A Martingale Framework for Concept Change Detection in Time-Varying Data

Stream

Ho Shen-Shyangsho@gmu.edu

Department of Computer ScienceGeorge Mason University

Preview:

● Problem: In a data streaming setting, data points are observed one by one. The concepts to be learned from the data stream may change infinitely often. ● How do we detect the changes efficiently?● Other Topics: Concept Drift, Anamoly detection, ... ...● Testing Exchangeability Online (Vovk et.al., ICML 2003)

Outline:

●Background: Strangeness, Martingale, Exchangeability, ●Martingale Framework - Two Tests●Theoretical Justifications●Additional Theoretical Results●Experimental Results

Strangeness Measure (Saunders et. al., IJCAI 1999)

● Support Vector Machine: Value of Lagrange Multipler or Distance from the hyperplane (we use SVM/Lagrange Multiplier – incremental SVM (Cauwenberghs and Poggio, NIPS 2000))

● K-nearest-neighbor rule: A/B whereA – Sum of the distance of a point from the k nearest points with the same labelB – Sum of the distance of a point from the k nearest points with different label

α: scoring how a data point is different from the rest.

Testing Exchangeability: Definitions

Let { Zi : 1 ≤ i < ∞ } be a sequence of r.v.

A finite sequence of r.v. Z1,..., Zn is exchangeable

if the joint distribution p(Z1,..., Zn) is invariant

under any permutation of the indices of the r.v.

A martingale is a sequence of r.v. { Mi : 0 ≤ i < ∞ }

such that Mn is a measurable function of Z1,..., Zn for

all n = 0, 1, ... (M0 is a constant, say 1) and the

conditional expectation of Mn+1 given M1,..., Mn is equal

to Mn, i.e. E(Mn+1 | M1,..., Mn ) = Mn

Testing Exchangeability (Vovk et. al., ICML 2003)

pn = V(Z U {zn}, θn)

=

where ε in [0,1] (say 0.92) and M0= 1

Performing Kolmogorov-Smirnov Test on the p-value distribution as data is observed one by one.

Skewed p-value distribution: small p-values inflate the martingale values

Martingale Framework: Test for Change Detection

Consider the simple null hypothesis H0: “no concept change in the data stream”

against the alternative hypothesisH1: “concept change occurs in the data stream”

Martingale Framework: Test for Change Detection

Martingale Test 1 (MT1)0 < Mn

(ε)< λ

where λ is a positive number. One rejects the null hypothesis when Mn

(ε) ≥ λ.

Martingale Test 2 (MT2)0 < | Mn

(ε) - Mn-1(ε) |< t

where t is a positive number. One rejects the null hypothesis when | Mn

(ε) - Mn-1(ε) | ≥ t.

Justification for Martingale Test 1: Doob's Maximal Inequality

Assuming that { Mi : 0 ≤ i < ∞ } is a nonnegative martingale,

the Doob's Maximal Inequality states that for any λ > 0 and 0 ≤ n < ∞,

Hence, if E(Mn) = E(M0) = 1, then

Justification for Martingale Test 2 Hoeffding-Azuma Inequality

Let c1, ..., cm be positive constants and let Y1, ..., Ym be

a martingale difference sequence with |Yk| ≤ ck for each k.

Then for any t ≥ 0,

At each n, the martingale difference is maximum and bounded when pn is 1/n for the deterministic martingale (θn=1 for all n)

Justification for Martingale Test 2:

When m = 1, the Hoeffding-Azuma Inequality becomes

Assuming that Mn-1(ε) = M0

(ε) = 1,

Comparison:

Some Theoretical Results for Martingale Test 1 (Ho & Wechsler,

UAI 2005)● Martingale Test based on the Doob's Inequality is an approximaton of the sequential probability ratio test.

●

Where α is the desirable size (type I error) and β is the probability of the type II error

● The mean delay time from the true change point is:

where

Experiments

Precision =

Recall =

Number of Correct DetectionsNumber of Detections

Precision: Probability that a detection is actually correctRecall: Probability that the system recognizes a true changeDelay time (for a detected change): the number of time unitsfrom a true change point to the detected change point, if any

Number of Correct DetectionsNumber of True Changes

Experimental Results: Synthetic Data Stream with noise (10-D Rotating Hyperplane) – Precision and Recall

Experimental Results: Synthetic Data Stream – Mean and Median Delay

Time

Experimental Results: Numerical (WaveNorm & TwoNorm)

and Categorical data streams (Nursery)

Experimental Results: Multi-class data streams (Modified USPS data-

set)

Dataset: 10 classes, 256 dimensions, 7291 data points

Data stream: 3 classes.

Experimental Results: Multi-class data streams (Modified USPS data-

set)

Conclusions:

● Our martingale approach is an efficient, one-pass incremental algorithm that

●Does not require a sliding window on the data stream●Does not require monitoring the performance of a base classifier as data is streaming●Works well for high dimensional, multiclass data stream●Theoretically justified.

Conclusions/Future (Current) Work:

● Previous works: Kifer et. al. (VLDB 2004), Fan et. al.(SDM 2004), Wald (1947), Page (1957) ......● Extension to Unlabeled and One-class data streams● Application: Keyframe Extraction, Anomaly Detection, Adaptive Classifier (Ho and Wechsler, IJCAI 2005)● Comparison using different classifiers (i.e. Different strangeness measure, also weak classifiers)● Comparison with other change detection algorithms.● http://cs.gmu.edu/~sho/research/change_detection.html Acknowledgement: Vladimir Vovk, Harry Wechsler.