Localized Supervised Metric Learning on ... - Researcher - IBM

<strong>Localized</strong> <strong>Supervised</strong> <strong>Metric</strong> <strong>Learning</strong> on Temporal Physiological Data
Jimeng Sun, Daby Sow, Jianying Hu, Shahram Ebadollahi
IBM T.J. Watson Research Center, New York, USA
{jimeng, sowdaby, jyhu, ebad}@us.ibm.com
Abstract
Effective patient similarity assessment is important
for clinical decision support. It enables the capture of
past experience as manifested in the collective longitudinal
medical records of patients to help clinicians assess
the likely outcomes resulting from their decisions
and actions. However, it is challenging to devise a patient
similarity metric that is clinically relevant and semantically
sound. Patient similarity is highly context
sensitive: it depends on factors such as the disease, the
particular stage of the disease, and co-morbidities. One
way to discern the semantics in a particular context is to
take advantage of physicians’ expert knowledge as reflected
in labels assigned to some patients. In this paper
we present a method that leverages localized supervised
metric learning to effectively incorporate such expert
knowledge to arrive at semantically sound patient similarity
measures. Experiments using data obtained from
the MIMIC II database demonstrate the effectiveness of
this approach.
1. Introduction
Medical records capture both observations of patients’
health status, and decisions and actions taken
by clinicians and care providers. Buried inside these
records are nuggets of insight on the temporal evolution
pattern of patient health status, and the effects of different
clinical decisions on the trajectory of a disease.
Tapping into this source of insight can be achieved by
developing techniques measuring cross patient similarities.
These techniques have the potential to improve
patients’ clinical outcomes as essential tools for diagnostic
and prognostic decision support.
Figure 1 illustrates several aspects of the scenario
that drives our research in this area. In this figure, a patient
with available observations up to a decision point
is presented to the system. The cohort of patients who
are clinically similar to the query patient are retrieved.
A clinician looks up the decisions and actions applied to
the retrieved cohort and their consequences and makes
up her mind about the best course of action for the current
patient. In addition, she can project the trajectory
of patient’s health status, as captured by the patient’s
clinical factors and biomarkers, under the regime of any
particular decision she makes.
There are three fundamental challenges that need to
be addressed before such decision support mechanism
can be materialized:
1. Alignment of the trajectories of patients’ temporal
characteristics to make the records amenable to
semantically and clinically sound comparison,
2. Devising similarity measures that can reflect the
clinical proximity or disparity between different
patients,
3. Coupling between decisions and their consequences
as manifested in patient prognosis.
The focus of this paper is on the second task. We propose
two different methods for feature generation over
multi-dimensional temporal patient data, and adopt a localized
supervised metric learning approach to arrive at
a semantically sound similarity measure for retrieving
patients represented in the multi-dimensional feature
space. The proposed method is tested using the MIMIC
II database, which consists of physiological waveforms,
and accompanying clinical data obtained for ICU patients
[1]. The study is carried out on 74 patients from
this database, categorized into 2 groups based on different
clinical conditions. Comparisons against unsupervised
metric learning approaches on classification and
retrieval accuracy are presented to illustrate the performance
benefit of the proposed approach.
2. Related Work
In [7], Saeed and Mark reported work on retrieving
similar patients using the same database, where
they employed a multi-resolution description scheme
for physiological temporal ICU data and used unsupervised
similarity metrics for retrieving patients. The focus
of that work was more on the symbolic representation
of the temporal data to make them amenable for

epresented by a N-dimensional feature vector x. Examples
of features are the mean and variance of the sensor
measures, or Wavelet coefficients. The prior belief
of physicians is captured as labels on some of the patients.
With this formulation, our goal is to learn a generalized
Mahalanobis distance between patient x i and
patient x j defined as:
√
d m (x i , x j ) = (x i − x j ) T P(x i − x j ) (1)
where P ∈ R N×N is called the precision matrix. Matrix
P is positive semi-definite and is used to incorporate
the correlations between different feature dimensions.
The key is to learn the optimal P such that the
resulting distance metric has the following properties:
Figure 1. Retrieving patients based on
their clinical similarity to a query patient
and using the retrieved patients to project
the evolution of patient’s clinical characteristics.
comparison. In this paper, we leverage both statistical
methods and Wavelet methods to extract features over
the temporal data.
<strong>Supervised</strong> metric learning has been studied in the
past [11, 12, 5]. The goal has been to learn a distance
metric such that samples in the same class are close and
those in different classes are far away. The common
treatment is to add constraints and regularization terms
into the objective function and then to solve it using optimization
methods. To avoid a large number of constraints,
in this paper we model this problem as trace
ratio problem which can be solved effectively (similar
to Wang et al. [9]).
3 <strong>Localized</strong> <strong>Supervised</strong> <strong>Metric</strong> <strong>Learning</strong>
In this section we present the supervised metric
learning problem in the context of patient similarity
measure. When a physician looks for similar patients
in a database, the similarity is often based not only
on quantitative measurements such as lab results, sensor
measurements, age and sex, but also on the physician’s
assessment of the disease type and stage. The
assessment would potentially influence the relative importance
a physician places on different measurements
or groups of measurements. To compute this specific
notion of similarity, we propose to learn a distance metric
that can automatically adjust the importance of each
numeric feature by leveraging the physician’s belief.
Formally, quantitative measurements of a patient are
• Within-class compactness: patients of the same label
are close together;
• Between-class scatterness: patients of different labels
are far away from each other.
To formally measure these properties, we use two kinds
of neighborhoods as defined in [10]: The homogeneous
neighborhood of x i , denoted as Ni o , is the k-nearest
patients of x i with the same label. The heterogeneous
neighborhood of x i , denoted as Ni e , is the k-nearest patients
of x i with different labels.
Based on these two neighborhoods, we define the local
compactness of point x i as
C i =
∑
d 2 m(x i , x j ) (2)
x j∈N o i
and the local scatter ness of point x i as
S i =
∑
x k ∈N e i
d 2 m(x i , x k ) (3)
The discriminability of the distance metric d m is defined
as
∑
J = ∑ i C ∑ ∑
i i x j∈N
(x
i
i S =
o i − x j ) T P(x i − x j )
∑ ∑
i i x k ∈N
(x
i
e i − x k ) T P(x i − x k )
(4)
The goal is to find a P that minimizes J , which is
equivalent to minimizing the local compactness and
maximizing the local scatterness simultaneously. In
contrast with linear discriminant analysis [4] , which
seeks for a discriminant subspace in a global sense,
the localized supervised metric aims to learn a distance
metric with enhanced local discriminability. To minimize
J , we formulate the problem as a trace ratio minimization
problem [9] and use the decomposed Newtown’s
method to find the solution [6].

Since P is a low-rank positive semi-definite matrix,
we can decompose the precision matrix as P = WW T ,
where W ∈ R N×d and d ≤ N. The distance metric
can be rewritten as d m (x i , x) = ‖W T x i − W T x j ‖.
Therefore, the distance metric is equivalent to euclidean
distance over the low-dimensional projection W T x.
4. Data Description and Feature Extraction
We have used the physiological data for 74 patients
obtained from the MMIC II database [1] in our experiments.
Each patient is represented with 5 streams
of sensor readings, sampled at 1 minute intervals: 1)
Sp02, 2) heart rate (HR), 3) mean ABP (ABPmean),
(4) systolic ABP (ABPSys), and diastolic ABP (ABP-
Dias). All patients belong to one of two groups H or C.
Those in group H (36 patients) had experienced Arterial
Hypotensive Episode (AHE) events during the forecast
window, whereas those in group C (38 patients) did not
experience any AHE within the forecast window. The
start of the forecast window is timestamped in the data
set (T 0 ) and its duration is 1 hour, in which an episode
of AHE can occur. For this study, we focus on a 2-
hour window around T 0 for each patient. Figure 2 illustrates
the data from two patients, in which samples in
H group show higher variability than those in C group.
Physicians actually use the variability level of ABP to
diagnose AHE [2].
We have used two different schemes to represent
the 2-hour temporal data for each patient: a statistical
time domain method and a wavelet domain method. In
the former, we compute the mean and variance of data
from each sensor for each patient. Thus, each patient is
represented in the time domain with a 10-dimensional
vector. In the latter, the wavelet coefficients of the 2-
hour window from each sensor are computed. We use
Daubechies-4 Wavelet [3] and keep the top-10 coefficients.
Finally, the coefficients from all 5 sensors are
vectorized into a 50-dimensional feature vector for each
patient.
5. Experiments
From the feature extraction step described in section
4, we obtain 74 N-dimensional feature vectors
where N = 10 for the statistic method and N = 50
for the Wavelet method. We then compare the following
three distance metrics using the leave-one-out
paradigm:
• Expert uses Euclidean distance of the variance of
the mean ABP as suggested in [2];
• PCA uses Euclidean distance over lowdimensional
points after PCA (an unsupervised
metric learning algorithm);
(a) Samples in H group
(b) Samples in C group
Figure 2. Examples of multivariate time
series data for H and C groups. H
group patients show higher variability
than those in C group.
• LSML using the localized supervised metric learning
method described in section 3.
Note that we do not make comparisons with global supervised
metric learning methods like LDA [4] because
as shown in [5, 8], localized metric usually performs
better. The performance metrics include k-NN classification
error rate and precision@10 retrieval results.
The precision@10 of a query point is computed by retrieving
10-nearest points with a specific distance metric
and then computing the percentage of those retrieved
points having the same label as the query point.
Performance Comparison To have a fair comparison,
both PCA and LSML project data into 1-
dimensional space since the Expert method only uses
one feature, i.e., the variance of mean ABP. Table 1
shows the classification results using 3-NN classifier,
and Table 2 shows the retrieval results. As can be
observed in both tables, LSML out-performs both expert
and PCA on both statistical and Wavelet features,

which confirms the importance of leveraging label information
into the distance metric. We also observe that
Wavelet features improve the performance significantly
for LSML, where the classification error drops by half
(from about 15% to less than 7%.)
Table 1. Classification error comparison
Expert PCA LSML
Statistic features 0.2295 0.2131 0.1475
Wavelet features NA 0.2295 0.0656
Table 2. Precision@10 retrieval results
Expert PCA LSML
Statistic features 0.6120 0.5355 0.6557
Wavelet features NA 0.5410 0.7869
Sensitivity Analysis There are two parameters in the
study: 1) the number of neighbors k in the k-NN classifier
and 2) the dimensionality d of the resulting lowdimensional
space (after PCA and LSML). Figure 3
shows the reuslts of sentivity analysis on these two parameters.
Figure 3(a) plots classification error vs. k for
all methods. Small k leads to lower classification error,
which confirms the need for a localized distance metric.
Figure 3(b) plots classification error vs. dimensionality
d for all methods except Expert, which confirms the
stability of LSML w.r.t. to different d.
6. Conclusion and Discussion
We have presented a method for deriving semantically
sound similarity measures for retrieving patients
represented by multi-dimensional time series. Our
method uses both statistical and wavelet based features
to capture the characteristics of patients, and leverages
localized supervised metric learning to incorporate
physicians’ expert domain knowledge. Experiments using
the MIMIC II database demonstrates the efficacy of
this appraoch. In future work we plan to explore ways
to explicitly incorporate temporal characteristics of the
data to further improve metric learning in this particular
context.
References
[1] MIMIC II Database.
http://physionet.org/physiobank/database/mimic2db/.
[2] X. Chen, D. Xu, G. Zhang, and R. Mukkamala. Forecasting
acute hypotensive episodes in intensive care
patients based on a peripheral arterial blood pressure
waveform. Computers in Cardiology, 36, 2000.
[3] I. Daubechies. Ten Lectures on Wavelets. SIAM,
Philadelphia, 1992.
(a) Stable with different k
(b) Stable with different d
Figure 3. LSML is stable with different parameter
values.
[4] K. Fukunaga. Introduction to Statistical Pattern Recognition.
Academic Press, San Diego, California, 1990.
[5] J. Goldberger, S. Roweis, G. Hinton, and R. Salakhutdinov.
Neighborhood component analysis. In NIPS, 2005.
[6] Y. Jia, F. Nie, and C. Zhang. Trace ratio problem revisited.
IEEE Transactions on Neural Networks, 2009.
[7] M. Saeed and R. Mark. A novel method for the efficient
retrieval of similar multiparameter physiologic time series
using wavelet-based symbolic representations. In
American Medical Informatics Association, 2006.
[8] M. Sugiyama. Dimensionality reduction of multimodal
labeled data by local fisher discriminant analysis. J.
Mach. Learn. Res., 8, 2007.
[9] F. Wang, J. Sun, T. Li, and N. Anerousis. Two heads
better than one: <strong>Metric</strong>+active learning and its applications
for it service classification. In ICDM, 2009.
[10] F. Wang and C. Zhang. Feature extraction by maximizing
the neighborhood margin. In CVPR, 2007.
[11] E. P. Xing, A. Y. Ng, M. I. Jordan, and S. Russell.
Distance metric learning, with application to clustering
with side-information. In NIPS, 2002.
[12] L. Yang. Distance metric learning: A comprehensive
survey. Technical report, Michgan State University,
2006.