Statistical Change Detection for MultiDimensional Data

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Statistical Change Detection for
Multi-Dimensional Data
Presented by Xiuyao Song Authors Xiuyao Song,
Mingxi Wu, Chris Jermaine, Sanjay Ranka
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Motivation example antibiotic
resistance pattern

• Culturing the specimen of E. coli bacteria and
  then testing its resistant rate to multiple
  antibiotic drugs.
• Typical data could be like (R resistant S
  susceptible, U undetermined)
• drug1 drug2 drug3 drug4
• R S U
  R
• We have a baseline data set and a recently
  observed data set.
• Question Does E.Coli show different resistance
  pattern recently?
• If a change is detected, it might be caused by
  the presence of new E. Coli strains. We will
  raise an alarm for further investigations.

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Problem definition
Multi-dimensional space
data set S
data set S
baseline data
recently observed data
Question FS FS ?
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Related work

• For uni-dimensional data, many existed tests,
  such as K-S test, chi-square test
• Only two tests to detect a generic distributional
  change in multi-dimensional space.
• Kdq-tree by Dasu et al relies discretization
  scheme, suffer from curse of dimensionality.
• Cross-match by Rosenbaum computationally
  expensive due to maximum matching algorithm

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hypothesis test framework
data set S
data set S
null hypothesis H0 FS FS
Null distribution ?
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Density test high-level overview
data set S
data set S
Step 1 Gaussian kernel density estimate of S1.
Step 3 derive the null distribution
Kernel Density Estimate
null distribution
Step 4 calculate the critical value and make a
decision.
KS1
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Step 1 Kernel Density Estimate (KDE)
--bandwidth selection

• Plug-in bandwidth asymptotically efficient, but
  not accurate.
• Data-driven bandwidth converge better to the
  true distribution.

bandwidth
correctness of density test can always be
guaranteed. power of test is increased when
estimate is accurate.
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Choose bandwidth by MLE/EM (maximum likelihood
estimation / Expectation Maximization)
kernel
adding constraint
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Effectiveness of EM bandwidth
Samples from the real distribution
Samples from KDE with scotts plug-in bandwidth
Samples from KDE with our EM bandwidth
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Step 2 define and calculate
data set S
data set S
Kernel Density Estimate
KS1
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Step 3 derive the null distribution
? normal By Central Limit Theorem
?1normal
?2 normal
Need to be estimated
Tk be r.v. with distribution FS
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Estimating
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Step 4 calculate critical value and
make a decision
estimated null distribution ?
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Density test all 4 steps
data set S
data set S
Step 1 Gaussian kernel density estimate of S1.
Step 3 derive the null distribution
Kernel Density Estimate
null distribution
Step 4 calculate the critical value and make a
decision.
KS1
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Run density test in 2 directions
the test is not symmetric, 2-way test may
increase the power. E.g.
FS
FS
S
S
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False positive
Data consists of low-D group and high-D
group. User-given p value is 8
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false negative on low-D group
false neg. ()
type of changes
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false negative on high-D group
false neg. ()
type of changes
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Scalability
density test has amortizable time cost (one-time
cost 84)
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Conclusion

• Our density test
• can correctly bound the type I error
• is most powerful on all 5 changes
• can easily scale to large data sets and has an
  amortizable time cost

Poster session ( 15)
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Thanks for your attention!