Data-driven window width Data-driven window width adaption adaption for robust for robust online moving window regression online moving window regression Matthias Matthias Borowski Borowski Fakultät Statistik, TU Dortmund COMPSTAT 2010, Paris 1
Situation Online-monitoring time series: systolic blood pressure Data-driven window width 180 adaption for robust online 160 moving window regression 140 Matthias x t Borowski 120 100 80 1 50 100 150 200 250 300 350 400 450 500 t General assumption: = + + X t µ t ε t η t data = signal + noise + outliers 2
Online filtering by Repeated Median (RM) regression Data-driven 12 window data ● width local linear regression 10 adaption for robust µ ^ t online 8 moving window 6 regression 4 Matthias Borowski ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● −2 0 20 40 60 80 100 t RM regression (Siegel, 1982): � x t − n + i − x t − n + j � ˆ β t = Slope med med i − j i ∈{ 1 ,..., n } j � = i , j ∈{ 1 ,..., n } � � x t − n + i − ˆ Level µ t = ˆ med β t · ( i − n ) i ∈{ 1 ,..., n } 3
Online filtering by Repeated Median (RM) regression Data-driven 12 window data ● width local linear regression 10 adaption for robust µ ^ t online 8 moving window 6 regression 4 Matthias Borowski ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● −2 0 20 40 60 80 100 t RM regression (Siegel, 1982): � x t − n + i − x t − n + j � ˆ β t = Slope med med i − j i ∈{ 1 ,..., n } j � = i , j ∈{ 1 ,..., n } � � x t − n + i − ˆ Level µ t = ˆ med β t · ( i − n ) i ∈{ 1 ,..., n } 4
Online filtering by Repeated Median (RM) regression Data-driven 12 window data ● width local linear regression 10 adaption for robust µ ^ t online 8 moving window 6 regression 4 Matthias Borowski ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● −2 0 20 40 60 80 100 t RM regression (Siegel, 1982): � x t − n + i − x t − n + j � ˆ β t = Slope med med i − j i ∈{ 1 ,..., n } j � = i , j ∈{ 1 ,..., n } � � x t − n + i − ˆ Level µ t = ˆ med β t · ( i − n ) i ∈{ 1 ,..., n } 5
Online filtering by Repeated Median (RM) regression Data-driven 12 window data ● width local linear regression 10 adaption for robust µ ^ t online 8 moving window 6 regression 4 Matthias Borowski ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● −2 0 20 40 60 80 100 t RM regression (Siegel, 1982): � x t − n + i − x t − n + j � ˆ β t = Slope med med i − j i ∈{ 1 ,..., n } j � = i , j ∈{ 1 ,..., n } � � x t − n + i − ˆ Level µ t = ˆ med β t · ( i − n ) i ∈{ 1 ,..., n } 6
Online filtering by Repeated Median (RM) regression Data-driven 12 window data ● width local linear regression 10 adaption for robust µ ^ t online 8 moving window 6 regression 4 Matthias Borowski ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● −2 0 20 40 60 80 100 t RM regression (Siegel, 1982): � x t − n + i − x t − n + j � ˆ β t = Slope med med i − j i ∈{ 1 ,..., n } j � = i , j ∈{ 1 ,..., n } � � x t − n + i − ˆ Level µ t = ˆ med β t · ( i − n ) i ∈{ 1 ,..., n } 7
Online filtering by Repeated Median (RM) regression Data-driven 12 window data ● ● ● width ● ● local linear regression 10 ● ● adaption ● ● ● ● ● for robust µ ^ t ● ● ● ● ● ● ● online 8 ● ● ● ● ● ● ● ● moving ● ● ● ● ● ● ● window ● ● ● ● 6 ● ● ● ● regression ● ● ● ● ● ● ● ● ● ● ● ● ● 4 ● ● Matthias ● ● ● ● ● ● ● Borowski ● ● ● ● ● ● ● 2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● −2 0 20 40 60 80 100 t RM regression (Siegel, 1982): � x t − n + i − x t − n + j � ˆ β t = Slope med med i − j i ∈{ 1 ,..., n } j � = i , j ∈{ 1 ,..., n } � � x t − n + i − ˆ Level µ t = ˆ med β t · ( i − n ) i ∈{ 1 ,..., n } 8
Repeated Median (RM) regression Data-driven window Trade-off problem for window width (ww): width adaption Large ww ⇒ smooth signal extraction for robust online moving Small ww ⇒ exact signal extraction window regression Matthias Borowski Approach: data-driven ww selection for RM At each time point t : Test: Is window width n adequate for RM fit? Yes: estimate signal No: decrease n 9
Methods Data-driven window width adaption for robust online moving window regression ADORE – ADaptive Online REpeated Median Matthias (Schettlinger et al., 2010) Borowski SCARM – Slope Comparing Adaptive Repeated Median (Borowski, 2010) 10
ADORE test Data-driven H 0 : ww n adequate, if window width adaption # neg. residuals ≈ # pos. residuals for robust online moving in right sub-window window regression Matthias ● 4 Borowski ● ● 20 pos. residuals ● ● 3 ● ● ● ● ● ● ● ● ● 2 ● ● ● ● ● ● x t RM fit 1 ● ● ● ● ● ● 0 ● 10 neg. residuals −1 ● ● ● −2 1 25 50 75 100 t 11
SCARM test Data-driven � β right � window β left − ˆ � ˆ � � width � adaption H 0 : ww n adequate, iff is small for robust � β left − ˆ Var (ˆ online β right ) moving window regression 4 Matthias Borowski 3 2 x t 1 RM fit left sub−window RM fit right sub−window 0 −1 −2 1 25 50 75 100 t 12
SCARM test Data-driven � β right � window β left − ˆ � ˆ � � width � adaption H 0 : ww n adequate, iff is small for robust � β left − ˆ Var (ˆ online β right ) moving window regression β left − ˆ Matthias Sophisticated estimation of Var (ˆ β right ) Borowski to increase power: Use that RM slope is unbiased (symm. noise) and regression equivariant Estimate noise variability by regression-free scale estimator (Gelper et al., 2009) 13
Power comparison ADORE vs. SCARM Data-driven Standard normal noise window width Significance level 0 . 01 adaption for robust Several sizes of level shifts and trend changes online moving window Level shifts Trend changes regression 1.0 1.0 Matthias Borowski 0.8 0.8 0.6 0.6 Power Power SCARM ADORE 0.4 0.4 0.2 0.2 SCARM ADORE 0.0 0.0 1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 Shift height Trend slope 14
Outlook Data-driven window width adaption for robust online ww adaption based on moving window Shift detection (Fried, Gather, 2007) regression Trend detection (Fried, Imhoff, 2004) Matthias Borowski Further comparisons: Other types of noise Effect of outliers SCARM in R package robfilter (Fried, Schettlinger, Borowski, 2010) 15
References Data-driven window width Borowski, M. (2010): Window width adaption for robust moving window adaption regression in online-monitoring time series. Discussion Paper, to appear . for robust online Fried, R., Gather, U. (2007): On rank tests for shift detection in time series. moving CSDA 52 , 221-233. window regression Fried, R., Imhoff, M. (2004): On the online detection of monotonic trends in time series. Biometrical Journal 46 , 90-102. Matthias Borowski Fried, R., Schettlinger, K., Borowski, M. (2010): robfilter: Robust Time Series Filters. R package version 2.6.1 Siegel, A. F. (1982): Robust regression using repeated medians. Biometrika 69 (1), 242-244. Gelper, S., Schettlinger, K., Croux, C., Gather, U. (2009): Robust online scale estimation in time series: a regression-free approach. J. Stat. Plann. Inf. 139 , 335-349. Schettlinger, K., Fried, R., Gather, U. (2010): Real time signal processing by adaptive repeated median filters. Int. J. Adapt. Control Signal Process. 24 , 346-362. 16
Estimation of Var ( D ) � β left − ˆ β right � ˆ Var ( D ) = Var � β left � � β right � ˆ ˆ ∗ = Var + Var *under H 0 for i.i.d. symmetric noise with zero median � � ˆ depends on ww n and on noise variance σ 2 : Var β � � ˆ =: V ( n , σ ) = V ( n , 1 ) · σ 2 Var β approximations ˆ V ( n , 1 ) =: v n for n = 5 , . . . , 300 by simulation estimate noise variance σ 2 on residuals? ⇒ small power! σ 2 by regression-free Q scale estimator better: obtain ˆ (Gelper et al., 2009)
Estimation of � β left � � β right � ˆ ˆ Var ( D ) = Var + Var ) 2 + v r · Q ( x right � ) 2 Var ( D ) = v ℓ · Q ( x left t t � � � � d � � → if | d ∗ | = � � � ) 2 + v r · Q ( x right � v ℓ · Q ( x left � ) 2 � � t t is ’too large’: reject H 0 Critical values for D ∗ ? D ∗ roughly N ( 0 , 1 ) distributed for several noise distributions simulation: quantiles of the emp. distribution of d ∗ for N ( 0 , 1 ) noise
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