# A Comparison of Functional Principal Component Analysis Methods with
Accelerometry Applications

Bohan Wu
Department of Statistics, Rice University
and
Bradley Van Allen
Department of Statistics, Rice University

Advisor: Dr. Daniel Kowal
Department of Statistics, Rice Univerity

(May 2021)

## Abstract

The association between a person’s physical activity and various health
outcomes is an area of active research. The National Health and Nutrition
Examination Survey (NHANES) data provide a valuable resource for studying
these associations. NHANES accelerometry data has been used by many to measure
individuals’ activity levels. A common approach for analyzing accelerometry
data is functional principal component analysis (FPCA). The first part of the
paper uses Poisson FPCA (PFPCA), Gaussian FPCA (GFPCA), and nonnegative and
regularized function decomposition (NARFD) to extract features from the count-
valued NHANES accelerometry data. The second part of the paper compares
logistic regression, random forests, and AdaBoost models based on GFPCA,
NARFD, or PFPCA scores in the context of mortality prediction. The results
show that Poisson FPCA is the best FPCA model for the inference of
accelerometry data, and the AdaBoost model based on Poisson FPCA scores gives
the best mortality prediction results.

##  1 Introduction

The association between a person’s daily physical activity levels and various
health outcomes is an active research area. The National Health and Nutrition
Examination Survey (NHANES) is a large annual survey conducted by the Centers
for Disease Control (CDC) to collect health, behavioral and nutritional
information of the US population (CDC, 2017). NHANES is one of the most
important national health surveys in the world in terms of size, scope and
availability. Furthermore, NHANES was the first study to make public a large
dataset containing the physical activity information measured using
accelerometers when they released accelerometry data from the 2003 - 2004 and
2005 - 2006 participants (Leroux et al., 2019).

The accelerometer is a hip-worn device that detects and records the magnitude
of acceleration or “intensity” of movement according to a specified time
interval. NHANES collected its 2003 - 2004 and 2005 - 2006 participants’
physical activity data with integer-valued "activity count" measured by the
accelerometers (CDC, 2017), and processed raw measurements into minute-by-
minute activity data. In this study, our focus is on the NHANES accelerometry
data from 2005 - 2006 samples. We are interested in extracting features from
NHANES accelerometry data and building a prediction model using the extracted
accelerometry features along with the demographic and comorbidity covariates
from NHANES.

A common feature extraction approach for accelerometry and wearable data is
functional principal component analysis (FPCA). Functional data are defined as
data consisting of random functions, where each function is considered as a
sample from a stochastic process (Shang, 2014). Functional Data Analysis (FDA)
is an emerging field that studies statistical methodologies that analyze
functional data, most notably by Ramsay and Silverman (1997). FDA is able to
extract the information shared across smooth functions, and it is most often
applied to high-dimensional time series that assume a latent continuous data
generative process. One of the most widely studied methods in FDA is
functional principal component analysis (FPCA). FPCA is an adaptation of
principal component analysis (PCA) to the analysis of function-valued
observations. Several previous studies have used FPCA to extract the
accelerometry features (Crainiceanu et al., 2009, Leroux et al., 2019). The
theoretical properties and computation of FPCA models have been studied in
great detail (Shang, 2014). However, most applications of FPCA have focused on
analyzing Gaussian functional data, with relatively little literature on other
types of functional data. In recent years, however, there have been rising
interests in modeling count-valued functional data in wearable technology
(Backenroth et al., 2020), ecology (Müller et al., 2005), and finance (Bening
and Korolev, 2012).

For NHANES accelerometry data, count-valued functional methods would be
advantageous for inference. The traditional approach is to transform the count
data and apply the Gaussian FPCA (Leroux et al., 2019). However, these
transformations are not well-defined for integer-valued data-generative
process, and log transformation has difficulty dealing with zero-inflation
(O’Hara et al., 2010). As a remedy, generalized FPCA extends the ideas from
Gaussian FPCA to distributions from the exponential family (Goldsmith, 2017).
The generalized FPCA family includes Poisson FPCA (PFPCA), which is
conceptually more well-defined for modeling functional count data by capturing
some aspects of the integer-specific properties through a Poisson generative
process. Besides FPCA approaches, another promising FDA method is nonnegative
and regularized function decomposition (NARFD). NARFD is a close alternative
to PFPCA that provides interpretable, local parameter estimates and
reconstruction for nonnegative count-valued functional data (Backenroth et
al., 2020).

Our first application of FPCA methods involves using GFPCA, PFPCA, and NARFD
to extract features from and conduct inference on the accelerometry data of
2005 - 2006 NHANES cohort. We provide detailed interpretations for the
functional components and scores from GFPCA, PFPCA, and NARFD, and determine
the most suitable model for interpreting the participants’ physical activity.
The second application involves using the functional scores from the three
FPCA methods to predict the five-year all-cause mortality outcomes of NHANES
2005 - 2006 participants in the year 2011 which was made available by linking
the 2005 - 2006 NHANES cohort to the year 2011 death certificates (Leroux et
al., 2019). The mortality outcome is defined as whether or not a given
participant had died from any cause within five years of the original survey.
We combine GFPCA, PFPCA, and NARFD scores with typical covariates in NHANES as
predictors in logistic regression, random forest, and AdaBoost methods to
predict mortality outcomes, and we discuss the strength and weakness of each
method in terms of predictive performance. Finally, we determine the variable
contribution to mortality and provide insights relevant to the medical
community.

##  2 Accelerometry Feature Extraction

We use Gaussian FPCA, Poisson FPCA, and NARFD to extract functional principal
component (FPC) scores from the accelerometry data. The FPC scores serve as
the summary measure of the participant’s physical activity. This allows for
the study of associations between individual physical activity, health
outcomes, and future mortality. Furthermore, we compare the interpretability
of the three methods to determine the most interpretable method for the
accelerometry data. In particular, we judge the model interpretability based
on the goodness of fit quantified with MAE, interpretability of the FPCs,
model assumptions, and how well the fitted values capture the data’s smooth,
functional patterns.

###  2.1 Data

The National Health and Nutrition Examination Survey (NHANES) is a large,
stratified, multistage, annualized survey conducted by the Centers for Disease
Control (CDC) that collects health and nutrition data on the non-
institutionalized US population. The NHANES website affirms that the data are
collected with the intent that it will be analyzed to “help develop sound
public health policy, direct and design health programs and services, and
expand the health knowledge for the Nation” (CDC, 2017). NHANES is one of the
largest and most influential public health surveys in terms of size, scope,
diversity, and public availability of the data. Furthermore, NHANES was the
first study to publicize a dataset that contains the physical activity
information measured with accelerometers when they released accelerometry data
from the 2003 - 2004 and 2005 - 2006 participants. The accelerometry data were
collected on the participants selected for the Mobile Examination Center (MEC)
portion of the study. For this study, we analyze the accelerometry data from
the 2005 - 2006 NHANES cohort.

####  2.1.1 Accelerometry Data

The accelerometry data consists of 5-minute level accelerometry data reported
as "activity counts" for NHANES 2005 - 2006 participants. It is a dataset with
6269 rows for 1011 subjects and 288 columns that represent 5-min time points.
The raw data come from the package rnhanesdata (Leroux et al., 2019) that
contains minute level accelerometry data reported as "activity counts" for
NHANES 2005-2006 participants. The accelerometry data are recorded each minute
and contains minute $0$ to minute $1440$ in a typical day. A related dataset,
also from the rnhanesdata package, contains a flag denoting whether or not the
accelerometer was worn at each minute of the day. To tackle missing values, we
first re-code activity counts which are considered "non-wear" to be 0. This
does not impact many data points as most estimated non-wear times already
correspond to 0 counts. To reduce the anomalies and missing values, we
aggregate the minute-level data into 5-min bins using summation, which leaves
$288$ time points. Then, we remove all $5$-min aggregated curves with missing
values.

Although each participant may have up to seven days of accelerometry records
in the raw data, we summarize the accelerometry records of each subject with
one record consisting of the rounded median activity account for the subject
at each 5-minute time point. Furthermore, we only include the participants
with no missing values in all NHANES covariate categories, with more details
in Section 3.1. The processed data contains $1011$ rows and $288$ columns. The
rounded median summary pairs each participant with one unique accelerometry
record and allows us to implement Poisson-type methods by preserving the
count-valued structure. For accelerometry feature extraction, all $1011$
subjects are included.

###  2.2 Methods

Extracting features from accelerometry data is challenging for three reasons.
The high dimensionality of the accelerometry data makes it difficult to apply
the usual dimensionality reduction techniques such as PCA. Moreover,
accelerometry data assumes a continuous data generative process that requires
the method to produce smooth fitted values. Finally, NHANES accelerometry data
are count-valued by design with properties such as zero-inflation and over-
and-under-dispersion. The ideal model needs to capture these count-specific
properties. To address those issues, we investigate three methods: 1) GFPCA
with log transformation, 2) PFPCA, and 3) NARFD.

####  2.2.1 Gaussian Functional Principal Component Analysis

Let $Y_{i}(t)$ be a real-valued response curves for $t\in\tau$ with mean
$\mu(t)$ and covariance function $\sigma(t,t^{\prime})$. In practice, the
nonegative count data follow a discrete time grid $Y_{i}(t_{j})$ for subject
$i=1,...,N$ at time $j=1,...,T_{i}$. The Gaussian functional principal
component analysis (GFPCA) model has the following generative process:

$Y_{i}(t)=\mu(t)+\sum_{k=1}^{K}s_{ik}\phi_{k}(t)+\epsilon_{i}(t)\quad\epsilon_{i}(t)\sim
N(0,\sigma^{2}),$ (1)

where $s_{ik}\sim N(0,\lambda_{k})$, and $(\lambda_{k}$ and $\phi_{k}(t))$ are
eigenvalue/eigenfunction of the covariance function with
$\lambda_{1}\geq\lambda_{2}\geq\dots\geq 0$. The set of functions
$\\{\phi_{j}(t)\\}_{j=1}^{K}$ is orthonormal and can be modelled as a linear
combination of basis functions such as splines. We implement a log
transformation on the responses by taking $log(Y+1)$ for each response curve
$Y$.

The Gaussian FPCA is fast and easy to implement. We implemented the Gaussian
FPCA via the "fpca.face" function in the refund package (Goldsmith, 2020).
However, the Gaussian method is not well-defined for count-valued data: the
data generative process of Gaussian FPCA cannot reproduce discrete responses,
which amplifies model misspecification, limits intepretability and harms the
reliability of the inference and predictive distributions (Kowal and Canale,
2020). For this reason, we would like to move beyond GFPCA and explore methods
that are well-defined for count-valued data.

####  2.2.2 Poisson Functional Principal Component Analysis

The Poisson functional principal component analysis (PFPCA) with a logarithmic
link function has the following generative process:

$Y_{i}(t)\sim
Pois[exp(\mu_{i}(t))],\quad\mu_{i}(t)=\mu(t)+\sum_{k=1}^{K}s_{ik}\phi_{k}(t)+\epsilon_{i}(t),$
(2)

where the parameter definitions are identical to those of GFPCA. The model
parameters of Poisson FPCA are estimated using a mixed model-based two-step
approach (Goldsmith et al., 2017). Implementation of Poisson FPCA and other
generalized FPCA methods in R is available in the registr package (Wrobel et
al., 2020).

Although Poisson FPCA has slower runtime than Gaussian FPCA, the Poisson model
is well defined for count-valued data, and the log link mitigates the Poisson
distribution’s disadvantage to model zeros. Poisson FPCA has all the inference
and prediction tools that Gaussian FPCA has, with less chance for model
misspecification and more reliable inference.

####  2.2.3 Nonnegative and Regularized Function Decomposition

The nonnegative and regularized function decomposition (NARFD) method is a
novel decomposition of nonnegative functional count data that draws on the
concept of nonnegative matrix factorization (Backenroth et al., 2020).
Compared to FPCA methods, NARFD uses a local, instead of global, estimation
scheme. Assume that the observations are Poisson-distributed. NARFD uses the
following generative process for functional observations:

$Y_{i}(t)\sim
Pois\\{\mu_{i}(t)\\},\quad\mu_{i}(t)=\sum_{k=1}^{K}s_{ik}\phi_{k}(t),$ (3)

where $\phi_{k}(t),k\in 1,...,K$ are functional prototypes, similar to
functional principal components (FPCs) in FPCA, and $s_{ik}$ are scores. The
estimated function $\mu_{i}(t)$ is a nonnegative latent process characterizing
both the mean and variance of $Y_{i}(t)$. Expanding the functional prototypes
in terms of spline basis functions with $M$ dimensions, we can express the
$\mu_{i}(t)$ term as a discrete-time process:

$\displaystyle\mu_{i}(t_{j})=\Theta_{i}(t_{j})\Phi s_{i}^{T},$

where $\Theta_{i}(t_{j})$ is a $T_{i}\times M$ spline matrix on the grid
$t_{j}$, $\Phi$ is the $M\times K$ "loading matrix" with columns $\phi_{k}$ as
spline coefficient vectors, and $s_{i}^{T}$ is the vector of scores for
subject $i$. The model parameters are estimated using an alternating
minimization method (Udell et al., 2016). The score vector $s_{i}$ can be
estimated using a Poisson generalized linear model with identity link and a
nonnegative constraint, with $Y_{i}(t_{j})$ being the functional responses and
$\Theta(t_{j})\Phi$ being the covariate matrix. When estimating the spline
coefficient matrix $\Phi$, a second derivative penalty is applied to ensure
smoothness. Implementation of NARFD in R is available on GitHub (Backenroth et
al., 2020).

Similar to PFPCA, NARFD assumes a Poisson data generative process which is
well defined for count-valued data. Unlike PFPCA, however, NARFD constrains
the functional prototypes to be non-negative. Furthermore, NARFD produces
highly-intepretable local functional prototypes which are added together in a
straightforward way to reconstruct the observed functions. In contrast, FPCA
methods produce components that vary across the entire functional domain
without contraints, and are added together via complex cancellations and
amplifications to reconstruct the observed functions. If fitted well, NARFD
results should be more intepretable than PFPCA results when we deal with
nonnegative count-valued accelerometry data.

###  2.3 Results

We extract features that characterize each participant’s daily physical
activity by fitting GFPCA, PFPCA, and NARFD on the participant’s accelerometry
record and taking the functional principal component (FPC) scores as the
features. The features extracted by FPCA facilitate interpretations of the
accelerometry data. For example, a change in one of the FPC scores may
indicate certain change of pattern in the participant’s daily activity. The
FPC scores estimated with the three methods will later be carried forward as
predictors in the mortality prediction models.

Figure 1 shows principal components for GFPCA, NARFD, and PFPCA. The PFPCA
principal components are smooth and the most interpretable. The GFPCA
principal components can capture the peaks and valleys at different time
points, but some principal components have unwanted non-smoothness. The
principal components from the NARFD adopt a local estimation approach, with
each one having a peak or valley in a specific area, depending on the score
for that component. When applied to accelerometry data, NARFD has been shown
to more intepretable than PFPCA (Backenroth et al., 2020). However, the
subject-wise median approach to the accelerometry data took away the local
estimation advantage from the NARFD functional principal estimation. As a
result, each of the six NARFD principal components in the example shows one
large, flat plateau during the daytime, which is not helpful for inference.
NARFD principal components are unable to characterize short-term activity
spikes and drops at different time points. The first four Gaussian FPCs have
similar interpretations as the Poisson FPCs, where the change in scores
indicates a change in the location of peaks and valleys. However, the fifth
and sixth Gaussian FPCs are not smooth and difficult to interpret. This makes
Gaussian FPCA a less attractive inference tool than Poisson FPCA. The
conclusion is that Poisson FPC scores characterize the NHANES accelerometry
data better than NARFD or Gaussian FPC scores.

Figure 1: First 6 principal components calculated on the population, 5 minute-
level NHANES accelerometry data using the GFPCA, NARFD and PFPCA.

In Figure 2, the pluses (and minuses) show what a specific principal component
would look like for an individual whose score is two standard deviations above
(below) the mean. For example, in the second principal component from the
PFPCA, we can see that a higher score for this component results in an upward
shift in the second half of the day and a downward shift in the first half of
the day. The interpretation is that a high score in the second principal
component denotes an individual with higher than average afternoon/nighttime
activity and lower morning activity. For the other Poisson functional
principal components, as the score changes, the locations of specific peaks
and valleys change. The scores for these components could be interpreted as
signifying the degree to which an individual follows the FPC patterns. The
same plots for NARFD and GFPCA are included in the Appendix Figure 8(a).

Figure 2:  First 6 Poisson FPCA principal components calculated on the
population, 5 minute-level NHANES accelerometry data using Poisson FPCA. Solid
lines represent the base line zero, +, - lines denote the effect of being 2
standard deviation from a score of 0 on the particular principal component.

Figure 3 shows GFPCA, PFPCA, and NARFD fits on four participants’ activity
curves. The samples are randomly selected. GFPCA fits are overly spiky, even
though the fitted values by GFPCA can capture the sudden spikes and dips in
the original data better than the PFPCA and NARFD. In contrast, PFPCA produces
smooth, reasonable fits that capture the general peaks, valleys, and zeros in
the data. NARFD fits are smooth and characterize the nonzero and zero levels
in the data, but NARFD fits fail to capture the peaks and valleys of the
original data. Overall, PFPCA provides the best fit out of the three models
because, unlike NARFD, PFPCA is flexible enough to capture multiple peaks and
valleys in the data, and PFPCA fits are less spiky than the GFPCA fits. Figure
4 depicts the mean absolute errors (MAE) of the three FPCA models fits for
each participant’s accelerometry data. The MAE per participant measures the
goodness of fit across three FPCA methods, as methods with lower MAE show
superior point estimations relative to other models. PFPCA is shown to be the
most accurate model in terms of goodness of fit, because the distribution of
PFPCA MAEs has a much lower mean and variance than GPFCA and NARFD and fewer
outliers.

Figure 3: Four randomly-chosen instances of GFPCA, NARFD, and PFPCA fits
Figure 4: MAE for FPCA models. Poisson FPCA has the best fiA comparison of
functional principal component analysis methods with accelerometry
applicationsat as it has the smallest MAE. Interestingly, Poisson FPCA
outperforms GFPCA and NARFD models by a wide margin, even though GFPCA and
NARFD have relatively similar performance.

##  3 Mortality Prediction

This section replicates the mortality prediction procedure in Leroux’s paper
(Leroux et al., 2019) with the methods in Section 2.2. We use Gaussian FPCA,
Poisson FPCA, and NARFD to extract FPC scores from the accelerometry data and
combine the extracted scores with the NHANES covariates to form a set of
predictors. The predictors are then used as the input for the survey-weighted
logistic regression, random forest, and AdaBoost models to predict the
mortality outcomes.
A training/test split is performed on the input data with a split ratio of
$70/30$. Given the class imbalance, we perform separate training/test split on
the data with "alive" outcomes and the data with "dead" outcomes. The training
data with "alive" and "dead" outcomes are combined to form the full training
data, likewise for the test data. After obtaining the prediction results from
the nine prediction models, we compare their predictive performance and select
the best model. Finally, we assess how each predictor contributes to the
classification of morality outcomes in the best model and tie the
interpretations back to the findings in Section 2.2.

###  3.1 Additional Data

In addition to the FPC scores extracted by the three methods in Section 2.3,
we utilize additional data including survey weights, processed covariates, and
the five-year all-cause mortality outcomes of the the 2005 - 2006 NHANES
cohort (Leroux et al., 2019). NHANES includes a survey weight for each
participant to account for the complex NHANES survey design, and it is
important to account for the survey weights in our analysis. The processed
covariates include demographic, lifestyle and comorbidity information of each
participant that we believe are relevant to predicting mortality. Moreover,
NHANES data can be linked to US national mortality data to obtain the
mortality outcomes of NHANES participants. The mortality data used in the
study comes from the R package rnhanesdata (Leroux et al., 2019), but the raw
mortality data was obtained by linking the 2005 - 2006 NHANES cohort to the
death certificate records from the National Death Index (NDI) (National Center
for Health Statistics, 2015). Each of these is discussed further below.

####  3.1.1 Survey Design & Weights

NHANES does not take a simple random sample from the US population; instead it
uses a complex survey design. Some of the sampling features include
oversampling, adjustment for non-response, and post-stratification.
Considering all these design features, NHANES assigns a sample weight to each
participant. The survey weight of a participant indicates the number of people
in the US population who he/she “represented”, and is the only survey design
variable relevant to our study. We need to account for survey weights in our
models. Otherwise, our conclusions may be skewed towards the segments of the
population that are over-sampled by NHANES, such as racial minorities and the
elderly. The raw survey weights come from the R package rnhanesdata (Leroux et
al., 2019). In rnhanesdata package, survey weighting was performed using the
${tableone}$ package (Yoshida and Bartel, (2020)) in R which interfaces with
the survey package (Lumley, 2020). We utilize the adjusted survey weights in
our analysis, defined as the raw survey weights divided by the mean survey
weights.

####  3.1.2 Covariate Data

NHANES covariate data contain survey sampling variables (i.e., survey weights)
and processed demographic and lifestyle variables for NHANES 2005-2006
participants. The data has 1011 rows and 26 variables with one row per
participant in the 2005-2006 wave. The demographic covariates include age,
gender, body mass index (BMI), ethnicity, and education level. The behavioral
covariates include smoking status and alcohol consumption. The comorbidities
include diabetes, coronary heart disease (CHD), congestive heart failure
(CHF), cancer and stroke. Other health covariates include blood pressure,
high-density lipoproteins (HDL) cholesterol, and total cholesterol. The
variables for blood pressure, HDL cholesterol and total cholesterol are
directly downloaded from the CDC website, while the remaining variables are
from the R package rnhanesdata (Leroux et al., 2019). All covariates are
included as predictors for mortality prediction. We only include participants
that had no missing values for any of the covariates. The processed dataset
contains $1003$ subjects after filtering out the $8$ participants with missing
mortality outcomes. See Table LABEL:tab:1 for more details.

Table 1: Characteristics of individuals included in the analysis | Unweighted | Survey-weighted
---|---|---
Sample Size | 1003 | 1003.00
Number of Weekday (mean (SD)) | 4.55 (0.78) | 4.57 (0.76)
Number of Weekend (mean (SD)) | 1.65 (0.60) | 1.67 (0.59)
Body Mass Index (%) |  |
Normal | 288 (28.7) | 288.7 (28.8)
Underweight | 11 (1.1) | 12.3 (1.2)
Overweight | 386 (38.5) | 380.0 (37.9)
Obese | 318 (31.7) | 322.1 (32.1)
Race (%) |  |
White | 606 (60.4) | 835.5 (83.3)
Black | 196 (19.5) | 75.6 (7.5)
Hispanic | 180 (17.9) | 65.6 (6.5)
Other | 21 (2.1) | 26.4 (2.6)
Gender (% Female) | 461 (46.0) | 518.0 (51.6)
Age (%) |  |
$[50,60)$ | 372 (37.1) | 524.6 (52.3)
[60,70) | 346 (34.5) | 279.6 (27.9)
$[70,80)$ | 210 (20.9) | 158.1 (15.8)
[80,85) | 75 (7.5) | 40.6 (4.1)
Education (%) |  |
Less than High School | 256 (25.5) | 144.7 (14.4)
High School | 246 (24.5) | 266.3 (26.6)
College and Above | 501 (50.0) | 592.0 (59.0)
Number of Alcoholic Drinks per Week (mean (SD)) | 2.87 (6.13) | 3.16 (6.31)
Cigarette Smoking (%) |  |
Never | 464 (46.3) | 472.1 (47.1)
Former | 377 (37.6) | 371.2 (37.0)
Current | 162 (16.2) | 159.7 (15.9)
Diabetes (% Yes) | 127 (12.7) | 98.9 (9.9)
Congestive Heart Failure (% Yes) | 27 (2.7) | 22.9 (2.3)
Coronary Heart Disease (% Yes) | 57 (5.7) | 59.6 (5.9)
Cancer (% Yes) | 134 (13.4) | 137.5 (13.7)
Stroke (% Yes) | 24 (2.4) | 20.9 (2.1)
HDL Cholesterol (mean (SD)) | 56.20 (16.58) | 56.82 (16.08)
Total Cholesterol (mean (SD)) | 204.71 (41.57) | 206.97 (40.66)
Systolic Blood Pressure (mean (SD)) | 131.14 (20.54) | 129.16 (19.81)

####  3.1.3 Mortality Data

The mortality data we use are from the package rnhanesdata (Leroux et al.,
2019). While the rnhanesdata package provides the processed mortality data,
the raw mortality data were obtained by linking the $2011$ death certificate
records provided by the National Death Index (NDI) to the NHANES record of
2005 - 2006 participants (Leroux et al., 2019). For each participant, the
mortality outcome is $1$ if the participant has died by the year $2011$, and
$0$ otherwise. The "alive" participants account for $94\%$ of all
participants, while the remaining $6\%$ are "dead." For mortality prediction,
we treat the mortality outcomes from the mortality data as the response
values, and $1003$ subjects that appear in both NHANES covariate and mortality
data are included in the analysis.

###  3.2 Methods

For each subject $i$, let $y_{i}$ be the five-year, all-cause binary mortality
outcome. In other words, we define

$y_{i}=\begin{cases}1\text{ if participant i is dead after 5 years, }\\\
0\text{ if participant i is alive after 5 years. }\end{cases}$

We use survey-weighted logistic regression, random forest, and AdaBoost
methods to predict the mortality outcomes. While each method has been well
studied in different aspects, this section introduces the three methods in a
way that is most natural and relevant to the mortality prediction problem.

####  3.2.1 Logistic Regression

For each subject $i$, we fit a survey-weighted logistic regression model of
the following form.

$logit(y_{i}|X_{i},s_{i})=\alpha+X_{i}^{T}\beta+s_{i}^{T}\theta$

where $y$ denotes the probability of death in 5 years,
$s_{i}=(s_{i1},...,s_{iK})^{T}$ is the $K\times 1$ dimensional vectors of
FPCA/PFPCA/NARFD scores for subject $i$, and $X_{i}$ is the predictor matrix
that contains the NHANES covariates. The logistic regression is fit using the
"svyglm" function from the survey package to incorporate the survey weights as
observation weights.

####  3.2.2 Random Forest

Because logistic regression is based on a linear relationship between
predictors and log odds of success, it often fails to capture non-linear
effects and interactions between predictors. Random forest is a more suitable
prediction model for dealing with non-linear effects. Random forests are based
on decision trees. In a decision tree, at each node, one predictor is used to
split the tree into two branches. This is done by selecting the predictor and
split which minimizes the impurity in the tree. In a random forest model, a
large number of decision trees are fit using a random subset of the predictors
for each tree. For regression, the predicted value is the average of all the
trees values, while for classification, the predicted value is either the most
commonly predicted class, or, in the case of two classes represented by 0 and
1, the average predicted value across all trees can be considered a
probability that the subject belongs to the class denoted by 1. The added
complexity of random forests comes at the cost of interpretability of the
model. To predict mortality using our data, we fit a random forest using the
"ranger" function from the ranger package (Wright and Ziegler, 2017). We chose
the ranger package over more commonly-used random forest packages such as
RandomForest because the function "ranger" has the capability of including
case weights.

####  3.2.3 AdaBoost

Random Forest and logistic regression models are known to perform poorly on
data with severe class imbalance. A boosting method is an excellent remedy to
the problem. One of the most commonly used boosting methods is AdaBoost.
AdaBoost is an iterative classification algorithm that trains decision trees
on the misclassified samples from the previous iteration in each iteration.
Previous studies have shown that AdaBoost models improve the prediction
accuracy of minority samples (Longadge et al., 2013). Therefore, boosting may
present an adequate solution to the class imbalance problem in mortality data.
We use an AdaBoost model that assumes an exponential loss on the $0-1$
mortality outcomes, with each observation weighted by the adjusted survey
weight. The implementation is done using function "gbm" from the R package gbm
(Greenwell et al., 2020).

###  3.3 Results

We use logistic regression, random forest, and AdaBoost methods as the main
prediction models, and all three models weight observations with the adjusted
survey weights. The input data are formed from the NHANES covariates combined
with the GFPCA, NARFD, or PFPCA scores. For the logistic regression models,
the FPC scores are selected using backward stepwise logistic regression with
AIC. For logistic + GFPCA (logistic regression with GFPCA scores) model, score
4 is included. For logistic + NARFD model, no score is included. For logistic
+ PFPCA model, FPC scores 2, 3, 4, 5, and 6 are included. After obtaining the
prediction results, we compare the nine models’ performance using survey-
weighted ROC curves and AUC scores. We also fit each model type with no FPC
scores as a base line. The survey-weighted ROC and AUC are implemented through
the function "WeightedROC" from the R package WeightedROC (Hocking, 2020).

The ROC curves for the prediction models are shown in Figure 5. For visual
clarity, and to focus on the FPCA results, the model with no scores was not
included in the ROC curve plot. The AUC scores are shown in Table 2. From most
to least accurate, the models are AdaBoost, logistic regression, and random
forest. The accuracy of AdaBoost results from the fact that AdaBoost
classifies a portion of "dead" outcomes accurately while logistic regression
and random forest fails to predict "dead" outcomes due to class imbalance.
Random forest and AdaBoost models outperform logistic regression models. From
best to worst, the submodels are GFPCA, PFPCA, NARFD, except for in the
logistic regression model. The difference in AUC between GFPCA and PFPCA
models is negligible under both AdaBoost and logistic regression.

Figure 5: ROC curves of the logistic regression/random forest/AdaBoost with GPFCA/NARFD/PFPCA scores for predicting 5-year all-cause mortality outcomes  Table 2: AUC scores for out-of-sample mortality prediction Model | GFPCA | NARFD | PFPCA | Baseline (no FPCA scores)
---|---|---|---|---
AdaBoost | 0.717 | 0.744 | 0.754 | 0.729
Logistic | 0.555 | 0.554 | 0.538 | 0.555
Random Forest | 0.724 | 0.703 | 0.722 | 0.733

We recommend AdaBoost + PFPCA as the go-to method for analyzing future NHANES
accelerometry data because PFPCA has better model interpretation than GFPCA
and a better AUC. In Section 2.3, the GFPCA models for the accelerometry data
have raised concerns over potential overfitting. On the other hand, PFPCA
provides smooth and reasonable fits for each participant’s accelerometry data.
Unlike Gaussian models, Poisson models theoretically account for the
accelerometry data’s count-valued structure. When it comes to classifying
mortality, AdaBoost + PFPCA has the highest AUC scores.

####  3.3.1 Logistic Regression Inference

Table LABEL:tab:coef1 is the coefficient table of the logistic + PFPCA model.
The table contains coefficients and confidence intervals (CI) for the NHANES
covariates and FPC scores selected via AIC; both the coefficients and the
confidence intervals are estimated using survey-weighted logistic regression
from the function "svyglm." Because our purpose is to infer the effect of each
variable on mortality using the coefficient table, the underlying logistic
regression is run on the full data rather than the training data. The
coefficients table of logistic + NARFD and logistic + GFPCA are included in
the Appendix.

Table 3: Coefficients table for logistic + PFPCA For categorical variable, the base level is given $0$ and $(0,0)$ as the coefficients and $95\%$ confidence intervals. Variable | Coefficient (95 % Confidence Interval)
---|---
Intercept | -6.769 (-16.924, 3.386)
Number of Weekdays | -0.066 (-0.493, 0.361)
Number of weekend | 0.153 (-0.742, 1.048)
Body Mass Index | -0.013 (-0.097, 0.071)
Race |
White | 0 $(0,0)^{*}$
Black | -0.012 (-0.918, 0.895)
Hispanic | -0.055 (-1.091, 0.981)
Other | -14.02 (-14.987, -13.052)
Gender |
Female | -0.007 (-0.552, 0.537)
Male | 0 $(0,0)^{*}$
Age | 0.107 (0.036, 0.178)
Education |
Less than High School | $0(0,0)^{*}$
High School | 0.379 (-0.45, 1.208)
College and Above | -0.329 (-1.097, 0.44)
Number of Alcoholic Drinks per Week | 0.034 (-0.018, 0.085)
Cigarette Smoking |
Never | 0 $(0,0)^{*}$
Former | 0.548 (-0.585, 1.681)
Current | 0.74 (-0.086, 1.566)
Has Diabetes | 0.58 (-0.49, 1.65)
Has Congestive Heart Failure | -0.018 (-1.808, 1.772)
Has Coronary Heart Disease | -0.221 (-1.748, 1.306)
Has Cancer | 0.611 (0.208, 1.014)
Has Stroke | -0.364 (-2.228, 1.5)
HDL Cholesterol | -0.013 (-0.031, 0.004)
Total Cholesterol | -0.002 (-0.011, 0.006)
Systolic Blood Pressure | 0 (-0.016, 0.017)
$2^{nd}$ score of Poisson FPCA | 0.006 (-0.002, 0.015)
$3^{rd}$ score of Poisson FPCA | -0.031 (-0.078, 0.016)
$4^{th}$ score of Poisson FPCA | -0.015 (-0.046, 0.017)
$5^{th}$ score of Poisson FPCA | 0.019 (-0.016, 0.055)
$6^{th}$ score of Poisson FPCA | 0.011 (-0.011, 0.032)

#### 3.3.2

Figure 6 shows the variable importance from the AdaBoost + PFPCA model. Other
variable importance plots are included in the Appendix. Among the most
important variables are age, the score for the second PFPCA component,
alcoholic drinks per week, blood pressure and total cholesterol level. Age is
to be expected, and is almost always the best predictor of mortality in
similar studies. The second PFPCA score is associated with more activity in
the late afternoon and less in the morning, as seen in Figure 1. PFPCA scores
3, 4, 5, and 6 also have nonzero importance, but are much lower in variable
importance and, due to their shapes (Figure 1), less interpretable than the
second score.

Figure 6: The variable importance plot from AdaBoost + PFPCA. The x-axis is
the relative importance. The y-axis is the variable names. Higher relative
importance means larger effect on the mortality outcome. The top five most
important variables are age, second PFPCA score, alcoholic drinks per week,
blood pressure and total cholesterol level. In particular, the second PFPCA
score is associated with more activity in the late afternoon and less in the
morning.

##  4 Discussion

###  4.1 FPCA Inference

The results from Section 2.3 show that PFPCA is the most suitable model for
interpreting the NHANES accelerometry data. From looking at the individual
FPCs in Figure 1, the Poisson scores, especially the first two, are easily
interpretable. The first score contributes to the overall level of activity,
while the second score corresponds to more activity in the afternoon and less
in the morning. For the other models, the interpretations are less
straightforward. For NARFD, the FPCs all have global effects with slightly
different shapes. This was slightly unexpected, as the NARFD model is based
off of local estimation. We thought the components would correspond to
different local changes, but this was not the case. The Gaussian principal
components have some interpretability; for example, the first component seems
to be similar to the second Poisson component: more activity in the afternoon
and less in the mornings. However, the Gaussian components are more erratic,
which may be a indication of over-fitting. This can also be seen in Figure 3.
The Poisson models give a nice middle ground between the overly erratic
Gaussian models and the overly smooth NARFD models. Due to the
interpretability, the balance between smoothness and fit, and the lowest MAE
of the Poisson FPCA, PFPCA is the best candidate of the three FPCA methods for
this data.

###  4.2 Mortality Prediction

We used logistic regression, random forest, and AdaBoost as the primary
mortality prediction models, each with three submodels that include GFPCA,
NARFD, or PFPCA scores. The results show that in terms of prediction accuracy,
AdaBoost + PFPCA is the best model. AdaBoost and random forest models
outperform logistic regression models. We recommend AdaBoost + PFPCA as the
go-to model for handling future NHANES mortality data, considering its
accuracy and inference power.
The most important variables according to the AdaBoost model were age, second
component score, number of drinks per week, blood pressure and total
cholesterol level. In the logistic model, age, the $2^{nd}$ Poisson FPC score
and the number of drinks per week all had positive coefficients, while total
cholesterol level and blood pressure has a near-zero coefficient. This may
indicate that age, second component score, and alcohol consumption are all
associated with higher risks of mortality for higher values. This lines up
with our intuition for age and drinks: given that age and number of drinks
have positive coefficients according to the logistic regression, our finding
confirms the intuitive idea that the elderly and heavy drinkers have higher
mortality risks.
The importance of the second score is very interesting, though. Because
AdaBoost is based on decision trees, we lose the interpretability of a
positive or negative effect of a variable. In the logistic regression,
however, the coefficient of the second PFPCA score was positive. It is
reasonable to conclude that, because of its positive coefficient in the
logistic regression and its high importance in the AdaBoost model, the second
PFPCA component is positively related to mortality. That is, people who are
more active in the evenings and less so in the mornings are at a higher risk
of mortality.

Random forest and logistic regression models both suffer from the class
imbalance problem. This is due to the fact that our mortality data has a lack
of death: $94\%$ of the mortality outcomes are classified as "alive", while
$6\%$ are classified as "dead". AdaBoost solves the class imbalance problem by
retraining the wrong classification outcomes, particularly the false "alive"
outcomes. For example, the AdaBoost + PFPCA model was able to predict $2/19$
death outcomes in the test data accurately while the other models failed to
produce any accurate death outcomes, which boosted AdaBoost models’ predictive
accuracy.

###  4.3 Further Work

There is a multitude of further work that could be done with this data. For
example, one could analyze which interactions exist between the variables and
include them in the logistic regression, which may lead to a logistic
regression model that actually predicts deaths. Another possible direction is
to explore the different sub-populations within the data. However, due to the
size of this data set and the already existing class imbalance, using a larger
data set, or waiting for the ten-year mortality data to ensure enough members
of each class would be beneficial, if not necessary, for any sub-population
analysis. Additionally, if more data was obtained, it would be useful to
perform out of sample predictions using a test data set, to see if the models
have generalized predictive power.

The methods for analyzing functional count data are not limited to GFPCA,
PFPCA, and NARFD. The Poisson distribution has well-known problems such as the
lack of flexibility, over- and under-dispersion. Other methods have been
proposed to address these problems, including the STAR framework (Kowal and
Canale, 2020) and Zero-inflated Negative Binomial (Neelon et al., 2019). In
the future, we hope to design an FPCA method using the STAR framework and
compare its inference with Poisson FPCA.

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##  6 Appendix

###  6.1 Further Analysis: Mixed Graphical Model

####  6.1.1 Method

Researchers are sometimes interested in testing the conditional independence
between sets of predictors, such as whether race is independent of having
diabetes given BMI. The mixed graphical model (MGM) is particularly suited for
the exploratory analysis and hypothesis testing with NHANES data via nodewise
regressions for categorical, count-valued, and continuous variables. For a
variable $X$, the nodewise regression of variable $X$ is the $L_{1}$ penalized
regression of $X$ on the other variables. The regression assumes
Gaussian/Poisson/multinomial distribution for continuous/count/categorical
data. The $L_{1}$ penalized nodewise regression selects the neighborhood of
associated variables for each covariate in the data. Previous studies have
demonstrated that MGM’s power for inferring the relationship between
covariates in high-dimensional multivariate data, such as genomics data (Ying-
Wooi Wan et al., 2015).
We use the function "mgm" from the R package mgm (Yang et al., 2015) . The
nodewise regression use function "glmnet" from the R package glmnet (Friedman
et al., 2011). It assigns multinomial, Poisson, Gaussian distributions to fit
categorical, count-valued, and continuous-valued variables. 10-fold cross-
validation is used to choose the optimal regularization parameter for each
regression.
The mixed graphical model in the example uses the NHANES covariates with all
six PFPCA scores. The graph visualizes the conditional dependency structures
between the variables, enabling one to learn between-covariate information
that may otherwise be obscured. For example, $6^{th}$ and $4^{th}$ Poisson FPC
scores are both positively related to the recorded number of weekends. One
possible meaning is that the scores 4 and 6 in PFPCA characterize weekend
activities. Furthermore, the number of drinks per week is conditionally
independent of age given gender. Given that number of drinks per week and age
are the two most important variables for mortality prediction, the graph shows
that their effects on mortality are independent given gender. HDL cholesterol
and total cholesterol are another pair of interesting predictors. The graph
shows that HDL cholesterol and total cholesterol are not closely associated,
even though the two variables both measure cholesterol levels. HDL cholesterol
positively associates with being female and negatively correlated with BMI. In
contrast, total cholesterol negatively associates with having diabetes,
coronary heart disease and positively correlated with being "other" races.
The graph also provides a fast and intuitive model selection if one wishes to
predict other NHANES variables. If one wishes to predict $X_{j}$ where $1\leq
j\leq n$, the neighborhood $Nr(X_{j})$ of $X_{j}$ represents the set of
covariates that are most indicative of $X_{j}$. For example, if
$X=CognitiveHeartFailure:Yes$,
$Nr(X)=\\{Diabetes:Yes,Stroke:Yes,CoronaryHeartFailure:Yes,SmokeCigsFormer,SystolicBloodPressure\\}$.
It follows that the history of coronary heart failure, diabetes, stroke,
whether the person smokes cigarettes and the person’s blood pressure are
enough to determine whether the person has/will have cognitive heart failure.
On the other hand, other covariates such as age, race, gender are independent
of having cognitive heart failure given the neighborhood variables.

Figure 7: Mixed Graphical Model for covariates in NHANES. Thicker green edges
indicate stronger positive correlation between the two variables; thicker red
edges indicate stronger negative correlation between the two variables.

###  6.2 Functional principal component plots for GFPCA and NARFD

(a) GFPCA

(b) NARFD

Figure 8: First 6 principal components calculated on the population, 5 minute-
level NHANES accelerometry data using GFPCA and NARFD methods. Solid lines
represent the base line zero, +,- lines denote the effect of being 2 standard
deviation from a score of 0 on the particular principal component. Figure $a)$
for GFPCA. Figure $b)$ for NARFD.

###  6.3 Coefficients Tables for logistic + GFPCA and logistic + NARFD

Table 4: Coefficents table for logistic + NARFD
*For categorical variable, the base level is given $0$ and $(0,0)$ as the coefficients and $95\%$ confidence intervals. Variable | Coefficient(95% Confidence Interval)
---|---
Intercept | -9.268 (-13.407, -5.129)
Number of Weekdays | -0.06 (-0.495, 0.374)
Number of weekend | 0.155 (-0.677, 0.987)
Body Mass Index | -0.007 (-0.092, 0.078)
Race |
White | 0 $(0,0)^{*}$
Black | 0.057 (-0.778, 0.893)
Hispanic | -0.009 (-1.027, 1.01)
Other | -13.992 (-14.855, -13.129)
Gender |
Female | 0.005 (-0.512, 0.522)
Male | 0 $(0,0)^{*}$
Age | 0.107 (0.036, 0.178)
Education |
Less than High School | $0(0,0)^{*}$
High School | 0.394 (-0.404, 1.191)
College and Above | -0.324 (-1.046, 0.399)
Number of Alcoholic Drinks per Week | 0.037 (-0.019, 0.092)
Cigarette Smoking |
Never | 0 $(0,0)^{*}$
Former | 0.558 (-0.586, 1.702)
Current | 0.701 (-0.112, 1.515)
Has Diabetes | 0.556 (-0.535, 1.648)
Has Congestive Heart Failure | 0 (-1.838, 1.838)
Has Coronary Heart Disease | -0.203 (-1.716, 1.311)
CancerYes | 0.604 (0.182, 1.025)
Has Stroke | -0.371 (-2.241, 1.499)
HDL Cholesterol | -0.013 (-0.031, 0.005)
Total Cholesterol | -0.003 (-0.012, 0.006)
Systolic Blood Pressure | 0 (-0.017, 0.016)
Table 5: Coefficents table for logistic + GFPCA
*For categorical variable, the base level is given $0$ and $(0,0)$ as the coefficients and $95\%$ confidence intervals. Variable | Coefficient(95% Confidence Interval)
---|---
Intercept | -8.634 (-12.826, -4.442)
Number of Weekdays | -0.096 (-0.543, 0.351)
Number of weekend | 0.112 (-0.708, 0.932)
Body Mass Index | -0.013 (-0.097, 0.071)
Race |
White | 0 $(0,0)^{*}$
Black | 0.098 (-0.78, 0.975)
Hispanic | 0.006 (-1.02, 1.033)
Other | -14.142 (-15.166, -13.119)
Gender |
Female | 0.081 (-0.505, 0.666)
Male | 0 $(0,0)^{*}$
Age | 0.103 (0.028, 0.177)
Education |
Less than High School | $0(0,0)^{*}$
High School | 0.411 (-0.403, 1.226)
College and Above | -0.302 (-1.031, 0.427)
Number of Alcoholic Drinks per Week | 0.036 (-0.021, 0.093)
Cigarette Smoking |
Never | 0 $(0,0)^{*}$
Former | 0.553 (-0.571, 1.677)
Current | 0.782 (-0.091, 1.655)
Has Diabetes | 0.523 (-0.604, 1.65)
Has Congestive Heart Failure | -0.013 (-1.786, 1.76)
Has Coronary Heart Disease | -0.17 (-1.733, 1.392)
Has Cancer | 0.6 (0.174, 1.026)
Has Stroke | -0.5 (-2.342, 1.342)
HDL Cholesterol | -0.013 (-0.033, 0.006)
Total Cholesterol | -0.003 (-0.012, 0.007)
Systolic Blood Pressure | -0.001 (-0.018, 0.015)
$4^{th}$ score of Gaussian FPCA | 0.057 (0.006, 0.109)