factorial switching linear dynamical systems for physiological

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

with Prof Neil McIntosh (Edinburgh Royal Infirmary)



Why model this data?

Artifact corruption, leading to false alarms

Our aim is to determine the baby’s state of health despitethese problems



Overview

Factors

Factorial switching Kalman filter

Inference

Parameter estimation

Results

Modelling novel regimes



Probes

1. ECG, 2. arterial line, 3. pulse oximeter 4. core temperature, 5.peripheral temperature, 6. transcutaneous probe.



Factors affecting measurements

The physiological observations are affected by differentfactors.Factors can be artifactual or physiological.An arterial blood sample (artifact):

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Sys

. BP

(m

mH

g)

0 200 400 600 800 10000

20

40

60D

ia. B

P (

mm

Hg)

Time (s)



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)



Common factor examples

Bradycardia (physiological)

0 20 40 60 80 10040

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80

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120

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160

180

HR

(bp

m)

Time (s)0 50 100 150

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HR

(bp

m)

Time (s)



Factorial Switching Kalman Filter

Artifactual state

Physiological state

Observations

Physiological factors

Artifactual factors



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.



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



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 .



Factor interactions



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.



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.



Gaussian Sum approximation

st−1= n

st−1= 1. .

.



Gaussian Sum approximation

st−1= n

st−1= 1. .

.st = 1



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

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Heart rate

Bradycardia0 20 40 60 80 100 120



The hidden continuous state in this application isinterpretable, and domain knowledge can be used to helpparameterize the dynamical models for each factor.



Parameter estimation example

0 100 200 300 400 500

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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.



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.



Known factor classification demo



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

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100

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200

HR

(bp

m)

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85

Hum

idity

(%

)



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)

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50

55

Dia

. BP

(m

mH

g)

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40

50

Time (s)

TD0 500 1000

Cor

e te

mp.

(°C

)

35

35.5

36

36.5

37

37.5

38



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



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



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

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150

160

Hea

rt r

ate

(bpm

)



Novel dynamics

There are many other factors influencing the data: drugs,sepsis, neurological problems...

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Heart rate

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Dia. BP

0 200 400 600 800 1000 12000

50

100

SpO2

?



Known Unknowns

Add a factor to represent abnormal dynamics



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



X-factor with known factors

The X-factor can be applied to the static data in conjunctionwith known factors (green):

y

p(y|

s)



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.



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



X-factor demo



More inference results

Classification of periods of clinically significant cardiovasculardisturbance:

X

Time (s)

True X0 500 1000 1500 2000 2500

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Sys

. BP

(m

mH

g)

70

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100

SpO

2 (%

)

X

Time (s)

True X0 2000 4000 6000

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55

Sys

BP

(m

mH

g)85

90

95

100

SpO

2 (%

)



EM for novel regimes

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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)



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)



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/



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


factorial switching linear dynamical systems for physiological

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