Regenerative Likelihood Ratio control schemes

Transcription

1 Regenerative Likelihood Ratio control schemes Emmanuel Yashchin IBM Research, Yorktown Heights, NY XIth Intl. Workshop on Intelligent Statistical Quality Control 2013, Sydney, Australia

2 Outline Motivation Basic monitoring setup Likelihood ratio (LR) approach Regenerative LR (RLR) approach Applications

3 Processes Subject to Abrupt Changes Selected Areas Data integrity monitoring Billing system auditing Network performance monitoring Fraud detection Yield management Related applications Real time monitoring & control Forecasting Periodic updates & report generation Retrospective data analysis

4 Main data analytics problems Detection Estimation of the current level of monitored processes Segmentation Filtering Diagnostics Settings: Fixed sample Sequential Traditional approaches: Mostly based on conventional Statistical Process Control (SPC) These do not take full advantage of information in the data and are difficult to use in environments involving massive data streams Newer approaches: based on theory of Sequential Analysis, Change-point theory

5 Sequence of observations: θ - subject to abrupt changes Basic Monitoring Setup X 1 X 2 f θ,,... ( i) Example: θ = ( ) E X t Observations t

6 Detection of Changes Control Scheme: is a set of criteria for testing, at any moment of time, whether the process is within acceptable variation { } X t Most popular scheme: Shewhart (typically used within the framework of Western Electric approach to monitoring) Example: Shewhart scheme with warning limits t

7 Search for better schemes: Criteria formulated in terms of Average Run Length (ARL) Protection against false alarms Sensitivity requirements ARL Curve: Control scheme A is more powerful than B: it provides better sensitivity for the same rate of false alarms

8 We need Good schemes for situations involving: Vector parameters Multivariate data Nuisance parameters Highly intensive data streams Serially correlated data Complex change patterns

9 Likelihood Ratio (LR) Approach: f ( i) 0 - good process f ( i) - bad process 1 LR Scheme: Signal at time T if for some m 1 T i= T m+ 1 ln[ f ( X ) / f ( X )] > A 1 t 0 t Alternative formulation (Page s scheme): Set f ( X ) s s s t 1 t 0 = 0, t = max t 1 + ln, 0, = 1, 2,... f0( X t ) and signal when st > A

10 Design principles: let us make a deal Performance of the monitoring system will depend on level of input that the user (real or virtual) is willing to provide Complete or partially complete input: Target, σ, Acceptable/Unacceptable levels, type of control, False Alarm (FA) rate. Minimal input: type of control (A), FA rate and spec deviation (B)

11 Ω 0 Generalized Likelihood Ratio (GLR) Approach: - acceptable region for θ Ω 1 - unacceptable region Current observation: X T Log-likelihood: Lm ( θ ) = ln fθ ( XT m+ 1,..., X T 1, XT ) Max log-likelihoods: Lm 0 = max θ Ω L ( ), 0 m θ Lm 1 = max θ Ω L ( ) 1 m θ Score: D = L L m m1 m0 GLR Scheme: Select signal level h > 0. Signal at time T if D > h for some m 1 m Main drawback: Very computing-intensive; no mechanism for discarding old history

12 Window-limited Schemes (WLS) Select window size: M WLS approach: Select signal level h > 0. Signal at time T if D > h for some 1 m M m Selection of best values for (M, h): Addressed in the literature

13 Regenerative Likelihood Ratio (RLR) Approach Regeneration point (RP): All info prior to this point is discarded Suppose that at time T: Last RP was recorded M T observations ago RLR approach: 1. Signal at time T if D > h for some m m m M 0 T 2. If D 0 for every m0 m MT, declare T m a new RP. m 3. Otherwise, denote by mt the maximal value of m in [ m, 0 MT ] for which D 0 and declare T m a new RP. m Note: If D > 0 for m = M, we keep the current RP. m T T

14 Establishing a new Regeneration Point (RP) Previous RP Current point in time New RP

15 RLR Control Chart 1. Define the value of scheme at time T by s T s = 0 max T m m M T D m s T 2. Plot on a control chart 3. Trigger a signal at time T if s 0 T m If declare a new RP. T m T T m T Otherwise, find and declare a new RP.

16 Simplified RLR Schemes 1. RLR(k): Regular RLR, but explores only value m = M T and values of m on a k-spaced grid Special case: k = (sequence of G SPRT) (a) Define the value of scheme s T at time T by s = D (b) Signal if st > h (c) If s 0, declare T a new RP. T 2. Same as RLR(k), but: Also explore m = m 0. When k = we obtain a generalized Cusum-Shewhart scheme T M T

17 Monitoring of Multivariate Normal Mean X,X,... N ( µσ, ); Assumption : Σ is known. 1 2 p ( p p)

18 Aim: Detect change from λ λ 0 to λ λ1, where λ = µ µ 0 Σ µ 2 µ 1

19 Optimization for a given window m Log-likelihood at time T for window of size m: m Lm = 0.5m ln[(2 π ) + p ( µ ) Σ ] 0.5 X µ Denote: = Value of µ that maximizes in region µ µ = λ i= 1 T i 1 Σ 1 µ ( m) = X( m) = m X 1 T i 1; λ ( m) ( m) 0 i= + = µ µ m µ ( m; λ) L ( µ ) 0 Σ m 2 Σ This optimization results in [ ( m) 0] λ µ ( m; λ) = µ 0 + µ µ λ ( m)

20 µ 2 µ 1

21 RLR Scheme for µ Based on λ = µ µ = ( µ µ )' Σ ( µ µ ) 1 0 Σ 0 0 Acceptable region: λ λ 0 Unacceptable region: λ λ1 (1)Trigger a signal at time T if for some 1 m MT k λ λ ( m) λ and m( λ 1 1 λ )[ 0 λ ( m) k ] h λ > λ or 2 λ ( m) λ and 0.5m[ λ ( m) λ 1 0] > hλ, where h λ = signal level and k λ = reference value = (λ 0 + λ 1 )/2 k λ (2) If ( m) for all m M then declare T a new RP. λ 1 T Otherwise, find m and declare T m a new RP. T T

22 ARL Comparisons Acceptable region: Dimension: p=2 λ λ 0 Unacceptable region: MC1: Proposed by Pignatiello & Runger (1990) T 2 : Shewhart s scheme applied to X µ λ λ 1 Assume: no data available prior to change-point (worst case analysis is more favorable to RLR) i 2 Σ

23 Conclusions 1. RLR schemes offer a number of advantages: (i) Statistical power (ii) Adjustable scheme complexity (iii) Single-parameter design procedure (iv) Computational efficiency 2. Open issues: (i) Efficient ARL and Run Length approximations (ii) Best window policy for LR exploration (iii) Strengths/Weaknesses in performance or RLR (relative to LR, RLR(k)) for various types of detection problems