validation dataset, known as the out-of-bag (OOB) error estimation.
The algorithm constructs each tree in the forest from a distinct bootstrap sample drawn from
the original dataset. The data points excluded from this sample, referred to as the OOB data,
serve to assess the performance of each tree. The OOB error, calculated as the average error of
predictions on the OOB samples across all trees, is an unbiased estimator of the model’s overall
performance.
Lastly, Random Forest is particularly adept in scenarios involving large datasets (Zainab
et al., 2020) with numerous features, as in our situation. It can manage thousands of input
variables without necessitating feature elimination and efficiently addresses missing data. Furthermore, the ensemble approach employed by Random Forest generally results in enhanced
performance and greater robustness than a single decision tree can achieve. A running example
can be seen in 2.3.
Algorithm 2.3: Random Forest Regressor example in Python.
from sklearn.ensemble import RandomForestRegressor
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error
import numpy as np
# Generate a simple dataset
np.random.seed(42)
X = np.sort(np.random.rand(100, 1) * 10, axis=0) # Features (sorted for
visualization)
y = np.sin(X).ravel() + np.random.randn(100) * 0.1 # Target with some
noise

Models

# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2,
random_state=42)
# Initialize the Random Forest Regressor
forest_regressor = RandomForestRegressor(n_estimators=100, max_depth=4,
random_state=42)
# Fit the regressor to the training data
forest_regressor.fit(X_train, y_train)
# Predict the target values for the test set
y_pred = forest_regressor.predict(X_test)
# Evaluate the model
mse = mean_squared_error(y_test, y_pred)
print(f"Mean Squared Error: {mse:.2f}")

2.4.4

XGBoost Regressor

XGBoost is one of the many gradient boosting frameworks, such as LightGBM 1 and CatBoost 2 . The gradient boosting process also comes from tree variation algorithms, such as
random forest. Still, instead of planting a load of independent trees at random (the bagging
technique we explained before), each tree created will be weighted to compensate for any residual error in the previous tree, along with the hyperparameter of max depth, n estimators and
now with a learning rate hyperparameter (where its range is between 0 and 1).
Also, XGBoost includes L1 (Lasso Regression) and L2 (Ridge Regression) regularization
terms in its cost function, which are not typical in standard implementations of gradient boosting that we cited before. This regularization helps reduce the overfitting process, making XGBoost particularly robust against model complexity, even when dealing with many features and
datasets with multiple dimensions.
Moreover, we can use XGBoost for regression, classification (binary and multi-class), ranking, and user-defined prediction scenarios. It supports user-defined objective functions and
evaluation criteria, adding a layer of flexibility for various scientific and business problems.
With such an enormous flexibility, as many tree-based models have, XGBoost can be applied
in different scenarios, such as health (Jyothsna et al., 2022), and education (Jeganathan et al.,
2022). A running example can be seen in 2.4.
Algorithm 2.4: XGBoost Regressor example in Python.
from xgboost import XGBRegressor
1
2

https://www.microsoft.com/en-us/research/project/lightgbm/
https://catboost.ai/

Models

from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error
import numpy as np
# Generate a simple dataset
np.random.seed(42)
X = np.random.rand(100, 1) * 10 # Features
y = np.sin(X).ravel() + np.random.randn(100) * 0.1
noise

# Target with some

# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2,
random_state=42)
# Initialize the XGBoost Regressor
xgb_regressor = XGBRegressor(n_estimators=100, max_depth=4, learning_rate
=0.1, random_state=42)
# Fit the regressor to the training data
xgb_regressor.fit(X_train, y_train)
# Predict the target values for the test set
y_pred = xgb_regressor.predict(X_test)
# Evaluate the model
mse = mean_squared_error(y_test, y_pred)
print(f"Mean Squared Error: {mse:.2f}")

2.4.5

Support Vector Regressor

The Support Vector Regressor (SVR) is a variation of the Support Vector Machine (SVM),
adapted for regression problems instead of classification. SVR aims to fit a function approximating the target values within a tolerance margin (ϵ). We use the ϵ-insensitive loss function,
so we do not penalize errors within the margin ϵ. The SVR algorithm identifies the best line or
hyperplane that fits the data while ensuring most data points fall within this margin. Only the
data points outside this margin, known as support vectors, influence the model. The objective of
SVR is to minimize the complexity of the model (i.e., the norm of the weights) while penalizing any deviations from the margin that exceeds ϵ, thereby achieving a balance between model
complexity and prediction accuracy.
Furthermore, one of the critical strengths of SVR, similar to SVM for classification, is its
ability to handle non-linear relationships in the data using the kernel trick. Kernels such as
polynomial, radial basis function (RBF), and sigmoid can be applied to transform the input
features into higher-dimensional spaces where a linear regression model can be more effective.

Models

Additionally, we need to tune several hyperparameters in SVR for optimal performance.
These include the regularization parameter C, which controls the trade-off between achieving a
low error on the training data and minimizing the model complexity. This ϵ parameter defines
the width of the insensitive zone and parameters associated with the chosen kernel (e.g., the
gamma parameter for the RBF kernel).
Furthermore, we can apply SVR in various scenarios, such as predicting environmental
disasters (Asim et al., 2018), financial accounting (Chakri et al., 2023), geology (Zhu et al.,
2018), and others. A running example can be seen in 2.5.
Algorithm 2.5: Support Vector Regressor example in Python.
from sklearn.svm import SVR
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
from sklearn.metrics import mean_squared_error
import numpy as np
# Generate a simple dataset
np.random.seed(42)
X = np.random.rand(100, 1) * 10 # Features
y = np.sin(X).ravel() + np.random.randn(100) * 0.1
noise

# Target with some

# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2,
random_state=42)
# Standardize the features (important for SVR)
scaler = StandardScaler()
X_train_scaled = scaler.fit_transform(X_train)
X_test_scaled = scaler.transform(X_test)
# Initialize the Support Vector Regressor
svr_regressor = SVR(kernel="rbf", C=1.0, epsilon=0.1)
# Fit the regressor to the training data
svr_regressor.fit(X_train_scaled, y_train)
# Predict the target values for the test set
y_pred = svr_regressor.predict(X_test_scaled)
# Evaluate the model
mse = mean_squared_error(y_test, y_pred)
print(f"Mean Squared Error: {mse:.2f}")

Models

2.4.6

K-Nearest Neighbors Regressor

The K-Nearest Neighbors (KNN) algorithm is a non-parametric method used for both classification and regression tasks. In numerical regression, KNN makes predictions by identifying
the K nearest data points (neighbors) to the given input based on a specified distance metric
(commonly Euclidean distance). The algorithm then predicts the target value for the input by
computing the average of the target values of these K nearest neighbors. This method leverages the intuition that data points with similar features will also have identical target values,
producing predictions that reflect local patterns in the dataset.
The KNN regression relies heavily on the choice of distance metric to identify the nearest
neighbors. While we commonly use Euclidean distance, we can also employ other distance
measures, such as Manhattan, Minkowski, or cosine distance, depending on the nature of the
data and the problem at hand.
Moreover, the performance of KNN regression depends on selecting the parameter K, which
determines the number of nearest neighbors to consider. A small value of K makes the model
sensitive to noise (over-fitting), while a significant value of K can smooth out the predictions
too much (under-fitting). We typically use cross-validation to choose an optimal K. One of the
many techniques to find the ideal K is the ”Elbow Technique,” where he can select a series of
K-values from a defined range and look at the result of each K, choosing the best K in the end.
KNN is another famous algorithm that we can apply in areas like energy prediction (Khan and
Byun, 2020), healthcare problems (Niemeijer et al., 2008), and physics (Durbin et al., 2021). A
running example can be seen in 2.6.
Algorithm 2.6: K-Nearest Neighbors Regressor example in Python.
from sklearn.neighbors import KNeighborsRegressor
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error
import numpy as np
# Generate a simple dataset
np.random.seed(42)
X = np.random.rand(100, 1) * 10 # Features
y = np.sin(X).ravel() + np.random.randn(100) * 0.1
noise

# Target with some

# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2,
random_state=42)
# Initialize the KNN Regressor
knn_regressor = KNeighborsRegressor(n_neighbors=5, weights=’uniform’)
# Fit the regressor to the training data

Models

knn_regressor.fit(X_train, y_train)
# Predict the target values for the test set
y_pred = knn_regressor.predict(X_test)
# Evaluate the model
mse = mean_squared_error(y_test, y_pred)
print(f"Mean Squared Error: {mse:.2f}")

2.4.7

Linear Regressor

The Linear Regression that we are using in this context is the Ordinary Least Squares (OLS)
method for Linear Regression, which aims to fit a linear model by determining the coefficients
β = (β1 , β2 , . . . , βp ) that minimize the residual sum of squares between the observed target
values in the dataset and the target values predicted by the linear model. This approach seeks
to find the best-fitting line (or hyperplane in higher dimensions) that reduces the discrepancy
between the actual and predicted values, thereby providing the most accurate approximation of
the underlying relationship between the dependent and independent variables.
Linear regression relies on several key assumptions. First, the assumption of linearity posits
that the relationship between the dependent and independent variables is linear. Second, the
independence assumption requires that each observation is independent of the others. Third,
homoscedasticity implies that the variance of the error terms remains constant across all levels
of the independent variables. Finally, the normality assumption asserts that the error terms
follow a normal distribution.
Linear Regression, as a classic regression structure, has been applied in the most different
areas, such as health (Nichols et al., 2022), politics (Han et al., 2023), and even in energy
consumption (Olu-Ajayi et al., 2022). A running example can be seen in 2.7.
Algorithm 2.7: Linear Regressor example in Python.
from sklearn.linear_model import LinearRegression
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error
import numpy as np
# Generate a simple dataset
np.random.seed(42)
X = np.random.rand(100, 1) * 10 # Features
y = 2.5 * X.squeeze() + np.random.randn(100) * 2

# Target with some noise

# Split the data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2,
random_state=42)

Autoencoder

# Initialize the Linear Regressor
linear_regressor = LinearRegression()
# Fit the regressor to the training data
linear_regressor.fit(X_train, y_train)
# Predict the target values for the test set
y_pred = linear_regressor.predict(X_test)
# Evaluate the model
mse = mean_squared_error(y_test, y_pred)
print(f"Mean Squared Error: {mse:.2f}")
# Display model coefficients
print(f"Coefficient: {linear_regressor.coef_[0]}")
print(f"Intercept: {linear_regressor.intercept_}")

2.5

Autoencoder

Although autoencoders are AI models, we have dedicated a separate section to them, as
they follow a unique structure within Deep Learning (DL) and differ from traditional models.
Autoencoders are primarily used for unsupervised learning to encode data efficiently. Designed
to learn a compressed representation, or encoding, of a dataset, autoencoders are often applied
for tasks such as dimensionality reduction or feature extraction. Their basic structure consists
of three main components: an encoder, which compresses the data; a latent space, which holds
the compact representation; and a decoder, which reconstructs the input data. Autoencoders
are widely applied in areas like image denoising (Tripathi, 2021), dimensionality reduction
(Lin et al., 2020), anomaly detection (Torabi et al., 2023), and feature extraction (Yu et al.,
2021), where they reveal hidden patterns and enhance computational efficiency by working with
lower-dimensional representations. Advanced variations, such as convolutional autoencoders
(Bedi and Gole, 2021), variational autoencoders (Kim et al., 2021) and Transformer Masked
Autoencoder (Mendonça Freire and Curi, 2024), further expand their functionality, making
them powerful tools for both generative tasks and complex data processing.

2.5.1

General Autoencoder Structure

The architecture of an autoencoder consists of three key components that are described in
the following items and can also be seen in the Figure 2.1.

Autoencoder

Figure 2.1: Autoencoder structure.
• Encoder: The encoder compresses the input data into a latent representation, reducing
its dimensionality and encoding it into a more compact form. Mathematically, the encoder can be represented as a function E(x) = z, where x denotes the input data and z
represents the encoded form. This process typically involves linear transformations, such
as z = W · x + b, where W and b are the weight matrix and bias vector, followed by
non-linear activation functions like ReLU, sigmoid, or tanh. In specific models, such as
transformers, an attention mechanism is integrated to help the encoder focus on the most
relevant parts of the input. The resulting output z is a reduced-dimensional representation
optimized to minimize a loss function that captures the essential structure of the input
data.
• Latent Space: The latent space is the compact, encoded representation of the input data,
containing the essential features necessary to reconstruct the original input. The latent
space is typically of a lower dimension than the input space, forcing the network to learn
efficient representations that are both meaningful and minimal.
• Decoder: The decoder reconstructs the input data from the encoded latent representation.
Conceptually, the decoder can be viewed as a function D(z) = x′ , where x′ is the reconstructed version of the input. The decoding process applies a transformation sequence

Autoencoder

that maps the latent representation z back to the original input space. These transformations may include linear operations, such as x′ = W ′ · z + b′ , where W ′ and b′ are the
weight matrix and bias vector specific to the decoder, followed by non-linear activation
functions to enhance reconstruction quality. In some models, the decoder also incorporates attention mechanisms to focus on relevant portions of the encoded representation.
The final output x′ aims to approximate the original input x closely, and the decoder is
trained to minimize a loss function measuring the difference between x and x′ , ensuring
that key features of the input data are retained.

2.5.2

Advantages and Disadvantages of Autoencoders:

Autoencoders have several strengths and limitations, particularly relevant when dealing
with complex data such as actigraphy.
- Advantages:
• Handling Non-Linear Relationships: Autoencoders can capture complex, non-linear
patterns, making them highly suitable for modeling intricate structures in actigraphy data.
• Dimensionality Reduction: Autoencoders achieve dimensionality reduction by learning compressed representations, minimizing noise, and focusing on features essential for
accurate imputation.
• Flexibility: They can be customized for different data types and optimized for specific
imputation tasks, offering flexibility in handling various missing data scenarios.
• Scalability: Autoencoders efficiently handle large datasets, making them ideal for extensive actigraphy datasets collected over extended periods.
• Automatic Feature Learning: Autoencoders learn features automatically from the data
without requiring extensive manual feature engineering, saving time and reducing the
need for domain-specific expertise.
- Disadvantages:
• Complexity: Training autoencoders can be computationally intensive and timeconsuming, particularly for large actigraphy datasets, requiring significant resources.
• Data Requirements: Autoencoders require large amounts of data to train effectively, and
performance may decline if the actigraphy dataset is limited.
• Overfitting: There is a risk of overfitting, where the autoencoder may learn noise rather
than meaningful patterns, leading to poor generalization on unseen data.

Autoencoder

• Hyperparameter Tuning: Autoencoder performance depends heavily on the choice of
hyperparameters, such as the number of layers, neurons, and learning rate, which can be
challenging and time-intensive to optimize.
• Interpretability: The internal representations learned by autoencoders are often difficult
to interpret, complicating understanding how the model imputes missing data.
• Imputation Bias: If the missing data pattern is non-random, the autoencoder may introduce bias in the imputation, potentially affecting subsequent analyses.

2.5.3

Transformed Masked Autoencoder

The Transformed Masked Autoencoder (TMAE) is an advanced variation of the traditional
autoencoder designed to enhance the handling of missing data by incorporating a masking
mechanism. In this approach, sections of the input data are masked during training, simulating missing data patterns. The autoencoder is then trained to reconstruct the complete data
from these partial inputs, thereby learning to handle gaps in the data effectively.
• Masking Mechanism: The TMAE applies a masking operation to random sections of
the input data, effectively hiding certain values. This masking allows the autoencoder to
be trained in a way that mimics real-world scenarios with incomplete data. By learning
to reconstruct missing segments, the TMAE is better equipped for imputation tasks, as it
has been explicitly trained to handle incomplete data.
• Transformation Layer: Besides masking, TMAEs often include a transformation layer
that processes the masked input, helping to reformat it in a way that preserves essential
data patterns. This transformation improves the model’s ability to detect and replicate underlying patterns, even in gaps. For instance, positional encodings or learned embeddings
may retain context and temporal structure within the data.
• Reconstruction Process: The decoder in a TMAE reconstructs the missing data by predicting the masked values based on the patterns learned during training. This process
enables the model to fill in missing values and capture temporal dependencies and correlations within the data, making it particularly effective for actigraphy datasets with
irregular missing patterns.
The TMAE’s design makes it especially suitable for tasks where missing data is standard,
as it has been optimized to reconstruct data with partial visibility. This model adapts well to
actigraphy applications, where data collection inconsistencies, such as off-wrist episodes, can
result in gaps. By explicitly training on masked data, the TMAE provides a more robust and
reliable solution for imputing missing values, capturing both linear and non-linear relationships
in the data.

Hyperparameter Optimization

2.6

Hyperparameter Optimization

Optimization plays a vital role in AI because the performance of algorithms is significantly
affected by the choice of hyperparameters. While each algorithm has a default set of hyperparameters, relying solely on these defaults is often insufficient, as they may not yield optimal
results for different problems. Consequently, exploring various hyperparameter combinations
is essential, considering available options, search time complexity, and resource consumption.

2.6.1

Grid Search

Grid Search is an exhaustive method for hyperparameter optimization, systematically evaluating every specified combination for the analyzed model. By default, Grid Search employs
cross-validation, which divides the dataset into multiple parts, training and testing the AI model
with different dataset splits. It ultimately selects the best configuration based on the desired
metric. In this way, we can resume the steps to:
• Defining the hyperparameters and their ranges.
• Creating a grid of all possible hyperparameter combinations.
• Training and evaluating the model for each combination using cross-validation.
• Selecting the best hyperparameters based on performance.

2.6.2

Randomized Search

Randomized Search can be a more efficient alternative to Grid Search, mainly when dealing with an ample hyperparameter space. Grid Search employs a discrete approach, evaluating
every possible combination of specified values, which can be time-consuming and resourceintensive. In contrast, a Randomized Search randomly selects hyperparameters from the specified values, continuing until it reaches a predefined number of iterations. The primary steps
involved include:
• Defining the hyperparameters and their distributions.
• Specifying the number of iterations (random samples).
• Randomly sampling hyperparameter combinations.
• Training and evaluating the model for each sampled combination using cross-validation.
• Selecting the best hyperparameters based on performance.

Metrics

2.7

Metrics

To evaluate each regressor that we are using to perform data imputation in each patient
dataset, we need statistical metrics to do so. In this context, we selected five metrics to do so,
which are R2 , MSE, RMSE, MAE, and MASE.

2.7.1

Coefficient of Determination

The coefficient of determination, also known as R², measures the proportion of variability
in the dependent variable explained by the model. It is defined by the Equation 2.5.
Pn
(yi − ŷi )2
R = 1 − Pi=1
n
2
i=1 (yi − ȳ)
2

(2.5)

where yi is the actual value of the dependent variable; ŷi is the predicted value of the dependent
variable; ȳ is the mean of the actual values yi ; and n is the number of observations.
The R2 value ranges from 0 to 1, where R2 = 1 indicates that the regression model perfectly
fits the data and R2 = 0 suggests that the model does not explain any of the variability in the
response variable around its mean. Higher values of R2 indicate a better fit of the model.

2.7.2

Mean Squared Error

The Mean Squared Error (MSE) is a standard measure of the average of the squares of the
errors. We define it in the Equation 2.6.
n

1X
(yi − ŷi )2
MSE =
n i=1

(2.6)

where yi is the actual value of the dependent variable; ŷi is the predicted value of the dependent
variable and; n is the number of observations.
The MSE measures the average of the squares of the errors, which are the differences between the actual and predicted values. Lower MSE values indicate a better fit of the model to
the data.

2.7.3

Root Mean Squared Error

The Root Mean Squared Error (RMSE) measures the differences between values predicted
by a model and the values observed. It is the square root of the average of the squared differences. We can define it as shown in Equation 2.7.
v
u n
u1 X
RMSE = t
(yi − ŷi )2
n i=1

(2.7)

SHapley Additive exPlanations (SHAP)

where yi is the actual value of the dependent variable; ŷi is the predicted value of the dependent variable and; n is the number of observations.
The RMSE quantifies the magnitude of the error. Lower values of the RMSE indicate a
better fit of the model to the data.

2.7.4

Mean Absolute Error

The Mean Absolute Error (MAE) is a measure of errors between paired observations expressing the same phenomenon. We can define it as shown in Equation 2.8.
n

1X
|yi − ŷi |
MAE =
n i=1

(2.8)

where yi is the actual value of the dependent variable; ŷi is the predicted value of the dependent
variable and; n is the number of observations.
The MAE measures the average magnitude of the errors in a set of predictions without
considering their direction. It is the average absolute difference between prediction and actual
observation over the test sample. Lower values of MAE indicate a better fit of the model to the
data.

2.7.5

Mean Absolute Scaled Error

The Mean Absolute Scaled Error (MASE) measures the accuracy of a forecast model. We
can define it as shown in Equation 2.9.
Pn
|yi − ŷi |
MASE = 1 Pni=1
i=2 |yi − yi−1 |
n−1
1
n

(2.9)

where yi is the actual value of the dependent variable; ŷi is the predicted value of the dependent
variable; n is the number of observations and; yi−1 is the actual value at the previous time step.
The MASE scales the mean absolute error by the in-sample mean absolute error from a
naive forecast model. It is scale-independent and can compare forecast accuracy across various
scales. Lower values of MASE indicate a better fit of the model to the data.

2.8

SHapley Additive exPlanations (SHAP)

SHAP, or SHapley Additive exPlanations, is an interpretation technique for AI models that
helps explain the predictions of complex models, including regression models, by assigning
each feature an importance value for a given prediction. Based on cooperative game theory,
SHAP provides a method to quantify each feature’s contribution to the prediction outcome fairly
and consistently. SHAP is widely used in different areas, such as Natural Language Processing

SHapley Additive exPlanations (SHAP)

(NLP) (Mosca et al., 2022), autoencoders (Antwarg et al., 2021), and ML models (Nohara et al.,
2022).
In regression models, where the goal is to predict a continuous outcome (e.g., activity levels,
temperature, or time spent in specific behaviors), SHAP can provide valuable insights into how
each feature influences the predicted value. This information is beneficial for:
• Understanding Feature Importance: SHAP identifies which features influence the
model’s predictions most. For example, in a model predicting daily activity levels, SHAP
values might show if variables like light intensity or sleep patterns significantly impact
the prediction.
• Quantifying Feature Contributions: Each SHAP value represents the contribution of
a particular feature to the model’s prediction for a specific instance. For example, if a
regression model predicts a patient’s activity level as 75, SHAP values could reveal that
light intensity contributed +10 to this prediction. In contrast, temperature contributed -5,
explaining how each feature influenced this outcome.
SHAP values are calculated by evaluating the impact of each feature across different combinations of feature subsets, resulting in an ”average” impact score. This process can be summarized in the following steps:
1. Baseline Prediction: SHAP starts with a baseline prediction, often the average prediction
across all instances in the dataset.
2. Feature Impact Calculation: SHAP calculates how including that feature changes the
prediction relative to the baseline for each feature. This is done by considering all possible
combinations of other features, ensuring that SHAP values are fair and consistent, akin to
Shapley values in game theory.
3. Summing Contributions: The final prediction is obtained by adding up the SHAP values
of each feature plus the baseline prediction. Thus, each feature’s SHAP value explains its
contribution to the final prediction.
SHAP values are useful for understanding both global and local feature importance in regression models:
• Global Interpretability: By averaging SHAP values across all instances in the dataset,
we gain insight into which features have the highest overall impact on predictions. This
is useful for identifying the importance of features in regression models, helping to determine which variables the model relies on most consistently.

SHapley Additive exPlanations (SHAP)

• Local Interpretability: SHAP values for individual predictions provide instance-specific
explanations, showing how each feature influenced a particular prediction. For example,
SHAP can reveal which features significantly affected a specific patient’s predicted activity level, providing detailed insights into the model’s behavior for each case.
Overall, SHAP enhances the interpretability of regression models by clearly separating feature contributions to each prediction. This transparency is crucial for validating model behavior
and understanding how specific features drive the model’s predictions.

2.8.1

Related Works

The work developed by Tonon et al. (2022) presents a method for handling OW episodes
in actigraphy time series of motor activity, using data from two records (one with a regular
sleep-wake cycle and another with an irregular cycle) collected over 14 consecutive days with
1-minute sampling intervals. To simulate real-life OW episodes, artificial missing value (NA)
intervals of various durations of hours (1, 2, 4, 6, 12, and 24) and random NA episodes following probabilistic rules were introduced into the data. These missing episodes were replaced
with different imputation methods: zeros (representing immobility), the mean, or the median
of the corresponding time points from the remaining 13 days. The results showed that missing
episodes up to 12 hours caused less than a 5% deviation from the original values. The study
recommends avoiding zero imputation when possible due to the risk of artificially reducing
variability. Instead, it suggests treating zeros as NAs to prevent inflating invariance. If specific
parameters cannot be calculated with NAs present, using the weekly mean of corresponding
time points is recommended as an alternative approach. This study brings great ideas and analysis, indicating the scenarios where it is worth performing data imputation or not in actigraphy
data. Although, it only uses classical statistical methods in the imputation phase.
The work developed by Jang et al. (2020) explores a DL approach for imputing missing values in actigraphy data, utilizing a denoising convolutional autoencoder trained on
accelerometer-based activity records from the National Health and Nutrition Examination Survey. The model was validated using external datasets, including daily activity counts from the
Korea National Health and Nutrition Examination Survey and a chronic cerebrovascular disease biobank. The evaluation involved introducing artificial missing data and comparing the
DL model’s imputation performance against traditional methods like mean imputation, zeroinflated Poisson regression, and Bayesian regression. Results indicated that the DL approach
achieved lower partial root mean square error (RMSE) and partial mean absolute error (MAE)
than the conventional methods, demonstrating superior accuracy in reconstructing missing values. The study highlights the potential of DL techniques for handling incomplete actigraphy
data, suggesting that such methods may outperform traditional approaches by learning complex
data patterns without relying on predefined statistical assumptions. Although the Autoencoder
outperforms the other methods, its performance still shows relatively high metrics, such as an

SHapley Additive exPlanations (SHAP)

RMSE of approximately 800. Since a lower RMSE indicates better accuracy, ideally closer to
zero, this result suggests room for further improvement.
The authors of Weed et al. (2022) aims to assess how the timing and duration of missing
data in actigraphy records affect the calculation of internal stability (IS) and intraday variability
(IV). Additionally, it evaluates the effectiveness of three imputation methods—linear interpolation, mean time of day (ToD), and median ToD imputation—in estimating these metrics. To
conduct the analysis, week-long actigraphy datasets with no original data gaps were systematically masked with zeros or ’Not a Number’ (NaN) values across different timings and durations,
simulating single and multiple missing data episodes. IV and IS calculations were performed
on the original (true), masked, and imputed datasets. Results showed that simulated missing
data led to significant deviations in IV and IS, particularly for longer gaps, around midday, and
when missing periods occurred at similar times across consecutive days. Among the tested
methods, median ToD imputation resulted in the most minor deviations, making it the most
reliable approach for estimating IV and IS when handling missing data durations of less than
24 hours.
Consequently, the study recommends using median ToD imputation to best preserve the
integrity of IS and IV calculations under these conditions. The study only evaluates three basic imputation methods, which limits its scope. More advanced techniques, such as machine
learning-based imputation (e.g., Random Forest or autoencoders), could have been included to
provide a broader assessment of available approaches.
The classification process presented in Haghayegh et al. (2020) shows the impact of different modes of operation and the potential benefits of a DL approach to evaluate the estimation
of sleep parameters using wrist actigraphy. The study aimed to answer two key questions: (1)
Does the mode of operation—Proportional Integrating Measure (PIM), which measures movement intensity, or Zero Crossing Mode (ZCM), which measures movement frequency—affect
sleep scoring accuracy? (2) Can a DL model that combines PIM and ZCM data outperform existing IAs? The research involved nighttime sleep data from 40 healthy adults, including ZCM
and PIM actigraphy data, alongside electroencephalographic (EEG) recordings as the reference.
The initial analysis applied several classic DL models to PIM-only, ZCM-only, and combined
PIM/ZCM data. Results indicated that the combined ZCM/PIM mode produced a higher agreement with the EEG reference for sleep/wake scoring than ZCM or PIM alone. The new DL
model achieved 87.7% accuracy, 94.1% sensitivity, 64.0% specificity, and 59.9% Kappa agreement, showing improvements over other IAs, particularly in specificity and Kappa agreement.
The model’s estimates for sleep onset latency (SOL), wake after sleep onset (WASO), total
sleep time (TST), and sleep efficiency (SE) did not significantly differ from the EEG reference,
with less bias and more minor detectable changes than the other IAs. While the DL model
shows high accuracy and sensitivity, the specificity (64.0%) and Kappa agreement (59.9%) are
relatively low, indicating potential challenges in distinguishing between sleep and wake states.
The works summarized above provide foundational methods for addressing missing data in

Final Considerations

actigraphy studies and have advanced data imputation techniques. However, these studies often
rely on traditional statistical approaches, limited sample sizes, or simplified models that restrict their ability to generalize across diverse actigraphy datasets. Classical statistical methods,
while helpful, may lack the sophistication needed to capture the nuanced patterns in complex
datasets, particularly for imputation during OW intervals. Similarly, some DL models, although
more accurate than conventional methods, display limitations in specific metrics, such as root
mean square error (RMSE) and specificity, indicating room for further improvement. The small
sample sizes in several studies also hinder the models’ applicability to broader populations, potentially reducing their effectiveness in diverse real-world scenarios. Furthermore, the limited
exploration of hybrid and patient-specific models suggests an opportunity to improve current
methods by incorporating advanced deep learning frameworks. These enhancements could lead
to more robust and reliable imputation techniques, ultimately supporting more accurate patient
activity and sleep pattern monitoring.

2.9

Final Considerations

In this chapter, we have laid the groundwork for understanding the theoretical elements
required for our data imputation methodology. We began by examining critical concepts in
actigraphy, emphasizing the significance of variables, such as activity and light metrics, and
specific actigraphy measures, like M10 and L5, used to assess circadian rhythms and activity
levels. Understanding these elements is essential, as they form the basis of the imputation
process in datasets with missing values due to off-wrist (OW) moments.
Next, we discussed Dynamic Time Warping (DTW) and Euclidean distance, two critical techniques for comparing time-dependent sequences. While Euclidean distance offers a
straightforward approach, it is only effective when the time sequences are aligned. DTW, however, is a flexible tool that allows non-linear alignment adjustments, making it suitable for analyzing time series data with temporal distortions, such as those found in actigraphy. This
capability is precious when comparing periods of activity that may vary in duration across days
or individuals, a common scenario in actigraphy-based studies.
We also introduced various machine learning models, mainly focusing on regressors. Treebased models, including Decision Trees, Random Forests, and XGBoost, offer robust predictive capabilities by constructing ensembles or combining multiple decision paths. Distancebased models like Support Vector Regression (SVR) and K-Nearest Neighbors (KNN) leverage
data point proximity, while linear regression provides a more straightforward, interpretable approach. Each model type brings unique advantages for handling actigraphy data, especially
when dealing with gaps due to OW periods, where imputing missing values accurately would
be meaningful for analysis.
Further, we covered autoencoders—a deep learning approach that compresses and reconstructs data through encoding and decoding processes. This approach is advantageous for com-

Final Considerations

plex imputation tasks, allowing us to capture non-linear relationships within the data, especially
beneficial for handling extensive and intricate actigraphy datasets.
Additionally, we reviewed essential data preprocessing steps, including normalization techniques like MinMaxScaler, and outlined hyperparameter optimization strategies, such as Grid
Search and Randomized Search, to fine-tune model performance. Finally, we discussed the
metrics, including R2 , MSE, RMSE, MAE, and MASE, that will be used to evaluate the effectiveness of our models in imputing missing data.
Overall, this theoretical foundation provides a comprehensive understanding of the tools
and techniques we will use to develop a robust data imputation pipeline for actigraphy data in
the upcoming sections.

Chapter 3
Methodology
Before detailing each step of the proposed methodology, we begin with an Exploratory
Data Analysis (EDA), which provided valuable insights that guided several key decisions, such
as identifying columns to drop and gaining a clearer understanding of the data structure. The
dataset contains records from 23 patients collected between 2018 and 2021 using the ActTrust
device developed by Condor. Each record includes multiple parameters that comprehensively
view each patient’s activities and conditions. Section 2, specifically subsection 2.1.1, provides
a complete description of these variables for each patient.
Figure 3.1 presents the correlation values between all the variables in each dataset. This
figure provides insight into the relationships and interactions among the various parameters,
aiding in the analysis and interpretation of the data. Also, Figures 3.2 and 3.3 show distribution
for both PIM and Temperature values, and Figure 3.4 shows the boxplots for ”Temperature x
PIM” indicating the interquartile range (IQR) for temperature within each PIM value category.
The horizontal line inside the box represents the median temperature. The whiskers extend to
the rest of the distribution, except for the outliers, plotted individually using dots.

Figure 3.1: Correlation Matrix - Patient 04.

Figure 3.2: Temperature distribution - patient 04.

Figure 3.3: PIM distribution - Patient 04.

Figure 3.4: Boxplots of Temperature x PIM - patient 04.
Our study involved 244,507 actigraphy recordings, all digitized at 1 sample per second.
Of these, 12,941 were classified as OW moments by the Random Forest Classifier (Pilz et al.
(2022)), which will be explained in the following subsection, resulting in 231,566 recordings
classified as ON moments. A specialist individually prescribed each patient’s duration of device
use, leading to variations in the total number of days recorded for each patient. This RFC uses

Random Forest Classifier (RFC)

entropy to build 30 trees and incorporates automatic feature selection via the out-of-bag score.
It mandates a minimum of 2 samples for leaf nodes and 5 for splits. This model identifies
extended OW periods and excludes days with over four hours of such intervals, resulting in a
dataset comprising 5.29% of OW data.
The methodology for reconstructing actigraphy data can be described in the steps presented
in Figure 3.5, where each step is described as:
• Random Forest Classifier: A identification of OW moments using the Random Forest
algorithm.
• Cleaning Data, Feature transform and scaling: Transforming and scaling the features
in the data to prepare it for the regression model.
• Removing OW Remaining Rows*: Specific data cleaning step to remove OW data on
the remaining training data. This step has to be aligned with the filtering data strategies
since, in some of those, the presence of the off-wrist moments in some training data can
represent the shift to another filtering strategy (as in the case of the Around Gap with
DTW method).
• Filtering Data Strategies: Application of strategies to filter and refine the data.
• Regression Strategies: Application of regression techniques to model the data.

Figure 3.5: Flowchart showing the methodology steps.

3.1

Random Forest Classifier (RFC)

The first problem we identify by analyzing our dataset is that we don’t have any label
marking our data as moments of OW, but only significant moments marked as 0 for column PIM.
As 0 values for column PIM can also represent that the patient does not realize any movement,
we can not classify every 0 register for column PIM as OW moments.
In this context, we decided to utilize the model developed by Pilz et al. (2022), which
consists of an RFC capable of classifying OW moments. This model is compatible with our
dataset, which comes from Condor, the ActTrust device, and adds a new column to our data
set, called M L Of f W rist P rediction, which contains values 0 and 1, where 1 represents
moments classified as OW moments and 0 as ON moments.

Cleaning Data, Feature transform and scaling

3.2

Cleaning Data, Feature transform and scaling

First, to clean the dataset, aiming not to process data that does not have an impact on the
model’s predictions, we analyzed some patterns and correlations between the variables (presented in the previous section, with the correlation matrix shown in Figure 3.1). First, some
variables such as M S, EV EN T , and ST AT E presented only 0 values for all patients (which
is a shame since a variable as state could indicate if the patient has OW moments on it or not,
according to the ActTrust manual and, since its always 0, we have to perform the RFC on the
data), so we only removed these columns. Furthermore, columns like T AT n, ZCM n, and
P IM n represented the normalized value of the columns T AT , ZCM , and P IM , respectively,
so they were also removed.
The M L Of f W rist P rediction was used only to identify the OW moments and, after the
filtering strategies, is also removed from the training and test data (since the training data will
have only values that were not marked as OW and test data will be marked as ON moments,
and the moments marked as OW are the ones who will be imputed). Moreover, the column
’DATE/TIME’ was decoded in two columns, hour, and day, to utilize the ’DATE/TIME’ object
information in the training process.
Finally, to perform the normalization process and ensure that each feature contributes
equally to the distance calculations, we utilized the MinMaxScaler. This tool efficiently scales
the data to a specified range, typically between 0 and 1, which is crucial for maintaining consistency across different feature scales and enhancing the performance of machine learning
algorithms that rely on distance metrics.

3.3

Removing Off-wrist remaining rows

This step is responsible for removing the remaining rows marked as OW depending on the
filtering data strategies used along with the model. It’s important to emphasize this statement
since some strategies rely on the presence of OW moments in close registers to the current OW
moment being analyzed. Still, other methods don’t rely on this, so all the remaining OW rows
can be removed in this step.

3.4

Filtering Data Strategies

3.4.1

Vigil periods and M10 approaches

In this approach, we integrated the RFC into our pipeline, assigning it the task of identifying
OW moments. In this approach, our primary objective was not to utilize the entire dataset,
excluding the OW periods, but to focus on the data from each day with the highest activity
levels for the patient. As previously mentioned, Vigil periods typically represent the times of

Filtering Data Strategies

highest activity for the patient. However, because fixed intervals define these periods, there is
a risk of missing data if the patient does not exhibit high activity levels within these intervals.
This limitation is mitigated by the M10 method, which dynamically defines the interval for each
patient each day, thereby accommodating individual activity patterns.
Both approaches have their respective advantages and disadvantages. Suppose the specialist
prefers the imputation method to focus on scenarios with a fixed vigil period, thereby maintaining the influence of low or high activity levels on other OW moments of the day. In that case,
using the fixed vigil period is likely the most appropriate approach. Conversely, suppose the
specialist aims for the imputation process to consistently consider the patient’s highest activity
levels, resulting in higher activity levels reflected in the imputation process. In that case, the
M10 metric as a clipping strategy is likely the best fit. However, it is essential to note that
both scenarios will introduce a bias into the imputation process, similar to all previously used
methods.
Code 1 shows the structure developed to filter the M10 data:
Algorithm 1: Get 10 hours of most excitement for each day
Input: DataFrame df , activity column activity column
Output: DataFrame with 10 hours of most excitement for each day
[”DAT E/T IM E”][0]
epoch ← df [”DAT E/T IM E”][1]−df
;
60
;
n epochs ← 10×60
epoch
days ← unique days from df [”DAT E/T IM E”];
activity data ← empty DataFrame;
foreach day in days do
data today ← filter df by day;
10 HOU RS IN ROW S = 600 number the minutes that represents 10 hours if
data today has less than 10 HOU RS IN ROW S rows then
”’ In cases that we have days with less then 10 hours, we won’t apply any
filtering to it. ”’
continue;
activity sum ← rolling sum of activity column over n epochs;
end index ← index of maximum value in activity sum;
start index ← end index − (n epochs − 1);
activity data today ← select rows from start index to end index;
concatenate activity data today to activity data;
return activity data;

3.4.2

Around Gap with DTW approaches

Our current strategy combines the Around Gap and DTW approaches. The choice between
these methods depends on whether the patient’s data exhibits continuous OW or spaced gap

Filtering Data Strategies

moments. The around-gap strategy is employed when the patient’s data contains a constant
gap, ensuring data availability with the same length as the gap before and after the OW period.
For instance, if a patient experiences an OW period of 1 hour between 4:00 PM and 6:00
PM, we can collect data for 1 hour before the gap (from 3:00 PM to 4:00 PM) and 1 hour
after the gap (from 6:00 PM to 7:00 PM). Thus, our training data will comprise 2 hours (120
minutes), resulting in 120 records, representing a one-minute interval.
In cases where data contains gaps during OW moments, it is not always possible to ensure
the availability of sufficient ON data before and after the gap. For instance, if there is an OW
period of 40 minutes, followed by 20 minutes of ON data, and then another 20 minutes of
OW time, the around-gap strategy cannot be employed because the after-gap data includes OW
moments.
In such scenarios, we utilize the DTW approach, which involves identifying similar periods
when the OW event occurred but on different days from other weeks. For example, if there is a
30-minute gap between 3:30 PM and 4:00 PM on a Thursday in the first week of the patient’s
data, we search for the same period on Thursdays of different weeks. If there are more than two
occurrences of Thursdays in the dataset, we compute the DTW between these days to identify
the most common day. The data from this selected period, corresponding to the OW time, is
then used as training data. Code 2 and 3 shows, respectively, the filtering strategy for around
gap and DTW:
Algorithm 2: Filter DataFrame by expanding around index gaps
Input: DataFrame df , DataFrame gaps df , Integer multiply f actor
Output: Filtered DataFrame and gaps df
indexes ← list of indexes from gaps df ;
range previous ← registers of same length before the gap;
range next ← registers of same length after the gap;
index ranges ← [range previous, range next];
df s ← empty list;
foreach (start, stop) in index ranges do
sliced df ← slice df from start to stop;
append sliced df to df s;
return concat(df s), gaps df ;

Filtering Data Strategies
Algorithm 3: Apply Dynamic Time Warping (DTW) filtering strategy
Input: DataFrame df , DataFrame with gaps gaps df , Integer
M IN IM U M DAY S = 2
Output: Tuple containing filtered DataFrames and gaps DataFrames
gaps df ← split df into gaps;
Add column ’day of week’ to df , calculated from df [”DAT E/T IM E”];
f inal gaps df ← empty list;
f inal f iltered df s ← empty list;
foreach gap df in gaps df do
day of week ← first element of gap df [”DAT E/T IM E”].dayof week;
matching days ← list of days in df where day of week matches and the date is
different from gap df ’s date;
if matching days is empty then
continue;
if len(matching days) ≤ M IN IM U M DAY S then
f inal day ← first matching day where ’ML OffWrist Prediction’ does not
contain 1;
else
most similar day ← day from matching days with minimum similarity
score to gap df using DTW;
f inal day ← copy of most similar day;
if f inal day is None then
continue;
start time, end time ← first and last times in gap df [”DAT E/T IM E”];
Expand start time and end time by the length of gap df ;
f inal day f iltered ← filter f inal day by start time and end time;
if 1 is not in f inal day f iltered[”M L Of f W rist P rediction”] then
Append gap df to f inal gaps df ;
Append f inal day f iltered to f inal f iltered df s;
Drop ’day of week’ from df ;
if f inal f iltered df s is empty then
Raise ValueError: ’No similar days found’;
return f inal f iltered df s, f inal gaps df ;

Regression Strategies

3.5

Regression Strategies

3.5.1

Autoencoder and TMAE

Our pipeline was developed using the autoencoder framework outlined by Fiorillo et al.
(2023). Our methodology utilized the autoencoder for data imputation by feeding the entire
dataset as input. Initially, we did not have access to the RFC, which we later used in other
pipelines focused on vigil and M10 periods. One major advantage of the autoencoder over
other methods is its ability to impute the PIM column—the primary focus of our work—and all
other columns in the dataset. This capability is due to the autoencoder having the same number
of neurons in its input and output layers.
Code 4 shows the building process of the autoencoder for each patient using the hyperparameter optimization technique:
Algorithm 4: Build Autoencoder Model
Input: Hyperparameters hp, DataFrame df with features
Output: Compiled Autoencoder model
input dim ← number of columns in df ;
encoding dim ← integer value from hp in the range [8, 64] with step sizeeight 8;
dropout rate ← float value from hp in the range [0.1, 0.5] with step size 0.1;
inputs ← input layer with shape (input dim);
encoded ← dense layer with encoding dim units and ReLU activation applied to
inputs;
encoded ← dropout layer with rate dropout rate applied to encoded;
decoded ← dense layer with input dim units and sigmoid activation applied to
encoded;
Create autoencoder model with inputs as input and decoded as output;
Compile autoencoder with Adam optimizer, mean squared error (MSE) loss, and the
following metrics: MAE, RMSE, R2 , MASE.
return autoencoder model;
The TMAE followed a similar structure, just changing the layers so it has the masking
operation, the positional encoder, and the transformer encoder and decoder. We masked the
data using values in a range of 15% to 40%, along with the hyperparameter tunning techniques.
Code 5 shows the training algorithm for the TMAE.

Regression Strategies
Algorithm 5: Train TMAE with Early Stopping and Validation
Input: model: Model, train loader: DataLoader, val loader: DataLoader,
num epochs: Integer, learning rate: Float, patience: Integer (default 10)
Output: model: Model, best val loss: Float
Initialize loss function criterion ← Mean Squared Error Loss;
Initialize optimizer optimizer ← Adam optimizer with learning rate;
Set device ← cuda if available else cpu; Move model to device;
best val loss ← ∞; patience counter ← 0; best model state ← None;
for epoch from 1 to num epochs do
Set model to training mode; foreach batch in train loader do
batch ← detach and move batch[0] to device;
masked batch, ← apply masking to batch with mask ratio;
reconstructed ← model(masked batch);
loss ← criterion(reconstructed, batch); Zero gradients in optimizer;
Backpropagate loss; Update optimizer;
Set model to evaluation mode; val loss ← 0; Initialize empty lists y true list,
y pred list; foreach val batch in val loader do
val batch ← detach and move val batch[0] to device; masked val batch, ←
apply masking to val batch with mask ratio;
reconstructed ← model(masked val batch);
val loss ← val loss + criterion(reconstructed, val batch);
Append val batch to y true list;
Append reconstructed to y pred list;
val loss ← val loss/ length of val loader;
y true concat ← concatenate y true list;
y pred concat ← concatenate y pred list;
Compute metrics mse, mae, rmse, r2, mase using y true concat and
y pred concat;
Print epoch details including loss, vall oss, and metrics;
if val loss < best val loss then
best val loss ← val loss; patience counter ← 0; best model state ←
current state of model;
else
patience counter ← patience counter + 1;
if patience counter ≥ patience then
Print “Early stopping at epoch epoch”;
break;
if best model state is not None then
Load best model state into model;
Print “Loaded the best model based on validation loss.”;
return model, best val loss;

Regression Strategies

3.5.2

Machine Learning with Hyperparameter Tunning

Firstly, it is essential to note that this step was utilized with the filtering methods: vigil periods,
M10, and the around Gap with DTW (Dynamic Time Warping). The ML approach employs
all the regression models, excluding the autoencoder mentioned in section 2, to function collectively as a grouped model. This grouped model leverages the strengths of multiple regression
techniques to improve the accuracy and robustness of the predictions. Each model within the
group is trained using the patient’s data. The most effective model, determined by statistical
metrics, also informed in section 2, is chosen as the final predictor for the data imputation process. Also, the DummyRegressor was trained for each patient only as a baseline method, so we
can compare the analyzed models and the basic techniques that the DummyRegressor brings.
Each model is subjected to the GridSearch and RandomizedSearch hyperparameter tuning
algorithms during this phase. These algorithms train the data using various combinations of
hyperparameters for each model, adhering to the optimization strategies of each respective algorithm. By fine-tuning the models through these hyperparameter optimization techniques, the
performance of each model is maximized, leading to more accurate and reliable predictions.
This meticulous tuning process is crucial for enhancing the model’s ability to generalize from
the training data to unseen data, thereby improving the overall effectiveness of the ML strategy
in the data imputation process. The hyperparameters for each model are presented in Table 3.1.
Table 3.1: Models Hyperparameters
Models

Hyperparameters

RFR
SVR
XGBRegressor

bootstrap, n estimators, max features
C, kernel, gamma
n estimators, learning rate, max depth, objective, min child weight
subsample, colsample bytree
max depth, min samples split
n neighbors, weights
fit intercept

Decision Tree
KNN Regressor
Linear Regressor

Code 6 shows the training process using KFold and training with both GridSearch and RandomizedSearch for all the presented models in Section 2.

Regression Strategies

Algorithm 6: Validate Models Using GridSearch and RandomizedSearch
Input: Training data X train, y train; Test data X test, y test; List of models with
parameter grids
Output: Best model, best score, second best model, second best score, worst model,
worst score
kf old ← K-fold cross-validation with five splits and shuffle enabled;
foreach model in models do
model name ← name of the current model;
model ← current model;
param grid ← parameter grid for GridSearch;
param dis ← parameter distribution for RandomizedSearch;
Perform GridSearchCV on model using param grid with scoring
’neg mean squared error’ and 5-fold CV;
Fit the GridSearch model on training data;
Predict values on test data using GridSearch model;
Compute performance metrics: MAE, RMSE, R², MASE for GridSearch
predictions;
if grid rmse > best score then
Update second best model and second best score with previous best values;
Set best model and best score to GridSearch best estimator and grid rmse;
else if grid rmse > second best score then
Update second best model and second best score with GridSearch best
estimator and grid rmse;
else if grid rmse < worst score then
Update worst model and worst score with GridSearch best estimator and
grid rmse;
Perform RandomizedSearchCV on model using param dis with scoring
’neg mean squared error’ and 5-fold CV;
Fit the RandomizedSearch model on training data;
Predict values on test data using RandomizedSearch model;
Compute performance metrics: MAE, RMSE, R², MASE for RandomizedSearch
predictions;
if randomized rmse > best score then
Update second best model and second best score with previous best values;
Set best model and best score to RandomizedSearch best estimator and
randomized rmse;
else if randomized rmse > second best score then
Update second best model and second best score with RandomizedSearch
best estimator and randomized rmse;
if randomized rmse < worst score then
Update worst model and worst score with RandomizedSearch best estimator
and randomized rmse;
return best model, best score, second best model, second best score,
worst model, worst score;

Regression Strategies

3.5.3

Final Considerations

In this chapter, we presented a methodology for reconstructing actigraphy data affected
by Off-Wrist (OW) periods. The approach began with Exploratory Data Analysis (EDA) to
understand the dataset’s structure and correlations, leading to the removal of irrelevant features
like M S, EV EN T , ST AT E, and normalized variables. We employed an RFC to identify OW
moments accurately, adding an M L Of f W rist P rediction column to distinguish between
OW and On-Wrist (ON) periods.
Data cleaning involved decoding the ’DATE/TIME’ column into ’hour’ and ’day’ components and normalizing features using MinMaxScaler. We implemented filtering strategies
tailored to our objectives: Vigil periods and M10 approaches focused on high-activity intervals, while the Around Gap with Dynamic Time Warping (DTW) method filled gaps by finding
similar activity patterns on different days.
For data imputation, we utilized an Autoencoder to reconstruct missing values across all
features and applied machine learning models—including Random Forest Regressor, Support
Vector Regressor, XGBoost, Decision Trees, and K-Nearest Neighbors—optimized through
hyperparameter tuning. This combination of strategies aimed to enhance the accuracy and reliability of actigraphy analyses by effectively handling missing data due to OW periods.
Our methodology offers a robust framework for improving patient activity monitoring, providing scalable solutions applicable to various research settings within sleep and circadian
rhythm studies.

Chapter 4
Results
All the results obtained for autoencoder were collected using the platform Google Colab
Pro (payed version of Google Colab), which had 12.67 GB of RAM, 107.7 GB of hard disk, and
his default GPU, which is an NVIDIA RTX Tesla K80 with 12GB of VRAM. The results using
the Essemble methods for M10 and Vigil times were using a laptop with an 11th Gen Intel(R)
Core(TM) i7-11800H @ 2.30GHz CPU, 16 GB DDR4 of RAM, and 768 GB of SSD. The M10
and DTW results were collected using a notebook with 64 GB of DDR5 and a processor from
Intel, the 13th Gen Intel(R) Core(TM) i7-13700HX.
Following the RFC step to determine OW duration, we included only 9 out of the 23
available patients in the study. We excluded fourteen patients because they either lacked OW
episodes or had more than four hours of OW time daily. Figure 4.1 illustrates the PIM (Patient
Impact Measure) values for Patient 7 (P7) across nine days. The RFC recorded 1,579 OW time,
totaling approximately 26 hours and 32 minutes, shown in Figure 4.2. The frequency of these
OW incidents could significantly affect a specialist’s evaluation, leading to the exclusion of the
days with more than 4 hours of OW for this patient. After this step, there were still some days
with less than 4 hours of OW moments, which made the patient still eligible for the analysis.

Figure 4.1: PIM original data value from Patient 7.

Figure 4.2: OW detected moments for Patient 7 using the RFC.
In the following subsections, we will present three tables showing, respectively, the statistical metrics for each imputed patient and the Wilcoxon tests (this type of test was chosen
since the PIM data does not follow a Gaussian distribution; we tested all the patients with
the Shapiro-Wilk test to identify if they followed a Gaussian distribution), and the Actigraphy
metrics provided by the Non-Parametric Measures of Actigraphy Data (nparACT), using the
language R.
The tables where we present the Wilcoxon results will show both T-Statistic and P-value
results. The T-Statistic is used to assess the evidence against the null hypothesis. In the case
of the Wilcoxon signed-rank test, the null hypothesis typically states that the median difference
between the paired samples is zero. A smaller stat value indicates more substantial evidence
against the null hypothesis, suggesting a significant difference between the paired samples.
The p-value is used to determine the statistical significance of the test results. A common
threshold for significance is 0.05:
• If P-Value < 0.05: There is sufficient evidence to reject the null hypothesis, suggesting a
statistically significant difference between the paired samples.

Autencoder with Full Data

• If P-Value >= 0.05: There is not enough evidence to reject the null hypothesis, suggesting
that any observed differences could reasonably occur by chance under the null hypothesis.

4.1

Autencoder with Full Data

The Autoencoder approach shows good performance metrics across most patients, as
shown in Table 4.2. The mean squared error (MSE) and mean absolute error (MAE) values
are low, with the R² values nearing 1.00, indicating a highly accurate model. Using complete
data with the Autoencoder allows for precise reconstruction, as evidenced by metrics such as
RMSE and MASE. The NaN value in MASE for patient 21 was caused by a division by zero
since the metric for calculating the MASE is a custom method we developed. Since the denominator represents the Mean Absolute Naive Error (MANE), calculated as the average of the
absolute differences between consecutive actual observations, normalizing the data could result
in all values becoming the same. In this case, the differences between successive observations
would be zero, creating a constant time series. Also, we have a different number of neurons and
weights for each patient for each layer, but a prevalent pattern of the following layers, with a
dense dropout and another dense layer, is shown in Table 4.1.
Table 4.1: Autoencoder Models Structure.
Patient

Input Neurons

Dense Neurons

Dropout Rate

Output Neurons

04
05
07
15
16
21
22
23
25

12
12
12
12
12
12
12
12
12

64
40
48
56
40
56
48
48
64

10%
10%
10%
10%
10%
10%
10%
20%
10%

12
12
12
12
12
12
12
12
12

The Wilcoxon test results, presented in Table 4.3, support the statistical significance of the
Autoencoder’s performance, with p-values indicating strong evidence against the null hypothesis across most patients. For instance, patient 07 has a solid result with a T-statistic of 0.0 and a
p-value of 1.3231 × 10−259 , indicating a highly significant difference.
Actigraphy metrics using nparACT in Table 4.4 reveal that the reconstructed data (indicated
by sub-index R) closely mirrors the original data (O), with minor variations in Interdaily Stability (IS) and Intradaily Variability (IV), which reflect the Autoencoder’s ability to capture
temporal patterns in activity data effectively.
Figure 4.3 shows the imputed values alongside the ON values from patient 4. For clarity, we
will mainly focus on presenting the results for patient 4 in this and the following subsections.

Autencoder with Full Data

Table 4.2: Autoencoder Performance Metrics for Patients. This method was developed using
and trained using cross-validation, with five folds and hyperparameter optimization techniques,
using Keras-tuner as a library.
Patient
04
05
07
15
16
21
22
23
25

MSE
−5

4.665 × 10
8.174 × 10−5
6.109 × 10−5
1.210 × 10−4
8.450 × 10−5
6.158 × 10−5
9.294 × 10−5
3.768 × 10−4
4.457 × 10−5

MAE

RMSE

R²

MASE

0.00345
0.00455
0.00371
0.00581
0.00455
0.00428
0.00493
0.01196
0.00307

0.00683
0.00904
0.00782
0.01100
0.00919
0.00785
0.00964
0.01941
0.00668

0.99907
0.99862
0.99860
0.99704
0.99862
0.99891
0.99827
0.99522
0.99898

0.00573
0.00595
0.00474
0.00684
0.00852
NaN
0.00706
0.02108
0.00457

Table 4.3: Wilcoxon Test for Patients Using Autoencoder
Patient
04
05
07
15
16
21
22
23
25

T-Statistic
0.0
0.0
0.0
0.0
0.0
557.0
0.0
0.0
0.0

P-Value
7.5778 × 10−12
1.7322 × 10−6
1.3231 × 10−259
6.6916 × 10−118
3.5595 × 10−13
1.2640 × 10−45
3.1480 × 10−179
1.6410 × 10−12
5.8354 × 10−19

The autoencoder method maintained the proportion between the maximum and minimum values
of the region without making the values too low or exceeding the maximum PIM value for that
patient in that region.

TMAE with Full Data

Table 4.4: nparACT Results for Autoencoder: the sub-indices O means Original Data and subindices R means Reconstructed Data. White and gray columns better visualize the same metrics
in original and imputed datasets.
Patient

ISO

ISR

IVO

IVR

RAO

RAR

L5O (mg)

L5R (mg)

M10O (mg)

M10R (mg)

04
05
07
15
16
21
22
23
25

0.3
0.28
0.49
0.51
0.53
0.6
0.4
0.31
0.47

0.28
0.49
0.44
0.53
0.6
0.37
0.4
0.5
0.47

0.47
1.21
0.76
0.91
0.78
1.03
1.36
0.73
0.83

1.21
0.76
0.93
0.77
1.03
1.12
1.36
0.63
0.82

0.92
0.81
0.9
0.92
0.97
0.87
0.92
0.83
0.85

0.81
0.9
0.95
0.97
0.87
0.87
0.92
0.89
0.85

158.48
348.66
227.43
143.97
80.46
206.4
148.5
339.42
210.04

348.66
227.43
89.96
80.46
206.4
302.7
150.53
146.94
210.04

3689.82
3304.07
4409.29
3306.32
5166.9
2855.47
3422.83
3749.18
2562.18

3304.07
4409.29
3583.15
5166.9
2863.02
4522.12
3435.47
2487.73
2565.95

Figure 4.3: Results for patient 4 using Autoencoder method.

4.2

TMAE with Full Data

The TMAE approach demonstrates consistent performance metrics across a range of patients, as shown in Table 4.5. A low MSE, MAE, and RMSE values across most patients should
indicate precise predictions. However, the R2 for all predictions is low or low, as in the case
of patient 23, which means a not-so-good fit of the model with the data. The MASE provides
additional insight into accuracy relative to naive predictions, with values above one indicating
that the model performs poorly. Different from the Autoencoder, the TMAE followed different
numbers of layers for transformer layers precisely but started with a dense dropout and finished
with another dense layer for all patients.
The Wilcoxon test presented in Table 4.7 shows statistically significant results across all

TMAE with Full Data

Table 4.5: TMAE Performance Metrics for Patients. This method was developed using and
trained using cross-validation, with also five folds and hyperparameter optimization techniques,
using Optuna as a library.
Patient

MSE

MAE

RMSE

R²

MASE

04
05
07
15
16
21
22
23
25

0.0028
0.0018
0.0013
0.0016
0.0009
0.0017
0.0026
0.0053
0.0020

0.0183
0.0175
0.0159
0.0184
0.0138
0.0174
0.0201
0.0399
0.0203

0.0532
0.0429
0.0364
0.0403
0.0305
0.0407
0.0509
0.0731
0.0444

0.5694
0.6789
0.7012
0.6959
0.6943
0.2649
0.7022
0.0894
0.6757

1.4925
1.1980
1.3683
1.1485
1.1678
1.2025
1.4857
1.5052
1.2084

Table 4.6: TMAE Models Structure.
Patient

Input Neurons

Dense Neurons

Dropout Rate

Transformer Encoder (Layers, Heads, Units)

Output Neurons

Total Parameters

04
05
07
15
16
21
22
23
25

12
12
12
12
12
12
12
12
12

32
48
64
40
56
64
48
72
80

10%
10%
10%
10%
10%
10%
10%
50%
10%

(2, 2, 32)
(3, 3, 48)
(4, 4, 64)
(2, 2, 40)
(3, 3, 56)
(4, 4, 64)
(3, 3, 48)
(5, 5, 72)
(6, 6, 80)

12
12
12
12
12
12
12
12
12

2,956
5,782
9,612
3,504
6,332
10,192
5,716
15,428
21,328

patients, with varying T-statistics, meaning there are still some significant differences between
the original and imputed data.
The actigraphy metrics in Table 4.8 demonstrate notable discrepancies between the original
and reconstructed data across most metrics. Significant changes are observed in IS, IV, RA, L5,
and M10 values, indicating that the reconstructed data deviates substantially from the original
measurements. These variations suggest potential challenges in maintaining data fidelity within
the reconstruction process for these metrics.
]
We can see the impact of a low R2 in Figure 4.4, which shows that the values for this patient
remained close to 0, keeping almost the same behavior as we have an OW.

ML Models with M10

Table 4.7: Wilcoxon Test for Patients Using TMAE
Patient

T-Statistic

P-Value

04
05
07
15
16
21
22
23
25

0.0
0.0
1134.0
107880.0
0.0
0.0
44775.0
1.0
0.0

5.835 × 10−19
1.734 × 10−6
1.137 × 10−258
7.939 × 10−4
3.559 × 10−13
3.676 × 10−48
1.517 × 10−129
1.718 × 10−12
5.835 × 10−19

Table 4.8: nparACT Results for TMAE
Patient

ISO

ISR

IVO

IVR

RAO

RAR

L5O (mg)

L5R (mg)

M10O (mg)

M10R (mg)

01
02
03
04
05
06
07
08
09

0.28
0.49
0.43
0.53
0.6
0.37
0.4
0.45
0.47

0.27
0.49
0.32
0.48
0.59
0.36
0.35
0.44
0.46

1.21
0.76
0.92
0.78
1.03
1.12
1.36
0.58
0.83

1.2
0.76
0.86
0.72
1.03
1.09
1.26
0.71
0.82

0.81
0.9
0.96
0.97
0.87
0.87
0.92
0.89
0.85

0.81
0.9
0.97
0.97
0.87
0.87
0.92
0.88
0.85

348.66
227.43
64.51
80.46
206.4
301.71
148.5
146.94
210.04

348.66
227.43
42.44
64.45
206.4
301.71
130.22
146.94
210.04

3304.07
4409.29
3544.86
5166.9
2855.47
4521.83
3422.83
2487.73
2562.18

3261.23
4367.62
3045.06
4927.33
2855.47
4509.21
3154.41
2228.26
2535.96

Figure 4.4: Results for patient 4 using TMAE method.

4.3

ML Models with M10

ML Models with the M10 filtering technique (Tables 4.9, 4.10, and 4.11) also demonstrate
high performance with minimal errors and high R² values. This method utilizes a combination
of models, predominantly XGBRegressor, which is the best model for most patients.
The performance metrics indicate that the ML method handles diverse patient data well, as

ML Models with M10

Table 4.9: ML Models with M10 Performance Metrics for Patients
Patient

MSE

MAE

RMSE

R²

MASE

Best Model

04
05
07
15
16
21
22
23
25

0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00

0.00
0.01
0.00
0.01
0.01
0.01
0.01
0.02
0.01

0.01
0.01
0.01
0.01
0.01
0.02
0.01
0.04
0.01

0.98
0.99
0.99
0.98
0.99
0.99
0.98
0.94
0.99

7645263612.40
4035495354.88
24719687124.48
165095448966.42
168646853054.68
22310086002.98
34967882319.29
2783996446055.05
68551396104.05

DecisionTreeRegressor
XGBRegressor
RandomForestRegressor
XGBRegressor
XGBRegressor
XGBRegressor
XGBRegressor
XGBRegressor
XGBRegressor

seen in the high R², low MAE, and RMSE across the board. Although all patients have a high
value of MASE, the forecasting model is performing poorly compared to the naive benchmark
for this method.
The Wilcoxon test shows statistically significant results across all patients, with varying Tstatistics, reinforcing the model’s effectiveness in improving predictive accuracy with the M10
filtering approach. For instance, patient 07 shows a T-statistic of 1608.0 with a p-value of
8.660 × 10−7 , indicating effective enhancement from the ML model.
Table 4.10: Wilcoxon Test for Patients Using ML Models with M10
Patient

T-Statistic

P-Value

04
05
07
15
16
21
22
23
25

0.0
0.0
1608.0
104.0
818.0
0.0
0.0
0.0
0.0

3.523 × 10−9
7.632 × 10−3
8.660 × 10−7
3.811 × 10−10
1.291 × 10−2
1.553 × 10−7
5.137 × 10−10
1.229 × 10−5
7.691 × 10−8

NparACT analysis indicates consistent data reconstruction with the original, with similar
values for metrics like IS and IV. The M10-filtered ML models tend to slightly under or overestimate specific metrics, such as L5 and M10, but maintain overall reliability.
Figure 4.5 shows that the values for this patient remained the same, close to 0, with the value
of 5.613946 exactly.

ML Models with Around Gap with DTW

Table 4.11: nparACT Results for ML model with M10: the sub-indices O means Original Data
and sub-indices R means Reconstructed Data. White and gray columns better visualize the same
metrics in original and imputed datasets.
Patient

ISO

ISR

IVO

IVR

RAO

RAR

L5O (mg)

L5R (mg)

M10O (mg)

M10R (mg)

04
05
07
15
16
21
22
23
25

0.28
0.3
0.49
0.53
0.6
0.4
0.31
0.47
0.6

0.1
0.3
0.49
0.32
0.6
0.4
0.31
0.47
0.6

1.21
0.47
0.76
0.78
1.03
1.36
0.73
0.83
0.71

1.4
0.47
0.76
0.95
1.03
1.36
0.73
0.83
0.71

0.81
0.92
0.9
0.97
0.87
0.92
0.83
0.85
0.96

0.3
0.92
0.9
0.34
0.87
0.92
0.83
0.85
0.96

348.66
158.48
227.43
80.46
206.4
148.5
339.42
210.04
149.74

1827.74
158.48
227.43
2422.98
206.4
148.5
339.42
210.04
149.74

3304.07
3689.82
4409.29
5166.9
2855.47
3422.83
3749.18
2562.18
6854.28

3421.05
3689.82
4409.29
4868.62
2855.47
3422.83
3749.18
2562.18
6854.28

Figure 4.5: Results for patient 4 using M10 method.

4.4

ML Models with Around Gap with DTW

The ML models using the Around Gap with DTW method, detailed in Tables 4.12, 4.13,
and 4.14, achieve competitive performance metrics with low MSE, MAE, and high R² values,
demonstrating its capacity to predict patient data accurately.
A notable observation is that while the MASE values are generally lower, indicating stable
predictive performance, there is a significant outlier in patient 05 with an extraordinarily high
MASE. This outlier suggests that the model might be sensitive to specific data characteristics
or distributional shifts.
The Wilcoxon test results corroborate the model’s performance, with significant p-values
across most patients, albeit with some variability in T-statistics. For example, patient 21 exhibits

ML Models with Around Gap with DTW

Table 4.12: Model Performance Metrics for Patients
Patient

MSE

MAE

RMSE

R²

MASE

Best Model

04
05
07
15
16
21
22
23
25

5.90
0.00
0.00
0.00
2.15
0.00
0.00
0.00
0.00

0.02
0.01
0.01
0.03
0.01
0.01
0.01
0.02
0.02

0.04
0.02
0.02
0.04
0.01
0.02
0.01
0.03
0.03

0.95
0.99
0.99
0.98
1.00
0.99
0.96
0.98
0.98

3.58
1241395892094.03
0.24
480016592680.68
1315259802431.74
0.10
0.39
5890949679407.39
0.48

LinearRegression
LinearRegression
XGBRegressor
XGBRegressor
LinearRegression
RandomForestRegressor
XGBRegressor
LinearRegression
DecisionTreeRegressor

a T-statistic of 0.0 with a p-value of 1.553 × 10−7 , confirming the method’s effectiveness in
aligning patient data with DTW.
Table 4.13: Wilcoxon Test for Patients Using ML Models with DTW
Patient

T-Statistic

P-Value

04
05
07
15
16
21
22
23
25

0.0
0.0
1608.0
104.0
818.0
0.0
0.0
0.0
0.0

3.523 × 10−9
7.632 × 10−3
8.660 × 10−7
3.811 × 10−10
1.291 × 10−2
1.553 × 10−7
5.137 × 10−10
1.229 × 10−5
7.691 × 10−8

NparACT metrics highlight the impact of DTW in aligning activity patterns, with reconstructed data generally matching the original. Variations in metrics like IS and IV are present
but do not significantly deviate from the expected values, indicating the proposed technique
may not present some differences from the original dataset after performing its steps.
Figure 4.6 illustrates that the values for this patient varied, fitting well within the data context. Low values followed this at the beginning of the imputed values, as they originated from
a drop in the values from the previous context and connected with the following context at the
end of the series.

SHAP Evaluations of ML Models

Table 4.14: nparACT Results for ML models with DTW and Around Gap: the sub-indices O
means Original Data and sub-indices R means Reconstructed Data. White and gray columns
better visualize the same metrics in original and imputed datasets.
Patient

ISO

ISR

IVO

IVR

RAO

RAR

L5O (mg)

L5R (mg)

M10O (mg)

M10R (mg)

04
05
07
15
16
21
22
23
25

0.3
0.28
0.49
0.53
0.6
0.4
0.31
0.47
0.6

0.48
0.28
0.49
0.57
0.61
0.16
0.53
0.47
0.6

0.47
1.21
0.76
0.78
1.03
1.36
0.73
0.83
0.71

0.68
1.21
0.76
0.77
1.05
1.43
0.76
0.83
0.71

0.92
0.81
0.9
0.97
0.87
0.92
0.83
0.85
0.96

0.92
0.81
0.9
0.97
0.87
1
0.95
0.85
0.96

158.48
348.66
227.43
80.46
206.4
148.5
339.42
210.04
149.74

189.59
348.66
231.42
79.11
206.4
240.86
120.3
210.04
149.74

3689.82
3304.07
4409.29
5166.9
2855.47
3422.83
3749.18
2562.18
6854.28

4751.14
3304.07
4409.29
5477.73
2858.54
196603.89
4733.17
2565.69
6862.58

Figure 4.6: Results for patient 4 using Around + DTW method.

4.5

SHAP Evaluations of ML Models

To better understand the impact of other variables in the learning process of the ML Models, we performed SHAP evaluation in the results for each one of the nine eligible patients. In
some cases, as for patient 04 and patient 21, the results seemed that every variable had a similar
impact in patient four since the values didn’t present any deviation for more or less as shown in
Figures 4.7 and 4.8 for patient four, and in Figures 4.9 and 4.10 for patient 21.

SHAP Evaluations of ML Models

Figure 4.7: Summary Plot for patient 4.

Figure 4.8: Decision Plot for patient 4.

SHAP Evaluations of ML Models

Figure 4.9: Summary Plot for patient 21.

Figure 4.10: Decision Plot for patient 21.
But in other cases, as for patient five and patient twenty-five, we can see how the variables
T AT , REDLIGHT , and ZCM performed an impact on the learning process of the model for
this patient, as presented in Figures 4.11 and 4.12 for patient 5, and in Figures 4.13 and 4.14 for

SHAP Evaluations of ML Models
patient 25.

Figure 4.11: Summary Plot for patient 5.

Figure 4.12: Decision Plot for patient 5.

SHAP Evaluations of ML Models

Figure 4.13: Summary Plot for patient 25.

Figure 4.14: Decision Plot for patient 25.
In most cases where certain variables dominate others, T AT and ZCM consistently emerge
as variables with a significant impact on prediction values. This is mainly due to their high correlation with the P IM , as both T AT and ZCM are indices of patient movement. Conversely,

SHAP Evaluations of ML Models

in other instances, luminosity-related variables such as REDLIGHT and LIGHT (a general combination of different light variables) significantly influence predictions. This indicates
that, in some cases, luminosity may substantially impact the prediction more than variables like
patient temperature or environmental temperature.

Chapter 5
Conclusion
The study aimed to evaluate various imputation methods for handling missing data in actigraphy
measurements, explicitly focusing on Off-Wrist (OW) periods. The methods assessed included
an Autoencoder with complete data, Temporal Moving Average Estimation (TMAE), Machine
Learning (ML) models with M10 filtering, ML models using Dynamic Time Warping (DTW)
with the Around Gap approach, and SHAP (SHapley Additive exPlanations) evaluations of ML
models. Including SHAP and TMAE adds depth to the analysis by providing interpretability
and a comparative baseline, respectively.

5.1

Autoencoder with Full Data

The Autoencoder demonstrated robust performance across most patients, evidenced by low
MSE and MAE values and high R2 scores nearing 1.00 (Table 4.2). This indicates the model’s
strong ability to reconstruct missing data accurately when complete data is available. The nparACT metrics further support this, showing minimal deviations between the original and reconstructed data (Table 4.4). The Autoencoder effectively preserved key circadian patterns, which
is crucial for accurate actigraphy analysis.
The NaN value observed in the MASE metric for patient 21, although an exclusive case, is
caused by a division by zero in the MASE calculation, highlighting sensitivity to data normalization that can result in constant time series. The Wilcoxon test results (Table 4.3) confirm the
statistical significance of the Autoencoder’s performance and the metrics the nparACT pointed
out.

5.2

TMAE with Full Data

The TMAE approach provided an even more robust strategy for the autoencoders, adding Transformer layers. While it yielded low MSE and MAE values (Table 4.5), the R2 values were
significantly lower than the Autoencoder, indicating a less accurate fit to the data. The higher

ML Models with M10 Filtering

MASE values above 1 suggest that TMAE performs worse than a naive benchmark, limiting its
effectiveness for precise imputation tasks.
Although the Wilcoxon test results (Table 4.7) showed significant differences between the
original and imputed data, the nparACT metrics (Table 4.8) revealed notable discrepancies in
circadian rhythm parameters, indicating that TMAE may distort essential behavioral patterns.
These findings suggest that while TMAE offers a computationally robust approach, it may not
suit the actigraphy context.

5.3

ML Models with M10 Filtering

The ML models combined with M10 filtering demonstrated high performance, low error metrics, and high R2 values (Table 4.9). Ensemble methods, particularly XGBoost, proved effective across most patients. The models maintained the integrity of crucial actigraphy metrics,
as shown by the nparACT results (Table 4.11), indicating their capability to preserve circadian
patterns during imputation.
However, the exceptionally high MASE values suggest that, despite the low MSE and high
R2 , the models perform poorly compared to a naive benchmark. This inconsistency highlights a
potential overfitting issue or sensitivity to specific data characteristics, which may not generalize
well across different patients or datasets. The Wilcoxon test results (Table 4.10) confirmed
statistical significance, but these factors may limit the models’ practical effectiveness.

5.4

ML Models with DTW and Around Gap Approach

The integration of DTW with the Around Gap approach in ML models aimed to enhance the
imputation by accounting for temporal dynamics and aligning similar patterns. The models
achieved competitive performance, with low MSE and MAE values and high R2 scores (Table
4.12). The nparACT metrics (Table 4.14) indicated that the reconstructed data closely matched
the original regarding circadian rhythm parameters.
Nevertheless, variability in MASE values, particularly the significant outlier for patient 05,
suggests that the method may be sensitive to individual differences in activity patterns. The
Wilcoxon test results (Table 4.13) showed statistical significance and variability in T-statistics
across patients. While the DTW approach can effectively capture temporal variations, it may
require patient-specific adjustments or further refinement to improve imputation accuracy consistently.

SHAP Evaluations of ML Models

5.5

SHAP Evaluations of ML Models

The application of SHAP provided insights into the interpretability of the ML models. For
patients 05 and 25, SHAP analyses revealed that features like Total Activity Time (TAT), Red
Light exposure, and Zero Crossing Mode (ZCM) significantly influenced the models’ predictions (Figures 4.11–4.14). This interpretability may be necessary for understanding the model’s
decision-making process and identifying potential biases or areas for improvement.
In contrast, for patients 04 and 21, SHAP evaluations indicated that the features had a uniform impact on the predictions (Figures 4.7–4.10), suggesting that the models relied equally on
all input variables or that the data lacked sufficient variability. This could point to overfitting or
an inability of the model to discern patterns specific to these patients, emphasizing the need for
tailored modeling approaches or feature engineering.

5.6

General Observations and Future Directions

The comparative analysis reveals that while advanced models like the Autoencoder and ML
methods with DTW offer high imputation accuracy, they also present challenges such as potential overfitting, sensitivity to individual patient data, and computational complexity. Complex
methods like TMAE, although less accurate, highlight the trade-off between computational efficiency and imputation precision.
The discrepancies in MASE values and the anomalies observed in certain patients underscore the importance of personalized approaches. Future work could focus on developing hybrid models that combine the strengths of different methods or incorporate adaptive algorithms
that adjust to individual patient patterns.
Furthermore, increasing the number of patients beyond the nine in the study may be necessary to enhance the generalizability of data imputation models in actigraphy. A larger, more
diverse sample would capture broader activity patterns and demographic variations, enabling
models to account for differences in variables and lifestyles. This would reduce overfitting to
a small, homogeneous group and ensure more robust, real-world applicability of imputation
methods in clinical and research contexts.
An additional direction for future research could involve testing median models or statespace models as alternatives to traditional regressors for data imputation. Median models, such
as SARIMA (Seasonal AutoRegressive Integrated Moving Average), offer robustness to outliers, computational efficiency, and the ability to handle seasonality in time-series data, making
them well-suited for clinical and actigraphy datasets.
State-space models, such as Kalman filters and hidden Markov models (HMMs), could
also be applied to the pipeline. These models excel in capturing temporal dependencies and
managing stochasticity. Kalman Filters are ideal for continuous data with linear or nonlinear
dynamics, while HMMs effectively model transitions between discrete states. Together, these

Conclusion

methods provide a promising toolkit for enhancing imputation accuracy in clinical and timeseries contexts.
Additionally, including SHAP analyses emphasizes model interpretability, which is essential for clinical applications where understanding the rationale behind predictions is as important
as understanding the predictions themselves. Further research could explore integrating SHAP
more deeply into the modeling process, such as retraining the models after identifying the most
important features for that patient later in training to enhance accuracy and interoperability.
Finally, Federated Learning (FL) offers a novel approach to addressing actigraphy data loss
by enabling collaborative model training across decentralized devices without sharing raw data.
FL can leverage the distributed nature of actigraphy data collected from wearable devices, allowing each device to contribute to model updates while preserving user privacy. By incorporating FL, imputation models can learn from patterns across a wide range of devices, enhancing
their ability to handle missing data effectively.

5.7

Conclusion

This work was able to propose a Pipeline to perform data imputation in actigraphy data. Both
filtering and regression methods were developed and integrated, along with other necessary
steps to fulfill the objective. The study demonstrates that imputation methods like the Autoencoder and ML models with advanced filtering techniques can effectively reconstruct missing
actigraphy data, preserving critical circadian metrics. However, challenges such as overfitting,
variability in patient performance, and the need for interpretability highlight the necessity for
continued refinement. Incorporating interpretability tools like SHAP may offer a balanced approach for handling missing actigraphy data in diverse patient populations.

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