Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
Modeling Clinical Time Series Using Gaussian Process Sequences
Zitao Liu Lei Wu Milos Hauskrecht
Department of Computer Science, University of Pittsburgh
Motivation Background (con’t) State Space Gaussian Process (con’t) Experiments
Goal
“Develop accurate models of complex clinical time series!”
Specifically, a prediction model that can:
1. Handle missing values
2. Deal with irregular time sampling intervals
3. Make accurate long term predictions
Problem Statement
We define the time series prediction/regression function for clinical time
series as: where is a sequence of past observation-time pairs
such that, , is a p-dimensional observation vector
made at time ( ), and n is the number of past observations; and is the
time at which we would like to predict the observation . Irregularly
sampled, .
obs:g t Y y obsY
obs 1( , )n
i i it Y y iy
it
10 i it t
1 1i i i it t t t
y
nt t
???
Time
Value (𝒚𝒊, 𝒕𝒊)
𝒕𝒊
𝒚𝒊 𝒚
Development of accurate models of complex clinical time series data is
critical for understanding the disease, its dynamics, and subsequently
patient management and clinical decision making.
• Gaussian Process (GP)
GP is an extension of a multivariate Gaussian to distributions over
functions. Defined by two components: . ( ( ), ( , '))m x k x x
Mean function:
Covariance function:
( ) [ ( )]m fx x
( , ) [( ( ) ( ))( ( ) ( ))]K f m f m x x x x x x
GP regression equations:
Estimated Mean :
Estimated Covariance :
12
*( , ) ( , )K x K I
x x x y
*( ( ))Cov f
*( )f1
2
* * * *( , ) ( , ) ( , ) ( , )K x x K x K I K x
x x x x
Time
Valu
e
???
• Discrete non-linear model (GPIL)
Y – time series of observations;
Z – hidden states driving the dynamics.
1 ( ) t t tr z z w ( ) t t tu y z v
1 1 1~ ( , ),Vz ~ (0, ),t Qw ~ (0, )t Rv
)(u
)(r )(r
)(u )(u
– unknown transition function;
– unknown measurement function.
)(r
)(u
Acknowledgement
This research work was supported by grants R01LM010019 and R01GM088224 from the
National Institutes of Health. Its content is solely the responsibility of the authors and does not
necessarily represent the official views of the NIH.
Future Work
• Study and model dependences among multiple time series
• Extend to switching-state and controlled dynamical systems
Reference • M. Hauskrecht, M. Valko, I. Batal, G. Clermont, S. Visweswaran, and G.F. Cooper, Conditional outlier
detection for clinical alerting, in AMIA Annual Symposium Proceedings, 2010, p. 286.
• Carl Edward Rasmussen and Christopher K. I. Williams, Gaussian Processes for Machine Learning, MIT
Press, 2006.
• R. Turner, M.P. Deisenroth, and C.E. Rasmussen, State-space inference and learning with Gaussian processes,
in AISTATS, vol. 9, 2010, pp. 868-875.
• Data
• Evaluation Metric
1/2
1 2
1
| |n
i i
i
RMSE n y y
Root Mean Square Error(RMSE):
• Results
• Choice of Covariance Functions( )
Mean Reverting Property:
Periodicity:
1 1 1exp( | |)K t t
2
2 2 2exp( sin ( ) )2
K
t t
1 2K K K
Figure 2. Time series for six tests from the Complete Blood Count(CBC) panel for one of the patients.
Figure 3. Root Mean Square Error(RMSE) on CBC test samples.
Figure 1. Graphical representation of the state-space Gaussian process model. Shaded nodes denote
(irregular) observations and shaded nodes denote times associated with each observation. Each rectangle
(plate) corresponds to a window, which is associated with its own local GP. is the number of observations in
each window. is Gaussian field. is
,i jy
,i jT
,i jf
• State Space Gaussian Process(SSGP) Model
( ) ( ) ( ) , ( ) ~ (0, ( , ))T
fq f f K t t h t β t t t
( ) ~ ( ( ) , ( , ) ( ) ( ))T
qq K Tt h t b t t h t h t
We consider the Gaussian process q(t) with the mean function formed
by a combination of a fixed set of basis functions with coefficients, β:
In this definition, f(t) is a zero mean GP , h(t) denotes a set of fixed
basis functions, for example, , and β is a Gaussian prior,
. Therefore, q(t) is another GP process, defined by:
2( ) (1, , , )h t t t
~ ( , )I b
Background
• Linear Dynamical System (LDS)
1( | ) ( , ),t t tp A Q z z z ( | ) ( , )t t tp C Ry z z
1 t t tA z z w t t tC y z v
1 1 1~ ( , ),Vz ~ (0, ),t Qw ~ (0, )t Rv
Y – time series of observations;
Z – hidden states driving the dynamics.
A A
C C C
???
Time
Valu
e
• Idea Illustration
Time
Va
lue
Time
Valu
e
Time
Valu
e
State Space Gaussian Process
• Learning
1 1 1log ( | ) 1 1Tr
2 2
Tp K KK K K
YY Y
Parameter Set: 1 1{ ,{ }, , , , , , }i A C R Q V β (Θ denotes covariance function parameters)
Learn Θ: gradient based methods( )
Learn Ω\Θ: EM algorithm with , [log ( , , )]p β z β z Y
1 1 ,
2 1 1 1
( ) ( , , ) ( ) ( | ) ( | ) ( | ) ism m m
i i i i i j i
i i i j
p D p p p p
z β Y z z z β z y βJoint distribution:
• Prediction
1. Split and t into windows.
2. For windows that do not contain t, extract the last values in those
windows as βs and feed them into Kalman Filter algorithms to infer
the most recent hidden state where k is the index of the last window
that does not contain t.
3. Get from and .
4. If t is in window k+1, use observations in window k+1 and
to make the prediction, where ;
otherwise find out the window index i where t belongs to. The
prediction at t is .
To support the prediction inference, we need the following steps:
obsY
1 1k kC β z1k kCA β z
kz
1
1 1 1 1 1 1( , ) ( , )( )k k k k k kK t t K t t
y β y β
1 1( , )k kt y
1k kA z z
i k
kCA y z
1kβ
Patient management Making decision Disease understanding