# Prognostics of Surgical Site Infections using Dynamic Health Data

Chuyang Ke  Yan Jin  Heather Evans  Bill Lober  Xiaoning Qian  Ji Liu  Shuai
Huang  Department of Computer Science, University of Rochester  Department of
Industrial $\&$ Systems Engineering, University of Washington  Department of
Surgery, University of Washington  Department of Biomedical Informatics and
Medical Education, University of Washington  Department of Electrical $\&$
Computer Engineering, Texas A$\&$M University

###### Abstract

Surgical Site Infection (SSI) is a national priority in healthcare research.
Much research attention has been attracted to develop better SSI risk
prediction models. However, most of the existing SSI risk prediction models
are built on static risk factors such as comorbidities and operative factors.
In this paper, we investigate the use of the dynamic wound data for SSI risk
prediction. There have been emerging mobile health (mHealth) tools that can
closely monitor the patients and generate continuous measurements of many
wound-related variables and other evolving clinical variables. Since existing
prediction models of SSI have quite limited capacity to utilize the evolving
clinical data, we develop the corresponding solution to equip these mHealth
tools with decision-making capabilities for SSI prediction with a seamless
assembly of several machine learning models to tackle the analytic challenges
arising from the spatial-temporal data. The basic idea is to exploit the low-
rank property of the spatial-temporal data via the bilinear formulation, and
further enhance it with automatic missing data imputation by the matrix
completion technique. We derive efficient optimization algorithms to implement
these models and demonstrate the superior performances of our new predictive
model on a real-world dataset of SSI, compared to a range of state-of-the-art
methods.

###### keywords:

Surgical Site Infections (SSI) , Machine Learning , mHealth , Risk Prediction

##  1 Introduction

Surgical Site Infection (SSI) is a national priority in healthcare research
[1, 2, 3]. It occurs in 3-5% of all surgical patients, and up to 33% of
patients undergoing abdominal surgery [4, 5]. More than 500,000 cases are
estimated to occur in the US annually, resulting in additional costs of up to
$20,000 per infection. It also results in worse health outcomes for patients,
such as prolonged length of hospital stay, increased mortality, and
compromised health-related quality of life [6, 7, 8]. SSI is overall the most
costly healthcare-associated infection, yet many of its associated costs are
non-reimbursable. Surveillance methods have been invented since the early
1980s to provide appropriate data (such as risk indicators, clinical
prediction rules) to surgeons to monitor how changes in practice can impact
SSI occurrence [9, 10, 11]. Many of these surveillance systems follow the
standard guidelines established in 1992 by the American Centers for Disease
Control and Prevention (CDC)’s National Nosocomial Infections Surveillance
(NNIS) system, and rely on volunteer surgical wards from public or private
hospitals that routinely collect nosocomial infection data. Although these
surveillance systems are continuously updated with successive risk indicators
being included, they are mainly based on individual or very few risk factors.
Furthermore, the quality of post-discharge data is poor, as active
surveillance is costly and rarely performed prospectively, relying on
retrospective review of clinical documentation. Recently, with the rapid
advances of sensing and information technologies, new tools have been
developed to provide patients the means to monitor their conditions and share
this data with their providers. One example is the mobile health (mHealth)
tool that we have developed to enable patient-initiated monitoring of surgical
wounds and improve patient-provider communication after hospital discharge.
Self-reported symptoms of pain, body temperature, and wound features, as well
as patient- or caregiver-generated images of the wound may be acquired to
assess the evolving condition of the wound and the likelihood of SSI.

While these emerging mHealth tools have provided unprecedented capacity to
closely monitor patients for signs and symptoms of SSI and measure the
evolution of the wound-related variables and other clinical factors, existing
prognostic models of SSI have not been able to take advantage of this rich and
accumulating information for SSI prediction. Prognostic models of SSI ([12,
13, 14, 15, 16]) in the literature have three main limitations. First, many of
these models were based on medical knowledge or heuristics, not data-driven by
nature. Thus, their utility is limited on some specific phenotypes or
particular time windows of the SSI progression process. Second, these models
only predict whether or not the patient will develop SSI, but do not predict
when. Third, most existing models only incorporate static variables known as
of the end of the operation, e.g., demographics, pre-operative laboratory
results, comorbidities, and operative factors. These models do not incorporate
the continuous observations of the patients and their wound.

Figure 1: An illustration of the dynamic data from a subject as a spatial-
temporal data matrix

To overcome the limitations of existing SSI prediction models, especially when
utilizing the dynamic data collected by mHealth tools (e.g., an illustration
of such dynamic data as a spatial-temporal data matrix is shown in Figure 1),
we develop a solution that is specifically designed to address the unique
analytic challenges from dynamic mHealth data. The collected dynamic data
includes continuous measurements on a range of wound variables and clinical
factors, for which we form a spatial-temporal data matrix for each individual
(Figure 1). Rather than predicting the development of SSI based on a snapshot
of these factors or the static variables known as of the end of the surgical
operation, we propose a learning formulation to predict the time to the
development of SSI directly based on these spatial-temporal matrices. In our
experience with real-world applications, one common data challenge is that
there are many patients who do not develop SSI during observational period (in
this paper we name them as “censored samples”). What is worse is that, the
number of censored samples is usually greater than the number of samples that
have observed outcomes (in this paper we name them as “complete samples”).
Thus, a learning formulation that can fuse both types of samples is needed.
Another data challenge is, as a well-known phenomenon in many healthcare
applications, we encounter many missing values that result in a challenge for
the learning formulation. We therefore propose a matrix completion approach to
overcome this challenge and derive a convex optimization formulation to solve
it. Further, with the spatial-temporal matrix representation for the dynamic
mHealth data as the input of the prediction model, we investigate the use of a
bilinear formalism to reduce the dimensionality of the prediction model. Our
main contributions include:

  * 1.

Developed a flexible learning formulation that can convert the dynamic
clinical signals of an individual (represented as the “spatial-temporal
matrix”) into accurate prognostics of SSI onset, even when some data have
censored outcome;

  * 2.

Proposed the use of matrix completion to mitigate the missing data problem
that has been found ubiquitous in many healthcare applications;

  * 3.

Developed efficient algorithms to implement the machine learning models;
specific optimization strategies were developed to ensure the feasibility and
robustness of the algorithms;

  * 4.

Conducted extensive numerical studies on a real-world dataset (that includes
860 patients observed in a time window ranging from 2 to 21 days while 167
developed SSI), which generated real-world experience and empirical evidence
for using the proposed method with dynamic mHealth data.

The remainder of the paper is organized as follows: In Section 2, a brief
review of related work will be provided. In Section 3, our method will be
presented with detailed explanations of the formulation, together with the
computational algorithm for solving the proposed formulation. The real-world
implementation of this method will be presented and discussed in Section 4.
Conclusions will be drawn in Section 5.

##  2 Related Work

Our work is closely related to the following two areas: prediction models of
SSI risk and machine learning methods that are related to the analysis of
spatial-temporal data.

Risk prediction models of SSI: Many current risk prediction models for SSI are
built upon expert opinions. For instance, the Ventral Hernia Working Group
(VHWG) has categorized patients into 4 grades: low risk, comorbid, potentially
contaminated, and infected. A grading system has thereafter been developed to
predict the risk of SSI, based on expert opinions but not directly on patient
data. This model has later been modified by researchers, including the 3-tier
system developed in [17], which are still knowledge-driven rather than being
data-driven. On the other hand, many data-driven risk scores for SSI have been
developed over the years, such as the ones in [12, 13, 14, 15, 16]. Among
these models, the National Nosocomial Infection Surveillance (NNIS) Risk Index
is a well-accepted risk-assessment tool, but it only includes three
predictors. Thus, it has limited capacity to utilize the much more information
that we could collect nowadays for SSI prediction. For more complex models,
such as the model developed in [12], a hierarchical multivariate logistic
regression model was used to discriminate SSI from non-SSI patients based on
risk factors such as some operative variables, preoperative clinical severity,
risk factors, comorbidities, and other variables related to the
hospitalization procedure. However, these existing models build on the static
measurements of some selected risk factors rather than the dynamic post-
operative symptom and wound observations.  Recently, [3] investigated the use
of the data of “last five days” (defined as the last 5 days prior to SSI onset
for SSI patients or the last 5 days for non-SSI patients) for predicting SSI
onset, which showed better performance than using only cross-sectional
measurements. Comparing with [3], our study focuses more on the methodological
issues of how to mitigate the statistical challenges in order to automate the
process of using the incomplete dynamic data to build and select the best
prediction model. More related works that could be applied on a range of
disease contexts can be found in [18], which has covered a wide range of
prognostic models.

Machine learning methods: Matrix completion [19, 20, 21, 22, 23] is an
important approach to estimate missing elements in a data matrix, without
making strong assumptions on data missing mechanisms. The authors in [19]
proved that the low-rank matrix completion technique only needs
$O(mr\log^{2}m)$ to exactly recover the missing values for a $m\times n$
matrix with rank $r$, assuming $m\geq n$. A related formalism that can be used
for building prediction models using spatial-temporal data could be the
bilinear models. The bilinear model is a popular approach to capture
information in two different dimensions, for example in text mining [24],
computer vision [25], and image processing [26]. A specific example is, for
classification using brain image data or EEG signals, the input data for
classification are represented as matrices with columns representing space
indices and rows representing time indices. It is always possible to apply
linear models by creating data vectors as stacking of the elements of the data
matrix; however, such stacking approaches can not exploit the spatial and
temporal structure in the data matrix and will inevitably lead to a high-
dimensional weight vector in the linear model. Thus, we adopt the bilinear
formulation to better exploit the data matrix structure to obtain a more
parsimonious representation of the weight vector. For instance, Singular Value
Decomposition (SVD) is a typical bilinear model that identifies representative
vectors (characterized as eigenvectors) in both dimensions to span the data
space. A general framework was presented in [24] for solving two-factor tasks
using bilinear models, which can characterize factor interactions and employ
efficient algorithms based on the singular value decomposition and
expectation-maximization. Although not directly applicable here, existing
matrix completion formulations and bilinear models provide a great resource
for us to develop a systematic data analytic pipeline by building on these
models and tackle the data challenges with the dynamic mHealth data for SSI
prediction.

##  3 Approach

###  3.1 An Overview of the Proposed Machine Learning System

The overall goal of the proposed machine learning system is to predict the
time to SSI onset using the spatial-temporal matrix data. There are several
major data challenges that need to be overcome, such as the fusion of both
complete samples and censored samples, missing values in the spatial-temporal
matrices, and the complex two-way interaction structure of the spatial-
temporal matrices.

To mitigate these challenges, Figure 2 illustrates an overall scheme of the
proposed machine learning system. It consists of a novel modeling formulation
to fuse both complete samples and censored samples, in which bounded matrix
completion is used to mitigate the missing data problem. The learning
formulation is further equipped with model regularization techniques such as
the bilinear formalism and sparse learning to mitigate the challenges from the
high dimensionality of the spatial-temporal matrix and to exploit its inherent
spatial and temporal structure. In the following subsections, each of the main
machine learning algorithms will be described in details.

Figure 2: Algorithm flow

###  3.2 Learning Formulation for Data Fusion

Denote the $i$-th spatial-temporal data sample in the training data as ${\bf
x}_{i}\in\mathbb{R}^{T\times P}$, where $T$ denotes the length of observations
and $P$ denotes the number of dynamic variables (such as those wound-related
variables) that are collected. Note that here, the optimal length of
observations $T$ is not a known parameter. Rather, it should be decided using
data-driven methods, which will be discussed in Section 4. As we have
mentioned, our training data consists of both complete samples and censored
samples. We use $\Omega$ to denote the index set of complete samples, and use
$\bar{\Omega}$ to denote the index set of censored samples. Missing data is
referred to the missing elements in the matrix ${\bf x}_{i}$. Further, we use
$y_{i}$ to denote the time to develop SSI for sample $i$ if this sample is in
$\Omega$, or the duration of observations if the sample $i$ is in
$\bar{\Omega}$. For example, suppose that the sample $i$ is the subject $i$’s
day 1 through day 5’s measurements. If the subject $i$ developed SSI on day 7,
which means $i\in\Omega$, $y_{i}=7-5=2$. Otherwise, if the subject $i$ did not
develop SSI in the observational period of 21 days ($i\notin\Omega$),
$y_{i}=21-5=16$. Then, the following learning formulation is proposed to learn
the prediction model based on both the complete and censored samples, under
the assumption that there is no missing data in ${\bf x}_{i}$ and the
prediction model is linear:

$\displaystyle\min_{{\bf w}\in\mathbb{R}^{T\times P},b\in\mathbb{R}}$
$\displaystyle\quad f({\bf w},b):=\sum_{i\in\Omega}\frac{1}{2}(\langle{\bf
x}_{i},~{}{\bf w}\rangle+b-y_{i})^{2}$ (1)
$\displaystyle+\sum_{j\in\bar{\Omega}}\frac{\lambda}{2}(\min(0,\langle{\bf
x}_{j},~{}{\bf w}\rangle+b-y_{j}))^{2}.$

Here, $\langle\textbf{x}_{i},\textbf{w}\rangle$ denotes the the sum of entry-
wise products of two matrices. Note that, the formulation consists of two
components in the objective function. The first component,
$\sum_{i\in\Omega}\frac{1}{2}(\langle{\bf x}_{i},~{}{\bf
w}\rangle+b-y_{i})^{2}$, is the least-square loss function that penalizes the
difference between the predicted values, i.e., $\langle{\bf x}_{i},~{}{\bf
w}\rangle+b$, with the real outcome data, i.e., $y_{i}$, from the complete
samples. This is the same with classic linear regression models. The second
component evaluates the squared hinge loss [27], which corresponds to the loss
function for the censored samples. The rationale for the use of the squared
hinge loss is that, if the predicted time to SSI onset is larger than the
censored time $y_{i}$, we should not penalize the model; otherwise, we
penalize the model. In this way, the censored samples can contribute to the
learning of the prediction model in an unbiased way. To balance the
contributions of these two types of samples, we further use a parameter,
$\lambda$, in the formulation, i.e., larger $\lambda$ will allow the censored
samples to give more influence on the model estimation. In practice, this
parameter can be selected by model selection methods such as cross-validation
to enable data-driven optimal model selection.

###  3.3 Bilinear Formulation for Spatial-Temporal Matrix

Note that the model proposed above imposes no structure on the model
parameters ${\bf w}$, ignoring the fact that the input data is a spatial-
temporal matrix that has an inherent correlation structure in both dimensions.
Here, we propose to adopt a bilinear formulation to capture the correlation
structure within the spatial-temporal matrix. This is inspired by the success
of the bilinear models in a range of applications such as [24, 26].
Specifically, here, by using the bilinear formulation, the prediction model
can be rewritten as

$\langle{\bf x}_{i},~{}{\bf w}\rangle+b-y_{i}=\sum_{r=1}^{R}{\bf
u}_{r}^{\top}{\bf x}{\bf v}_{r}+b-y_{i},$ (2)

where ${\bf u}\in\mathbb{R}^{T\times R}$ and ${\bf v}\in\mathbb{R}^{P\times
R}$ with ${\bf u}_{r}$ and ${\bf v}_{r}$ denoting their corresponding $r$th
column. Actually, the bilinear formulation can be interpreted from another
point of view for dimension reduction by noting that $\sum_{r=1}^{R}{\bf
u}_{r}^{\top}{\bf x}{\bf v}_{r}=\langle{\bf u}{\bf v}^{\top},{\bf x}\rangle$.
In other words, to learn the bilinear prediction model characterized by ${\bf
u}\in\mathbb{R}^{T\times R}$ and ${\bf v}\in\mathbb{R}^{P\times R}$, it is
equivalent to learn the optimal solution of ${\bf w}$ if we restrict ${\bf
w}$’s rank by $R$ in the formulation (1). This leads to the following
transformed learning formulation:

$\displaystyle\min_{{\bf w}\in\mathbb{R}^{T\times P},b\in\mathbb{R}}$
$\displaystyle\quad f({\bf w},b):=\sum_{i\in\Omega}\frac{1}{2}(\langle{\bf
x}_{i},~{}{\bf w}\rangle+b-y_{i})^{2}$ (3)
$\displaystyle+\sum_{j\in\bar{\Omega}}\frac{\lambda}{2}(\min(0,\langle{\bf
x}_{j},~{}{\bf w}\rangle+b-y_{j}))^{2}$ s.t.
$\displaystyle\quad\text{rank}({\bf w})\leq r.$

Hence, besides the bilinear formalism, we note that this formulation can also
be interpreted as a rank-regularized formulation. The low-rank constraint of
${\bf w}$ is a reasonable treatment here, since due to the potential spatial
correlations (i.e., correlations between the variables) and temporal
correlations (correlations between the time points), the spatial-temporal data
inherently lie on a low-dimensional manifold. Therefore, ${\bf w}$ has the
corresponding low rank. Thus, the bilinear formulation induces the rank-
regularized formulation of ${\bf w}$.

###  3.4 Preconditioning Reformulation and Optimization

To solve the formulation in (3), first, we rewrite the formulation by
vectorizing ${\bf w}$ and each spatial-temporal data ${\bf x}_{i}$.
Particularly, we concatenate the rows of the matrix to convert the matrix to
be a vector. Similarly, to generate the corresponding design matrices
$X_{\Omega}$ and $X_{\bar{\Omega}}$ for the complete samples and censored
samples respectively, a column vector can be constructed by concatenating the
rows of a spatial-temporal matrix ${\bf x}_{i}$. A graphical illustration of
this concatenation process will be presented in Figure 3. Also, ${\bf
y}_{\cdot}$ denotes the corresponding vector of outcomes or censored outcomes.
Then, we can rewrite (3) as the following vectorized form:

$\displaystyle\min_{{\bf w}\in\mathbb{R}^{T\times P},b\in\mathbb{R}}$
$\displaystyle\frac{1}{2}\left\|X_{\Omega}^{\top}\text{vec}({\bf
w})-{\bf{1}}b-{\bf y}_{\Omega}\right\|^{2}$ (4) $\displaystyle+$
$\displaystyle\frac{\lambda}{2}\left\|\min\left({\bf{0}},X_{\bar{\Omega}}^{\top}\text{vec}({\bf
w})-{\bf{1}}b-{\bf y}_{\bar{\Omega}}\right)\right\|^{2}$ s.t.
$\displaystyle\quad\text{rank}({\bf w})\leq r,$

where $\mathbf{1}$ denotes the vector whose entries are all one. The objective
function in (4) is convex, and the projection onto the rank constraint has a
closed form. It is tempting to apply the projected gradient descent (PGD)
approach ([28]) to solve this convex formulation. However, it would be
inefficient if one directly applies PGD. The reason is that, due to the large
spatial and temporal correlations in the spatial-temporal matrix, the
optimization problem could be ill conditioned (recall the way by which we
generated the matrix $X_{\Omega}$ and $X_{\bar{\Omega}}$). Thus, we propose
the following preconditioning reformulation to enable the application of the
PGD on solving for (4).
Let $\text{vec}(\hat{{\bf w}})=(X_{\Omega}X_{\Omega}^{\top})^{1\over
2}\text{vec}({\bf w})$. We can substitute $\text{vec}({\bf w})$ by
$\text{vec}(\hat{{\bf w}})$ in the objective function in (4). The challenging
issue is how to substitute $\text{vec}({\bf w})$ by $\text{vec}(\hat{{\bf
w}})$ in the rank constraint. In fact, Lemma 1 indicates that
$\text{rank}(\hat{{\bf w}})=\text{rank}({\bf w})$ as long as
$(X_{\Omega}X_{\Omega}^{\top})^{1\over 2}$ is invertible, i.e., in other
words, as long as $X_{\Omega}$ has full column rank.

######  Lemma 1.

For any matrix ${\bf w}$, its rank remains the same after the vectorization
and a full ranked linear transformation, that is, $\text{rank}(\hat{{\bf
w}})=\text{rank}({\bf w})$, where $A$ is any full rank square matrix and

${\color[rgb]{0,0,0}\text{vec}(\hat{{\bf w}})=A\cdot\text{vec}({\bf w}).}$

###### Proof.

We first prove that $\text{range}(\hat{{\bf w}})\subset\text{range}({\bf w})$.
Note that the range of a matrix is defined as the span (set of all possible
linear combinations) of its column vectors. It is easy to see that any column
of $\hat{{\bf w}}$ is the linear combination of all columns in ${\bf w}$. So
the range (or column) space of $\hat{{\bf w}}$ is a subspace of the range (or
column) space of ${\bf w}$. Next, we show $\text{range}({{\bf
w}})\subset\text{range}(\hat{{\bf w}})$. Since $A$ is invertible, we can write
$\text{vec}({\bf w})=A^{-1}\cdot\text{vec}(\hat{{\bf w}})$. For the same
reason above, the range space of matrix ${\bf w}$ is the subset of the range
space of matrix $\hat{{\bf w}}$. Then, we can complete the proof by combining
both conclusions. ∎

Then, ignoring the constant terms, we can finalize the preconditioning
reformation as follows:

$\displaystyle\min_{\hat{{\bf w}}\in\mathbb{R}^{T\times P},b\in\mathbb{R}}$
$\displaystyle\quad\frac{1}{2}\left\|\text{vec}(\hat{{\bf
w}})\right\|^{2}-({\bf{1}}b+{\bf
y}_{\Omega})^{\top}X_{\hat{\Omega}}^{\top}(X_{\Omega}X_{\Omega}^{\top})^{-{1\over
2}}\text{vec}(\hat{{\bf w}})+{1\over 2}\|{\bf{1}}b+{\bf y}_{\Omega}\|^{2}$ (5)
$\displaystyle+\frac{\lambda}{2}\left\|\min\left({\bf{0}},X_{\bar{\Omega}}^{\top}(X_{\Omega}X_{\Omega}^{\top})^{-{1\over
2}}\text{vec}(\hat{{\bf w}})-{\bf{1}}b-{\bf
y}_{\bar{\Omega}}\right)\right\|^{2}$ s.t.
$\displaystyle\quad\text{rank}(\hat{{\bf w}})\leq r.$

Now we can apply PGD to solve (5). As summarized in Algorithm 1, first, we run
gradient descent with a fixed step size $\eta$ on the smooth part (objective
function); and then, solve the proximal mapping (projection) with Singular
Value Decomposition (SVD) and recover ${\bf w}$ by $\text{vec}({\bf
w})=(X_{\Omega}X_{\Omega}^{\top})^{-{1\over 2}}\text{vec}(\hat{{\bf w}})$.
Note that, in Algorithm 1, the criterion of convergence is met when the
decrease rate of the objective function value is less than 0.0001 in an
iteration.

Algorithm 1 Projected Gradient Descent

0: $X_{\Omega}$, $X_{\bar{\Omega}}$, $\lambda$, $r$ and $\eta$

0: ${\bf w}$, $b$

$A=(X_{\Omega}X_{\Omega}^{\top})^{-{1\over 2}}$

while not converge do

$\nabla_{\hat{{\bf w}}}=\text{vec}(\hat{{\bf w}})-({\bf{1}}b+{\bf
y}_{\Omega})^{\top}X_{\Omega}^{\top}A+\lambda
AX_{\bar{\Omega}}\min({\bf{0}},X_{\bar{\Omega}}^{\top}A\cdot\text{vec}(\hat{{\bf
w}})-b-{\bf y}_{\bar{\Omega}})$

$\nabla_{b}={\bf{1}}(b+1)+{\bf
y}_{\Omega}+\lambda\min({\bf{0}},X_{\bar{\Omega}}^{\top}A\cdot\text{vec}(\hat{{\bf
w}})-b-{\bf y}_{\bar{\Omega}})$

$\text{vec}(\hat{{\bf w}})=\text{vec}(\hat{{\bf w}})-\eta\nabla_{\hat{{\bf
w}}}$

$b=b-\eta\nabla_{b}$

$[U,\Sigma,V]=\text{svd}(\hat{{\bf w}})$

$\hat{{\bf w}}=U_{r}\Sigma_{r}V_{r}$

end while

$\text{vec}({\bf w})=A\cdot\text{vec}(\hat{{\bf w}})$

One remaining issue in the implementation of the PGD approach is the selection
of the step size $\eta$. It can be chosen as a sufficiently small value, or
the AMIGO rule [29] can be adopted to dynamically decide the value of $\eta$.
No matter which approach is used, it has been shown in [28] that the key issue
is to ensure that the objective function is decreasing iteratively, which is
not a difficult task empirically.

###  3.5 Bounded Matrix Completion (BMC)

In this section, we will develop a matrix completion approach to mitigate the
missing data issue in our problem. Missing data has been ubiquitous in many
healthcare applications. We found no exception in our study, since the data is
collected by patients and for many reasons the patient compliance in data
collection has always been a challenge. We propose to use the matrix
completion approach to fill in the missing values, rather than using simple
heuristic approaches such as using mean values to replace the missing values.
Our rationale is, again, the correlations in the spatial-temporal matrix
essentially imply that the spatial-temporal matrix is low dimensional. For
instance, some variables that are used to measure the wound could be
correlated, such as the color and temperature of the wound. Due to this
reason, by using their correlations, we could recover many missing values as
long as we have observed some values in the matrix. To exploit this idea, we
assume that there exist a few basis vectors that span the daily observation
vector of each individual (i.e., in the same spirit of many dimensionality
reduction methods). Let $\mathcal{T}_{n,t}$ denote the $t$-th day observation
vector of the $n$-th individual. We assume that $\mathcal{T}_{n,t}$ can be
represented as a linear combination of the basis vectors
$\\{U_{1},U_{2},\cdots,U_{r}\\}$:

$\mathcal{T}_{n,t}\approx\sum_{i=1}^{r}\alpha_{i}U_{i}.$

Given observed measurements in $\mathcal{T}_{n,t}$, the optimal weights
$\\{\alpha_{1},\alpha_{2},\cdots,\alpha_{r}\\}$ can be obtained by minimizing
the difference between the corresponding elements in
$\sum_{i=1}^{r}\alpha_{i}U_{i}$ with the observed measurements in
$\mathcal{T}_{n,t}$. In other words, we seek to identify the optimal weights
that can best match the projected vector $\sum_{i=1}^{r}\alpha_{i}U_{i}$ with
the observed incomplete vector $\mathcal{T}_{n,t}$. This rationale leads to
the following formulation:

$\displaystyle\min_{X_{\Phi},M}\quad$ $\displaystyle||X-M||^{2}_{F}$ s.t.
$\displaystyle\text{rank}(M)\leq r$ $\displaystyle
X_{i,j}\in[l_{j},u_{j}]\quad\forall(i,j)\in\Phi$

where $M$ encodes the information of $\\{U_{1},U_{2},\cdots,U_{r}\\}$ and
$\Phi$ denotes the set of the missing values. Note that, in this bounded low-
rank matrix completion problem, we further restrict the missing elements to be
within a certain range $[l_{j},u_{j}]$ for the corresponding variable $j$.
This is to ensure that the missing values won’t be filled in with unreasonable
values. Actually, the lower bound and the upper bound can be decided based on
the observed data to ensure the clinical validity of the estimated values:

$\displaystyle l_{j}:=$
$\displaystyle\min_{k\in\\{i:~{}(i,j)\notin\Phi\\}}X_{k,j};$ $\displaystyle
u_{j}:=$ $\displaystyle\max_{k\in\\{i:~{}(i,j)\notin\Phi\\}}X_{k,j}.$

Note that these bounds are obtained from the observed data in $\Phi^{c}$, the
complement of $\Phi$.

To solve the BMC problem, we apply the coordinate descent algorithm that
iteratively solves the optimization by alternately updating $X_{\Phi}$ or $M$
while fixing the other one. Since the value of the objective function
decreases iteratively, this algorithm is guaranteed to converge. It is worthy
of mentioning that, there is a closed-form solution in each step to solve for
$M$ or $X_{\Phi}$. We summarize the BMC algorithm in Algorithm 2.

Algorithm 2 Bounded Matrix Completion (BMC)

0: $X_{\Phi^{c}}$, $r$, $l_{j}$ and $u_{j}$

0: $X_{\Phi}$

Initialize $X_{\Phi}$

while not converge do

$[U,\Sigma,V]=\text{svd}(X)$

$M=U_{r}\Sigma_{r}V_{r}$

$X_{ij}=\max(l_{j},\min(u_{j},M_{ij}))\quad\forall(i,j)\in$ $\Phi$

end while

###  3.6 Missing Value Estimation for the Testing Data

As shown in the previous section, the BMC formulation is used to fill in the
missing values in the training data. The rationale is to learn a few basis
vectors $U_{i}$’s, and then, for each incomplete observation vector
$\mathcal{T}_{n,t}$, the algorithm seeks to identify the optimal weights that
can best match the projected vector $\sum_{i=1}^{r}\alpha_{i}U_{i}$ with
$\mathcal{T}_{n,t}$, where $U=[U_{1},U_{2},\ldots,U_{r}]$ is obtained by
applying SVD to the learned low-rank imputation matrix: $M=U\Sigma V^{\top}$.
Then, the missing values in $\mathcal{T}_{n,t}$ can be filled in by the
corresponding elements in $\sum_{i=1}^{r}\alpha_{i}U_{i}$. Thus, it is
straightforward to use the results from the BMC formulation learned from the
training data to fill in the missing values in the testing data as well.
Denote the coming testing data as $z$. For the new testing data $z$, its
missing elements indexed by $S$ can be estimated by solving the following
convex optimization problem:

$\displaystyle\min_{\alpha,z_{S}}\quad$ $\displaystyle\|z-U\alpha\|^{2}$ s.t.
$\displaystyle z_{j}\in[l_{j},u_{j}]\quad\forall~{}j\in S.$

Then, the best estimation can be found by updating $z_{S}$ and $\alpha$
iteratively by following the formulas shown below:

$\alpha=U^{\top}z,\quad
z_{j}=\min(U_{j},\max(l_{j},U_{j\cdot}\alpha)),\quad\forall j\in S.$

Note that, many of the existing missing data imputation methods have been
built on different assumptions of the mechanism of why data is missing, such
as the Missing Completely At Random (MCAR) and the Missing At Random (MAR).
While this may lead to more tailored missing data imputation methods, here, we
have difficult in identifying and validating the missing data mechanism in our
study. Hence, we pursue this low-rank model that places less strict
assumptions on the missing data mechanism, which leads to the development of
the low rank based estimation methods. Also, the low rank based estimation
method can be applied to scenarios where a large portion of elements (e.g.,
50% elements) are missing.

##  4 Experiments

###  4.1 Study population

Table 1: Baseline data from patient cohorts with and without inpatient SSI |  | Without SSI
---
(N=684; 80.6%)
| With SSI
---
(N=167; 19.4%)
p-value
Patient factors |  |  |
Age, mean, [CI], years |  56.18 [55.09-57.27] |  57.48 [55.45-59.51] |  0.29
Male sex, N (%) [CI] |  247 (36.1) [0.33-0.40] |  62 (37.1) [0.30-0.45] |  0.81
Procedure-related |  |  |
Type of operation |  N (%) |  N (%) |  $<$0.001
abdominal wall |  44 (6.4) |  3 (1.8) |
gastroduodenum |  27 (3.9) |  4 (2.4) |
gall bladder/bile duct |  31 (4.5) |  4 (2.4) |
liver |  101 (14.8) |  19 (11.4) |
spleen/adrenal gland |  29 (4.2) |  4 (2.4) |
small bowel |  35 (5.1) |  18 (10.8) |
kidney |  179 (26.2) |  23 (13.8) |
vascular |  50 (7.3) |  6 (3.6) |
esophagus |  75 (11.0) |  25 (15.0) |
large bowel |  69 (10.1) |  35 (21.0) |
pancreas |  44 (6.4) |  26 (15.6) |
Risk factors |  |  |
|  N (%) [CI] |  N (%) [CI] |
Smoking |  287 (42.0) [0.38-0.46] |  59 (35.3) [0.28-0.43] |  0.12
Diabetes mellitus |  83 (12.2) [0.10-0.15] |  21 (12.6) [0.08-0.19] |  0.88
Chronic lung disease |  58 (8.5) [0.07-0.11] |  22 (13.2) [0.08-0.19] |  0.063
Systemic corticosteroid |  79 (11.6) [0.09-0.14] |  25 (15.0) [0.10-0.21] |  0.23
Chemotherapy |  46 (6.7) [0.05-0.09] |  12 (7.2) [0.04-0.12] |  0.83
Radiotherapy |  12 (1.8) [0.01-0.03] |  3 (1.8) [0.00-0.05] |  0.97
Ascites present |  16 (2.3) [0.01-0.04] |  10 (6.0) [0.03-0.11] |  0.014
Infection (non-SSI) |  75 (11.0) [0.09-0.14] |  14 (8.4) [0.05-0.14] |  0.33
Alcohol use |  311 (47.1) [0.43-0.51] |  70 (45.5) [0.37-0.54] |  0.71
Alcohol quantity |  4.49 [3.85-5.12] |  5.24 (3.54-6.94) |  0.34
Body Mass Index |  |  |  0.65
Underweight |  19 (2.8) |  6 (3.6) |
Normal |  317 (45.2) |  80 (47.9) |
Overweight |  220 (32.2) |  59 (35.3) |
Class 1 obesity |  80 (11.7) |  20 (12.0) |
Class 2 obesity |  20 (2.9) |  8 (4.8) |
Class 3 obesity |  8 (1.2) |  2 (1.2) |
Figure 3: The data structure

A prospective cohort study of 1,000 open abdominal surgery patients was
conducted at a 1200-bed academic teaching hospital in the Netherlands,
described previously [30]. Patients who didn’t undergo surgery (n=33) or with
$<$ 2 days of wound observations (n=107) were excluded from analysis, leaving
860 patients in total. Subjects were prospectively assessed for CDC-defined
superficial, deep and/or organ space infections [9]. Superficial SSI are
infections that occur within 30 days after the operation and involve only skin
or subcutaneous tissues of the incision and at least one of the following: 1)
purulent drainage from the incision, with or without confirmation by
laboratory tests; 2) organisms isolated from an aseptically obtained culture
or aspiration of the incision; 3) at least one of the following signs or
symptoms: pain or tenderness, localized swelling, redness, or heat and
superficial incision opened by the surgeon; and 4) diagnosis of superficial
SSI by the surgeon or attending physician. Stitch abscesses and infected burn
wounds are excluded. Deep incisional SSIs are infections occurring within 30
days of surgery, involving the deep soft tissues (muscle and fascia) and at
least one of the following: 1) purulent drainage from the deep incision; 2)
incision spontaneously dehisces or is opened by the surgeon when the patient
has at least one of the following: fever (38°C), localized pain, or
tenderness; 3) an abscess or other evidence of infection involving the deep
incision is found on examination, reoperation, or histopathologic or radiology
examination; and 4) deep SSI is diagnosed by the surgeon or attending
physician. Finally, an organ space infection is an infection occurring within
30 days of surgery; the infection appears related to the surgery and involves
any part of the anatomy (organs or spaces), other than the incision and at
least one of the following: 1) purulent drainage from drain; 2) organisms
isolated from an aseptically obtained culture of fluid or tissue; 3) diagnosis
by the surgeon or attending physician; and 4) an abscess or other evidence of
infection involving the organ space found on direct examination, reoperation,
or radiologic examination. Patients’ characteristics are reported in Table 1
for both SSI and non-SSI patients, where the p-value is computed by the t-test
for comparing the two groups.

###  4.2 Data collection

Subjects in the dataset were examined daily, using a previously described
protocol [31], from post-operative day 2 until discharge or 21 days, whichever
was earlier. Follow-up was performed at 30 days through clinic visit, phone,
or letter to ascertain post-discharge infections. Table LABEL:conti_data_demo
shows the dynamic wound data that was collected, including definitions of
categorical wound score variables [31]. For analysis purposes, we defined the
SSI group as having any of the 3 types of SSI due to the small numbers of deep
and organ-space infections. In addition, though a patient may have developed
multiple types of SSI during the observation period, we only include their
first infection in this analysis. The non-SSI group was defined as having not
developed any kind of infection, but may have had a post-discharge infection.
For patients who didn’t develop SSI during the observational period, their
days of observation are 21. For patients who developed SSI during the
observational period, the summary statistics of their observational time is
shown in Table 2.

Minimum | 1st Quantile | Median | Mean | 3rd Quantile | Maximum
---|---|---|---|---|---
3 | 5 | 7 | 8.244 | 11 | 21
Table 2: Days of post-operative observation for patients developing in-
hospital SSI

Further, Figure 3 schematically illustrates the process by which we translated
the original data into the data structure that will be used by the proposed
machine learning system. The basic idea is to use a sliding window to segment
each individual’s time series measurements into blocks of equal size $T$.

Table 3: Repeated data collected Variable | Scale |  Details
---|---|---
Primary wound variables |  |
Induration amount (mm) | 0 |  $>$5 mm
| 1 |  3-4 mm
| 2 |  1-2 mm
| 3 |  0 mm
Wound edge distance (mm) | 0 |  0 mm
| 1 |  1-2 mm
| 2 |  3-5 mm
| 3 |  6-10 mm
| 4 |  11+ mm
Slough/necrosis type | 0 |  none visible
| 1 |  white/grey nonviable tissue
| 2 |  loosely adherent yellow slough
| 3 |  adherent, soft, black eschar
| 4 |  firmly adherent, hard, black eschar
Slough/necrosis amount | 0 |  None visible
| 1 |  $<$25% of wound bed covered
| 2 |  25 to 50% of wound covered
| 3 |  $>$50% and $<$75% of wound covered
| 4 |  75 to 100% of wound covered
Granulation/epithelialization score | 0 |  Skin intact
| 1 |  75 to 100% of wound filled
| 2 |  25 to 75% of wound filled
| 3 |  $<$25% of wound filled
| 4 |  no granulation or epithelialization present
Exudate type | 0 |  none or bloody
| 1 |  serosanguineous: thin, watery, pale red/pink
| 2 |  serous: thin, watery, clear
| 3 |  purulent: thin or thick, opaque, tan/yellow
| 4 |  foul purulent: thick, opaque, yellow/green with odor
Exudate amount | 0 |  none (tissue is dry)
| 1 |  scant (non measurable amount)
| 2 |  small (exudate spread over wound, gauzes 25% wet)
| 3 |  moderate (exudate irregularly spread over wound)
| 4 |  large (large amount, widespread, gauzes $>$75% wet)
Wound edge color | 0 |  pink or normal for ethnic group
| 1 |  bright red and/or blanches to touch
| 2 |  white or gray pallor or hypopigmented
| 3 |  dark red or purple and/or nonblanchable
| 4 |  black or hyperpigmented
Temperature $10\,^{\circ}\mathrm{C}$ |  |  Wound
|  |  1 cm from wound edge (left/right)
|  |  3 cm from wound edge (left/right)
|  |  5 cm from wound edge (left/right)
Wound malodor |  |  Yes/no
Other wound variables |  |
Hematoma* |  |  Yes/no
wound mass palpable* |  |  Yes/no
seroma* |  |  Yes/no
wound culture* |  |  Yes/no
visual analogue pain scale | 1-100 |  Wound pain
wound length (cm) | count |  …cm
Vital signs |  |
heart rate | count |  …bpm
diastolic RR | count |  …mmHg
systolic RR | count |  …mmHg
tympanic temperature $10\,^{\circ}\mathrm{C}$ | count |  …$10\,^{\circ}\mathrm{C}$
Other observations |  |
cough* |  |  Yes/no
productive cough* |  |  Yes/no
vomiting* |  |  Yes/no
ventilator* |  |  Yes/no
antibiotics* |  |  Yes/no
reoperation* |  |  Yes/no
nasogastric tube* |  |  Yes/no
Suspicion of ileus* |  |  Yes/no
serial operation number | count |
* in previous 24 hours |  |

###  4.3 Parameter tuning and validation

Cross validation is adopted to tune the parameters in each model that we will
build in this study. Specifically, we randomly split the samples into multiple
sets with equal size. In our implementation, we take a training-testing ratio
of 4:1. Then we choose 4 sets as the training data, and the one left as the
testing data. For each proposed model, we repeat the training-testing
procedure 5 times, to ensure that every set is used as the testing data once.
We use the mean absolute prediction error (MAE), i.e.,
$\sum_{i\in\Omega}{|\Omega|}^{-1}|\langle{\bf x}_{i},~{}{\bf
w}\rangle+b-y_{i}|$, to evaluate the prediction performance of each model in
predicting the onset time of the individuals.

Figure 4: The MAE of different methods across different $\lambda$ Figure 5:
The MAE of different models across different duration of observation

###  4.4 Comparison with other models

We will compare our proposed learning system with closely related benchmark
approaches. The first one is the classic regression model [32] that assumes
linear and additive predictive effects of the predictors on a continuous
outcome variable. It has been used in many risk prediction models in a wide
range of applications. The second one is the Support Vector Regression (SVR)
model [33]. It is a more robust model than the classic regression model by
using a soft margin loss function, representing the state-of-the-art
performance. The performance comparison of these models are shown in Figures 4
and 5. Note that, Figure 4 is to compare the methods under different values of
$\lambda$ which essentially reflects the weight of the censored samples in the
learning formulation. Since we have an unbalanced dataset, using weights to
balance the contribution of both the complete and censored samples has been a
common approach in the use of the existing methods. From both Figures 4 and 5,
it can be observed that our methods perform significantly better than both
benchmark methods across all the comparisons. Also, it seems that with rank =
2, our method performs better than its companion that has rank = 3, indicating
that for this dataset, rank = 3 might result in overfitting. Note that, in
generating this result, we didn’t include the selection of the rank into the
cross-validation procedure. For a given rank, we adopted cross-validation to
select for the remaining parameters. Actually, later in Section 4.6, we will
show that the “full-scale” cross-validation study that includes the selection
of the rank actually selected rank = 2 as the optimal model, which indicates
that the cross-validation can help control the model complexity and reduce the
risk of over-fitting.

###  4.5 Evaluation of the matrix completion method for missing data
imputation

We further evaluate the performance of the matrix completion method for
missing data imputation in comparison with other methods. For example, for the
“column mean” method, to fill in the missing value on feature $j$ of
individual $i$ at a certain time point, the mean value of feature $j$ of all
the individuals across all the time points can be used based on the observed
measurements. Other methods we used for comparison include the K-Nearest-
Neighbor (KNN) method that has been widely used for missing data imputation
[34, 35] and a recent Bayesian approach [36]. We train the models using
different missing data imputation methods under different values of $\lambda$
and report the results as shown in Figure 6. It clearly shows that our low-
rank matrix completion method could lead to superior performance on missing
data imputation, providing better prediction performance of the model.

Figure 6: The MAE of different models that use matrix completion and other
procedures for missing data imputation, respectively

###  4.6 Optimal configuration of the proposed learning system

To identify the optimal configuration of the proposed learning system, we
build a sequence of models to investigate the performances of different
configurations. Critical parameters include the duration of the dynamic data
we could use for building the prediction model, the rank of the model
parameters ${\bf w}$, and the value of $\lambda$. Among these parameters,
probably the duration of the dynamic data is the most important parameter.
Apparently, there is a trade-off between accuracy and cost. With longer
duration, more information could be accumulated which will lead to better
prediction accuracy. On the other hand, it will result in higher cost, not
only in data collection, but also in the healthcare cost due to the potential
delay of identifying the patients who will develop SSI soon. Therefore, it is
a practical and important decision to find a reasonable length for observation
and prediction. Here, based on the cross validation with the training set
ratio set to be 4:1, the prediction performances of the models are shown in
Table 4. This demonstrates that the best model could be built if the duration
of the dynamic wound data is 5-days and the rank of $w$ is 2. A worse
performance was generated when the length of observation is 6. This is
probably because that the sample size for training the model drops
significantly when the length of observation increases. In Figure 7, we also
present the important variables of the best model in Table 4 with their
coefficients in decreasing order. We further show the distribution of the
predicted onset day for both the complete and censored populations using our
best model in Figure 8, and demonstrate that our model can effectively
separate the two groups.

Figure 7: Selected important variables with their coefficients |  |  $\lambda$-value
---|---|---
Duration | rank(${\bf w}$) | 0.01 | 0.05 | 0.10
3 | 2 | 3.720192 | 3.246942 | 3.206233
3 | 3.817237 | 3.395871 | 3.364114
4 | 2 | 3.260273 | 2.857273 | 2.863779
3 | 3.334785 | 2.935544 | 2.962186
5 | 2 | 2.998079 | 2.677925 | 2.718011
3 | 3.087649 | 2.794683 | 2.847655
6 | 2 | 3.085667 | 2.787124 | 2.779734
3 | 3.080365 | 2.739127 | 2.487222
Table 4: Prediction performance of different model configurations Figure 8:
Distribution of the predicted onset day for both the complete and censored
samples

##  5 Conclusions

Accurate determination of the SSI risk of a particular patient is essential
for deciding whether or not particular preventive strategies, i.e., enhanced
prophylactic antibiotics, decolonization, or other emerging interventions
should be adopted. Furthermore, uncovering developing SSI at the earliest
possible time affords the opportunity for anticipatory intervention,
preventing progression of infection and its morbid sequelae. It also holds
great potential to reduce the healthcare cost since delayed diagnosis of post-
discharge SSIs has significant financial and life quality costs, with more
than half of patients who develop post-discharge SSI readmitted to the
hospital [37]. In this paper, we investigated the problem of predicting the
time to the onset of SSI using the dynamic wound data collected on
individuals. The dynamic data of an individual, represented as the spatial-
temporal matrix, include the continuous measurements of many wound-related
variables. Many of the wound variables are applicable in a post-discharge
setting, allowing monitoring of these wounds using automated image analysis to
predict infections in real time. Thus, this study aligns well with the
emerging mHealth tools that are developed to closely monitor SSI patients.
While most existing SSI prediction models only use preoperative and operative
variables, [12, 13, 14, 15, 16], to the best of our knowledge, we are the
first team who developed a systematic treatment to utilize the dynamic wound
data of an individual for SSI risk prediction. This study used a unique
dataset, coming from a hospital in the Netherlands, which had a significantly
longer average length of stay than in the US, allowing us to characterize a
longer spectrum of the SSI progression process.

Our study is subject to limitations. First, we have only investigated the use
of the dynamic wound data for SSI risk prediction. However, it has been found
that many preoperative and operative variables are important in predicting the
SSI risk. Knowledge of the potential impact of these predictors may help guide
physicians with both preoperative and operative decision-makings, and help
develop personalized post-discharge care plans and interventions as well. For
instance, previous studies have demonstrated that poor nutrition is associated
with increased rates of SSI, probably due to a depressed immune system [38].
Thus, these physiologically impaired patients might benefit from timely and
appropriate nutritional support. It has also been found that smokers have
higher SSI risk than non-smokers, probably due to the altered perfusion
mechanics at the surgical site. There are many other preoperative and
operative factors such as age, surgery type, operative time, glucose, length
of stay [39, 38], that could significantly increase our capability to predict
the SSI risk if being combined with the dynamic wound data. Second, SSI was
classified into 3 categories based on the depth of infection [9], which
include superficial, deep, and organ space SSI. It might be possible that
different kinds of SSI would have different risk profiles. Third, many
variables that have been found predictive of SSI risk in the literature are
not collected in our dataset. For instance, specifics of the operative
procedure, such as timing of antibiotics, core body temperature, blood glucose
level, are not currently available. On the other hand, we have not included
the image-based wound variables, which theoretically can be extracted from the
wound images that have been available in the dataset, into our prediction
model. The challenge to extract those image-based wound variables from the
wound image data includes the enormous variations in the parameters regarding
how the images are acquired by the individual patients, i.e., with different
angles and different light conditions, etc. Thus, this is also one of our
future research directions to extract these image-based wound variables and
incorporate them into our prediction model. Finally, although our dataset has
longer observation duration than many other datasets studied in the
literature, it is still a short period of time considering that the typical
follow up period for SSI is thirty days after discharge. This interval of
observation is more administratively derived than biologically derived, as SSI
could potientilaly occur at longer periods after the surgical event itself.

In summary, in this study, we have developed a systematic treatment of the
data challenges associated with the dynamic wound data. The model can predict
not only yes-or-no, but also when the SSI will be developed. In addition, this
model has potential use in real-time mHealth monitoring systems that can
incorporate patient-reported outcomes and automate image analysis to predict
post-discharge SSI. In our future research, we will investigate how to
integrate the dynamic wound data with some static risk factors such as some
preoperative and operative factors to further improve the SSI risk prediction.
We will also develop robust image feature extraction method to extract SSI-
related features from the wound image data that is available in our dataset.
Also, it is of interest to investigate the course of SSI development and
evolution for different types of SSI and surgery procedures. We notice that
previous works such as [40] raised an interesting question from a conceptual
point of view, which is, whether one could generate meaningful features or
episodes from variables over time. It is a very interesting question to derive
such features or patterns to better understand the characteristics of the
underlying health condition. While the pathogenesis and natural history of SSI
has been largely unknown, a reasonable assumption is that there is a
progression process of SSI that leaves a trail of clinical patterns/features.
Through this study, we demonstrate that using dynamic data could improve SSI
prediction. It will be our future work to thoroughly study the dynamics of the
clinical signals and its connection/implication of the underlying disease
progression process. Last but not least, note that we assumed the prediction
model is linear. Despite the appearance as a linear model, proper
transformations could be applied to extend the proposed linear model for
characterizing potential nonlinearity. On the other hand, if the
transformation is needed, we do need to base the linear models on
interpretable and meaningful variable transformation. Besides the
methodological issues for data fusion and predictive modeling, how to
integrate the monitoring tool in real surgical settings is definitely a very
important problem. We have studied some important aspects of this problem in
our previous work [41]. Usability research is also part of our future research
directions, e.g., in [41] we only focused on a unique post-acute surgical use
case that may not perfectly generalize to medical or chronic care settings.

## Acknowledgments

Chuyang Ke and Ji Liu are in part supported by the NSF grant CNS-1548078.

## References

  * [1] N. N. Mahmoud, R. S. Turpin, G. Yang, and W. B. Saunders.  Impact of surgical site infections on length of stay and costs in selected colorectal procedures.  Surg Infect (Larchmt), 10(6):539–544, 2009.
  * [2] N. N. Mahmoud, R. S. Turpin, G. Yang, and W. B. Saunders.  The impact of surgical-site infections in the 1990s: attributable mortality, excess length of hospitalization, and extra costs.  Infect Control Hosp Epidemiol, 20(11):725–730, 1999.
  * [3] P. C. Sanger, G. H. van Ramshorst, E. Mercan, S. Huang, A. Hartzler, C. A. L. Armstrong, R. J. Lordon, W. B. Lober, and H. L. Evans.  A prognostic model of surgical site infection using daily clinical wound assessment.  Journal of the American College of Surgeons, 2016.
  * [4] T. D. Pinkney, M. Calvert, D. C. Bartlett, A. Gheorghe, V. Redman, G. Dowswell, W. Hawkins, T. Mak, H. Youssef, C. Richardson, et al.  Impact of wound edge protection devices on surgical site infection after laparotomy: multicentre randomised controlled trial (rossini trial).  BMJ, 347(7):f4305, 2013.
  * [5] B. R. Krieger, D. M. Davis, J. E. Sanchez, J. J.L. Mateka, V. N. Nfonsam, J. C. Frattini, and J. E. Marcet.  The use of silver nylon in preventing surgical site infections following colon and rectal surgery.  Dis Colon Rectum, 54(8):1014–1019, 2011.
  * [6] E. N. Perencevich, K. E. Sands, S. E. Cosgrove, E. Guadagnoli, E. Meara, and R. Platt.  Health and economic impact of surgical site infections diagnosed after hospital discharge.  Emerg Infect Dis., 9(2):196–203, 2003.
  * [7] J. T. Dipiro, R. G. Martindale, A. Bakst, P. F. Vacani, P. Watson, and M. T. Miller.  Infection in surgical patients: Effects on mortality, hospitalization, and postdischarge care.  Am J Heal Pharm, 55(8):777–781, 1998.
  * [8] E. Zimlichman, D. Henderson, O. Tamir, C. Franz, P. Song, C. K. Yamin, C. Keohane, C. R. Denham, and D. W. Bates.  Health care-associated infections: A meta-analysis of costs and financial impact on the us health care system.  AMA Intern Med, 173(22):2039–2046, 2013.
  * [9] T. C. Horan, R. P. Gaynes, W. J. Martone, W. R. Jarvis, and T. G. Emori.  Cdc definitions of nosocomial surgical site infections, 1992: a modification of cdc definitions of surgical wound infections.  Infect Control Hosp Epidemiol, 13(10):606–608, 1992.
  * [10] R. W. Haley, D. H. Culver, J. W. White, W. M. Morgan, T. G. Emori, van P. Munn, and T. M. Hooton.  The efficacy of infection surveillance and control programs in preventing nosocomial infections in us hospitals.  Am J Epidemiol, 121(2):182–205, 1985.
  * [11] M. Perennec-Olivier and P. Jarno.  Surveillance of surgical site infections.  French Institute for Public Health Surveillance.
  * [12] E. H. Lawson, B. L. Hall, and C. Y. Ko.  Risk factors for superficial vs deep/organ-space surgical site infections: implications for quality improvement initiatives.  JAMA surgery, 148(9):849–858, 2013.
  * [13] R. L. Berger, L. T. Li, S. C. Hicks, J. A. Davila, L. S. Kao, and M. K. Liang.  Development and validation of a risk-stratification score for surgical site occurrence and surgical site infection after open ventral hernia repair.  Journal of the American College of Surgeons, 217(6):974–982, 2013\.
  * [14] C. van Walraven and R. Musselman.  The surgical site infection risk score (ssirs): a model to predict the risk of surgical site infections.  PLoS one, 8(6):e67167, 2013.
  * [15] V. P. Ho, S. L. Stein, K. Trencheva, P. S. Barie, J. W. Milsom, S. W. Lee, and T. Sonoda.  Differing risk factors for incisional and organ/space surgical site infections following abdominal colorectal surgery.  Diseases of the Colon & Rectum, 54(7):818–825, 2011.
  * [16] L. Saunders, M. Perennec-Olivier, P. Jarno, F. L’Hériteau, A. Venier, L. Simon, M. Giard, J. Thiolet, J. Viel, et al.  Improving prediction of surgical site infection risk with multilevel modeling.  PLoS one, 9(5):e95295, 2014.
  * [17] A. E. Kanters, D. M. Krpata, J. A. Blatnik, Y. M. Novitsky, and M. J. Rosen.  Modified hernia grading scale to stratify surgical site occurrence after open ventral hernia repairs.  Journal of the American College of Surgeons, 215(6):787–793, 2012\.
  * [18] N. Peek and A. Abu-Hanna.  Clinical prognostic methods: trends and developments.  Journal of Biomedical informatics, 48:1, 2014.
  * [19] E. J. Candès and B. Recht.  Exact matrix completion via convex optimization.  Foundations of Computational mathematics, 9(6):717–772, 2009.
  * [20] J. Cai, E. J. Candès, and Z. Shen.  A singular value thresholding algorithm for matrix completion.  SIAM Journal on Optimization, 20(4):1956–1982, 2010.
  * [21] H. Ji, C. Liu, Z. Shen, and Y. Xu.  Robust video denoising using low rank matrix completion.  In Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on, pages 1791–1798. IEEE, 2010.
  * [22] J. Liu, P. Musialski, P. Wonka, and J. Ye.  Tensor completion for estimating missing values in visual data.  IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(1):208–220, 2013.
  * [23] M. Signoretto, R. Van de Plas, B. De Moor, and J. AK Suykens.  Tensor versus matrix completion: a comparison with application to spectral data.  Signal Processing Letters, IEEE, 18(7):403–406, 2011.
  * [24] J. B. Tenenbaum and W. T. Freeman.  Separating style and content with bilinear models.  Neural computation, 12(6):1247–1283, 2000.
  * [25] W. T. Freeman and J. B. Tenenbaum.  Learning bilinear models for two-factor problems in vision.  In Computer Vision and Pattern Recognition, 1997. Proceedings., 1997 IEEE Computer Society Conference on, pages 554–560. IEEE, 1997.
  * [26] B. A. Olshausena, C. Cadieub, J. Culpepperc, and D. K. Warlandd.  Bilinear models of natural images.  2007\.
  * [27] P. L. Bartlett and M. H. Wegkamp.  Classification with a reject option using a hinge loss.  The Journal of Machine Learning Research, 9:1823–1840, 2008.
  * [28] Y. Nesterov.  Introductory lectures on convex optimization.  Springer Science & Business Media, 87, 2004.
  * [29] K. J. Åström and T. Hägglund.  Revisiting the ziegler–nichols step response method for pid control.  Journal of process control, 14(6):635–650, 2004.
  * [30] G. H. van Ramshorst, M. C. Vos, D. den Hartog, W. C. J. Hop, J. Jeekel, S. E. R. Hovius, and J. F. Lange.  A comparative assessment of surgeons’ tracking methods for surgical site infections.  Surg Infect (Larchmt), 14(2):181–187, 2013.
  * [31] G. H. van Ramshorst.  Wound failure in laparotomy: New insights.  Erasmus University Rotterdam, 2014.
  * [32] M. H. Kutner, C. J. Nachtsheim, and J. Neter.  Applied linear regression models.  McGraw-Hill/Irwin, Boston, 4th Edition, 2004.
  * [33] A. Smola and B. Schölkopf.  A tutorial on support vector regression.  NeuroCOLT Technical Report NC-TR-98-030, Royal Holloway College, University of London, UK, 1998.
  * [34] K. Kim, B. Kim, and G. Yi.  Reuse of imputed data in microarray analysis increases imputation efficiency.  BMC bioinformatics, 5(1):1, 2004.
  * [35] J. M. Keller, M. R. Gray, and J. A. Givens.  A fuzzy k-nearest neighbor algorithm.  IEEE transactions on systems, man, and cybernetics, (4):580–585, 1985.
  * [36] A. Gelman and J. Hill.  Opening windows to the black box.  Journal of Statistical Software, 40, 2011.
  * [37] A. Gibson, S. Tevis, and G. Kennedy.  Readmission after delayed diagnosis of surgical site infection: a focus on prevention using the american college of surgeons national surgical quality improvement program.  Am J Surg. Elsevier Inc, 207(6):832–839, 2014.
  * [38] D. L. Malone, T. Genuit, J. K. Tracy, C. Gannon, and L. M. Napolitano.  Surgical site infections: reanalysis of risk factors.  J Surg Res, 103(1):89–95, 2002.
  * [39] M. Haridas and M. A. Malangoni.  Predictive factors for surgical site infection in general surgery.  Surgery, 144(4):496–501, 2008.
  * [40] T. Toma, R. Bosman, A. Siebes, N. Peek, and A. Abu-Hanna.  Learning predictive models that use pattern discovery - a bootstrap evaluative approach applied in organ functioning sequences.  Journal of biomedical informatics, 43(4):578–586, 2010.
  * [41] P. Sanger, A. Hartzier, R. Lordon, C. Armstrong, W. Lober, H. Evans, and W. Pratt.  A patient-centered system in a provider-centered world: challenges of incorporating post-discharge wound data into practice.  Journal of the American Medical Informatics, 0:1–13, 2016.