A Wavelet-based Anomaly Detector for Disease Outbreaks

Thomas LotzeGalit Shmueli

University of Maryland College Park

Sean MurphyHoward Burkom

Johns Hopkins University Applied Physics Lab

Outline

● Motivation● Wavelet method● Difficulties● Preconditioning● Results

Related Work

● Bakshi Wavelets in

Chemical SPC● Zhang

Baseline wavelets Normalize syndromic

baseline● Goldenberg, et. al.

Wavelets in syndromic surveillance

Motivation● Detecting disease

outbreaks Bioterrorist attacks Virulent diseases Early detection

saves lives!● Syndromic Data will

show outbreaks● Anomaly detection

to find outbreaks faster

Wavelets

● Models a series as a sum of “wavelets”

● Wavelets are at different scales

● Wavelets are local (change over time)

Goldenberg et. al., 2002

Xt

SPC( )

AR(aLX)

AR(dLX)

AR(dmX)

AR(d1X)

WT

1ˆtx

Decompose the series

with desired wavelet

Use an AR at each of the detail levels and coarsest approximation

level to forecast the next point

Reconstruct series and obtain next day

forecast

Compare the forecast with the actual value.

Use a control chart to monitor the discrepancy

11ˆ tt xxW

Difficulties

● Holidays● Non-stationary

Day of week Seasonal

● Noisy● Outbreaks are not

labeled● Outbreak pattern not

known in advance

Preconditioning

● Differs from Goldenberg, et. al

● Replace holidays One week previous

● Day-of-week Ratio to moving

average

Evaluation: Simulated Outbreaks

● Real data from 5 cities, Resp and Gi● Simulated outbreak patterns inserted● Specific pattern of additional syndromes over

several days● Size is normalized by standard deviation of

recent days● Inserted at different starting points within the

sample data● Average detection rates vs. false alarm rates

can be determined to create ROC curves

Results

● Comparable to Holt-Winters

● Not amazing

Results

● Preconditioning is important

● Detection is much better when preconditioned

Results

● Easier to detect on some days than others

● Days with low counts

● Daily preconditioning not sufficient

Summary

● Wavelets are a fairly good detection method

● Preconditioning is very important

● Day-of-week not fully accounted for

Questions?

● More details on wavelets method?

● Difficulties?● Other outbreak

signals?● Future work?● Will Microsoft

survive Bill Gates' stepping down?

Bonus: More on Wavelets

● Level 1:– Run the data through a low-

pass filter. This gives the approximation coefficients

– Run the data through a high-pass filter. This gives the detail coefficients

– Down-sample– Reconstruct approximation

and detail by up-sampling and running “reconstruction” filters.

● Level 2 and on:– Repeat the steps by applying

them to the previous level approximation coefficients.

Bonus: Wavelets on Cough Medication Sales

100 200 300 400 500

10

15

20

a1

81012141618

a2

81012141618

a3

8

10

12

14

16

a4

8

10

12

14

16

a5

10

15

20

s

Signal and Approximation(s)

10

15

20

s

cfs

Coefs, Signal and Detail(s)

54321

-2

0

2

d5

-1

0

1

d4

-1

0

1

d3

-1

0

1

d2

100 200 300 400 500-2

0

2

d1

Haar Wavelet:

h = [1/sqrt(2), 1/sqrt(2)]

g = [1/sqrt(2), -1/sqrt(2)]

Downsample

Upsample

h* = [1/sqrt(2), 1/sqrt(2)]

g* = [-1/sqrt(2), 1/sqrt(2)]

In general:

s = a5 + d1 + d2 +… + d5

Bonus: Wavelet Prediction

● Additional details:● 5 level decomposition

Can be performed with more or fewer● SWT: Fill in “holes”

Perform a decomposition for every possible position

Series are no longer independent● Edge issue

Prediction is not possible at all time steps Solution: construct wavelets “backwards” from

most recent observations

Bonus: Ratio-to-Moving-Average

● Way of normalizing day-of-week effects● 1: Determine moving averages

a(i)=(x(i-3) + x(i-2) + ... + x(i+3)) /7● 2: Determine ratio (“raw seasonal”) for each

day r(i)=x(i)/a(i)

● 3: Determine avg. ratio for each day r(Mon)=sum(r(i): i is Mon) / count(i is Mon)

● 4: Normalize ratios to sum to 1 r'(Mon)=r(Mon) / (r(Mon) + ... + r(Sun))

● 5: Divide each day by its ratio x'(i)=x(i)/r(Mon)

Bonus: Possible Extensions

● Multivariate wavelets● Each day-of-week as a separate series● Different wavelet shapes● Different wavelet scale basis● Different preconditioning● Different sizes, lengths of outbreaks● Don't normalize outbreak by standard

deviation of recent days Show when outbreaks are harder to detect

● Estimate confidence based on experience● Boosting

Bonus: Wavelet Prediction

● Decompose into timescales

● Use AR or EWMA to predict for each timescale

● Reconstruct prediction from predicted timescales

● Monitor deviations from prediction

Bonus: Alternative Preconditioning

● Regression using day-of-week predictors● 7-day differencing● Holt-Winters as preconditioner● Seasonal preconditioning

Bonus: Other Outbreak Signals

● Normalized by total size● Lognormal, exponential, step● Spike is much easier than the others