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Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Factorial Switching Linear Dynamical Systems forPhysiological Condition Monitoring
Chris Williamsjoint work with John Quinn, Neil McIntosh
Institute for Adaptive and Neural ComputationSchool of Informatics, University of Edinburgh, UK
June 2008
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Premature Baby Monitoring
with Prof Neil McIntosh (Edinburgh Royal Infirmary)
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Why model this data?
Artifact corruption, leading to false alarms
Our aim is to determine the baby’s state of health despitethese problems
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Overview
Factors
Factorial switching Kalman filter
Inference
Parameter estimation
Results
Modelling novel regimes
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Probes
1. ECG, 2. arterial line, 3. pulse oximeter 4. core temperature, 5.peripheral temperature, 6. transcutaneous probe.
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Factors affecting measurements
The physiological observations are affected by differentfactors.Factors can be artifactual or physiological.An arterial blood sample (artifact):
30
40
50
60
Sys
. BP
(m
mH
g)
0 200 400 600 800 10000
20
40
60D
ia. B
P (
mm
Hg)
Time (s)
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Common factor examples
Transcutaneous probe recalibration (artifact)
0
10
20
30
TcP
O2 (
kPa)
0 1000 2000 3000 4000 50000
5
10
TcP
CO
2 (kP
a)
Time (s)
0
10
20
30
TcP
O2 (
kPa)
0 2000 4000 60000
5
10
15
TcP
CO
2 (kP
a)
Time (s)
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Common factor examples
Bradycardia (physiological)
0 20 40 60 80 10040
60
80
100
120
140
160
180
HR
(bp
m)
Time (s)0 50 100 150
40
60
80
100
120
140
160
180
HR
(bp
m)
Time (s)
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Factorial Switching Kalman Filter
Artifactual state
Physiological state
Observations
Physiological factors
Artifactual factors
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
FSKF notation
st is the switch variable, which indexes factor settings, e.g.‘blood sample occurring and first stage of TCP recalibration’.
xt is the hidden continuous state at time t. This containsinformation on the true physiology of the baby, and on thelevels of artifactual processes.
y1:t are the observations.
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Kalman filtering
Continuous hidden state affects some observations:
xt ∼ N (Axt−1,Q)
yt ∼ N (Cxt ,R)
Kalman filter equations can be used to work computep(x1:t |y1:t)
Done iteratively by predicting and updating
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Switching dynamics
The switch variable st selects the dynamics for a particularcombination of factor settings:
xt ∼ N (A(st)xt−1,Q(st))
yt ∼ N (C(st)xt ,R(st))
For each setting of st , the Kalman filter equations give apredictive distribution for xt .
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Factor interactions
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Factor interactions
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Factor interactions
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Factor interactions
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Factor interactions
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Related work
Switching linear dynamical models have been studied by manyauthors, e.g. Alspach and Sorenson (1972), Ghahramani andHinton (1996).
Applications include fault detection in mobile robots (deFreitas et al., 2004), speech recognition (Droppo and Acero,2004), industrial monitoring (Morales-Menedez et al., 2002).
A two-factor FSKF was used for speech recognition by Maand Deng (2004). Factorised SKF also used for musicaltranscription (Cemgil et al., 2006).
There has been previous work on condition monitoring in theICU, though we are unaware of any studies that use a FSKF.
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Inference
For this application, we are interested in filtering, inferringp(st , xt |y1:t).
Exact inference is intractable.
Using two inference methods:
Gaussian Sum (Alspach and Sorenson, 1972), analyticalapproximationRao-Blackwellised particle filtering.
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Gaussian Sum approximation
st−1= n
st−1= 1. .
.
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Gaussian Sum approximation
st−1= n
st−1= 1. .
.
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Gaussian Sum approximation
st−1= n
st−1= 1. .
.st = 1
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Parameter estimation
We need to estimate a dynamical model for each continuousstate variable for each setting of the factors
We use AR/ARMA/ARIMA modelling, e.g. an AR(p) process
xi (t) =
p∑j=1
αijxi (t − j) + εt
Fortunately, annotated training data is available
60
80
100
120
140
160
180
Heart rate
Bradycardia0 20 40 60 80 100 120
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
The hidden continuous state in this application isinterpretable, and domain knowledge can be used to helpparameterize the dynamical models for each factor.
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Parameter estimation example
0 100 200 300 400 500
33
34
35
36
37
Core temp.
Incubator air temp.
For example, we know that the falling temperaturemeasurements caused by a probe disconnection will follow anexponential decay
Therefore we can model these dynamics as an AR(1) process,and set parameters by solving the Yule-Walker equations.
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Learning stable physiological dynamics
Each observation channel has different dynamics when thebaby is ‘stable’ (self regulating) and no artifactual factors areactive
By analysing examples of stable data, dynamical models canbe found for each channel with the Box-Jenkins approach andEM.
For example, a hidden ARIMA(2,1,0) model is a good fit tobaseline heart rate data.
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Known factor classification demo
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Inference results
Inference of bradycardia and incubator open factors. Notethat heart rate variation while incubator is open is attributedto handling of the baby (BR factor suppressed)
BR
Time (s)
IO0 500 1000 1500
50
100
150
200
HR
(bp
m)
65
70
75
80
85
Hum
idity
(%
)
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Inference results
Can examine variance variance of estimates of true physiologyx̂t , e.g. for blood sample (left) and temperature probedisconnection (right):
Time (s)
BS0 50 100 150 200 250
Sys
. BP
(m
mH
g)
35
40
45
50
55
Dia
. BP
(m
mH
g)
20
30
40
50
Time (s)
TD0 500 1000
Cor
e te
mp.
(°C
)
35
35.5
36
36.5
37
37.5
38
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Quantitative Evaluation
3-fold cross validation on 360 hours of monitoring data from15 babies.
FHMM has the same factor structure as the FSKF, with nohidden continuous state.
Inference type Incu. open Core temp. Blood sample Bradycardiaauc 0.87 0.77 0.96 0.88
GSeer 0.17 0.34 0.14 0.25auc 0.77 0.74 0.86 0.77
RBPFeer 0.23 0.32 0.15 0.28auc 0.78 0.74 0.82 0.66
FHMMeer 0.25 0.32 0.20 0.37
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
0 0.5 10
0.2
0.4
0.6
0.8
1(a) Incubator open
1 − specificity
sens
itivi
ty
GSRBPFFHMM
0 0.5 10
0.2
0.4
0.6
0.8
1(b) Core temp. disconnect
1 − specificity
sens
itivi
ty
0 0.5 10
0.2
0.4
0.6
0.8
1(c) Blood sample
1 − specificity
sens
itivi
ty
0 0.5 10
0.2
0.4
0.6
0.8
1(d) Bradycardia
1 − specificity
sens
itivi
ty
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Comparison with FHMM model
FSKF can handle drift in baseline levels:
BR FSKF
Time (s)
BR FHMM0 200 400 600 800 1000 1200 1400 1600
100
110
120
130
140
150
160
Hea
rt r
ate
(bpm
)
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Novel dynamics
There are many other factors influencing the data: drugs,sepsis, neurological problems...
50
100
150
200
Heart rate
40
50
60
70
Dia. BP
0 200 400 600 800 1000 12000
50
100
SpO2
?
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Known Unknowns
Add a factor to represent abnormal dynamics
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Known Unknowns
Add a factor to represent abnormal dynamics
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
X-factor for static 1-D data
For static data, we can use a model M∗ representing‘abnormal’ data points.
y
p(y|
s)
The high-variance model wins when the data is not wellexplained by the original model
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
X-factor with known factors
The X-factor can be applied to the static data in conjunctionwith known factors (green):
y
p(y|
s)
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
X-factor for dynamic data
xt ∼ N (Axt−1,Q)
yt ∼ N (Cxt ,R)
Can construct an ‘abnormal’ dynamic regime analogously:
Normal dynamics: {A,Q,C,R}
X-factor dynamics: {A,ξQ,C,R}, ξ > 1.
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Spectral view of the X-factor
f
S y(f)
0 1/2
Plot shows the spectrum of a hidden AR(5) process, andaccompanying X-factor
More power at every frequency
Dynamical analogue of the static 1-D case
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
X-factor demo
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
More inference results
Classification of periods of clinically significant cardiovasculardisturbance:
X
Time (s)
True X0 500 1000 1500 2000 2500
30
40
50
60
Sys
. BP
(m
mH
g)
70
80
90
100
SpO
2 (%
)
X
Time (s)
True X0 2000 4000 6000
35
40
45
50
55
Sys
BP
(m
mH
g)85
90
95
100
SpO
2 (%
)
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
EM for novel regimes
10
20
30
40
50
60
70
Sys
. BP
(m
mH
g) (
solid
)D
ia. B
P (
mm
Hg)
(da
shed
)
Blood sampleX
ξ=5 (initial setting)
Time (s)
Blood sample0 500 1000 1500 2000 2500 3000
X
ξ=1.2439 (EM estimate after 4 iterations)
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Conclusions
FSKF successfully applied to complex physiological monitoringdata
FSKF can be applied more generally to condition monitoringproblems
Interpretable structure
Knowledge engineering used to parameterize dynamic models
Allows monitoring of known and novel dynamics (supervisedand unsupervised learning)
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
References
Factorial Switching Kalman Filters for Condition Monitoring inNeonatal Intensive Care.Christopher K. I. Williams, John Quinn, Neil McIntosh. In Advancesin Neural Information Processing Systems 18, MIT Press (2006)
Known Unknowns: Novelty Detection in Condition Monitoring.John A. Quinn, Christopher K. I. Williams. Proc 3rd IberianConference on Pattern Recognition and Image Analysis (IbPRIA2007)
Both available fromhttp://www.dai.ed.ac.uk/homes/ckiw/neonatal/
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring
Premature Baby Monitoring Factors Factorial Switching Kalman Filter Inference Parameter estimation Results Unknown conditions
Models for stable physiology
A fitted model can be verified by comparing real physiologicaldata against a sample from that model, e.g. for heart rate:
0 100 200 300 400 500−20
−10
0
10Physiological
0 100 200 300 400 500−10
0
10
20ARIMA(2,1,0), e=0.001182
Chris Williams Factorial Switching Linear Dynamical Systems for Physiological Condition Monitoring