Change-point methods for anomaly detection in fibrous media joint - PowerPoint PPT Presentation

Change-point methods for anomaly detection in fibrous media joint work with E. Spodarev, C. Re- denbach, D. Dresvyanskiy, T. Karaseva Vitalii Makogin | 10.10.2019 | Institute of Stochastics, Ulm University and S. Mitrofanov

Seite 2 Problem setting | Change-point methods for anomaly detection | 10.10.2019 Fibre materials Random sequential adsorption (RSA) Visualization of layered RSA image with 200 × 200 × 300 voxels (left) and homogeneous RSA image with 500 × 500 × 500 voxels (right), radius 4

Seite 3 Problem setting | Change-point methods for anomaly detection | 10.10.2019 Problem setting - A fibre γ is a simple curve { γ ( t ) , t ∈ [ 0 , 1 ] } in R 3 of finite length. - The collection of fibres forms a fibre system φ . - The length measure φ ( B ) = � γ ∈ φ h ( γ ∩ B ) , where h is the length of fibre in window B ⊂ R 3 . - A fibre process Φ is a random element with values in the set D of all fibre systems φ with σ -algebra D generated by sets of the form { φ ∈ D : φ ( B ) < x } .

Seite 4 Problem setting | Change-point methods for anomaly detection | 10.10.2019 Classification - Let w ( x ) be some characteristic of a fibre at point x : fibre local direction, curvature, etc. - A weighted random measure � Ψ( B × L ) = B ✶ { w ( x ) ∈ L } Φ( dx ) . - If the fibre process Φ is stationary, then E Ψ( B × L ) = λ | B | f ( L ) , where λ is called the intensity of Ψ , - A probability measure f on S 2 is called the directional distribution of fibres.

Seite 5 Problem setting | Change-point methods for anomaly detection | 10.10.2019 Testing H 0 : Φ is stationary with intensity λ and directional distribution f vs. H 1 : There exists a compact set A ⊂ W with | A | > 0 and | W \ A | > 0 such that � � 1 1 λ | A | E ✶ { w ( x ) ∈ ·} Φ( dx ) � = λ | W \ A | E ✶ { w ( x ) ∈ ·} Φ( dx ) . A W \ A If H 1 holds true, the region A is called an anomaly region.

Seite 6 Data and samples | Change-point methods for anomaly detection | 10.10.2019 Data - The dilated fibre system Φ ⊕ B r ∩ W in window W ⊂ R 3 is observed as a 3D greyscale image. - Reconstruction of µ CT image by MAVI: Modular Algorithms for volume Images, provided by Fraunhofer ITWM. , , ... , ⇒

Seite 7 Data and samples | Change-point methods for anomaly detection | 10.10.2019 Local directions and clustering criteria - A separation of fibres (and estimation of their directions) requires large com- putational resources for 3D images. - An estimation of local direction is much faster but produces dependent sample. - In each � W l , the “average local directi- on” is computed using SubfieldFibreDi- rection in MAVI. - We group � W l in classification windows W l . - For each W l we assign a classification attribute: entropy, average direction.

Seite 8 Data and samples | Change-point methods for anomaly detection | 10.10.2019 Entropy estimation - Entropy of an absolutely continuous S 2 − valued random � variable with density f is E X = − S 2 log ( f ( x )) f ( x ) σ ( dx ) , where σ is the spherical surface measure on S 2 . - Plug-in estimators required large samples. In simulations for uniform distribution on a sphere N > 50 3 . - Nearest neighbour estimator (KL-estimator, 1987) � N E = d ˆ log ρ i + log ( c ( N − 1 )) + γ. N i = 1 - We propose modification of the nearest neighbour estimator: N � E M = d ˆ ✶ { ρ i > ρ 0 } log ρ i + log ( c ( N M − 1 )) + γ, N M i = 1 where N M = � N i = 1 ✶ { ρ i > ρ 0 } , ρ 0 is a penalty value.

Seite 9 Data and samples | Change-point methods for anomaly detection | 10.10.2019 Entropy estimation: homogeneous RSA (random sequential adsorption) image Histogram of frequencies of the local entropy 2000 × 2000 × 2100 voxels

Seite 10 Data and samples | Change-point methods for anomaly detection | 10.10.2019 Entropy estimation: layered RSA image Histogram of frequencies of the local entropy 2000 × 2000 × 2100 voxels

Seite 11 Change-point detection in random fields | Change-point methods for anomaly detection | 10.10.2019 Random fields with inhomogeneities in mean - Let be { ξ k , k ∈ Z 3 } an integrable, centered, stationary, real-valued random field. - { ξ k , k ∈ Z 3 } is m − dependent, and there exist H , σ > 0 such that E | ξ k | p ≤ p ! 2 H p − 2 σ 2 , p = 2 , 3 , . . . - Let Θ be a finite parametric space. For every θ ∈ Θ we define subspace of anomalies I θ ⊂ Z 3 . - For some θ 0 ∈ Θ we observe s k = ξ k + µ + h ✶ { k ∈ I θ 0 } , k ∈ W . - Let Θ 0 correspond to the significant anomalies, i.e, for γ 0 , γ 1 ∈ ( 0 , 1 ) we let Θ 0 = { θ ∈ Θ : γ 0 | W | ≤ | I θ | ≤ γ 1 | W |} . - Then Θ 1 = Θ \ Θ 0 corresponds to the extremely small or large anomalies, i.e, Θ 1 = { θ ∈ Θ : | I θ | < γ 0 | W | , or | I θ | > γ 1 | W |} .

Seite 12 Change-point detection in random fields | Change-point methods for anomaly detection | 10.10.2019 Testing the change of expectation - The change-point hypotheses for the random field { s k , k ∈ W } with respect to its expectation H 0 : E s k = µ for every k ∈ W (i.e. h = 0) vs. H 1 : There exists θ 0 ∈ Θ 0 such that E s k = µ + h , k ∈ I θ 0 , h � = 0 , and E s k = µ, k ∈ I θ c 0 . - Analogue of CUSUM statistics � � Z ( θ ) = 1 s k − 1 s k | I c | I θ | θ | k ∈ I c k ∈ I θ θ � | I θ ∩ I θ 0 | � � � − | I c = 1 ξ k − 1 θ ∩ I θ 0 | ξ k + h . | I c | I c | I θ | θ | | I θ | θ | k ∈ I c k ∈ I θ θ - Test statistics: T W = max θ ∈ Θ 0 | Z ( θ ) | .

Seite 13 Change-point detection in random fields | Change-point methods for anomaly detection | 10.10.2019 Test statistics - Critical values y α via the probability of the 1st-type error: � �   � � � � � � 1 ξ k − 1  max � �  ≤ α. P H 0 ( T W ≥ y α ) = P ξ k ≥ y α � � | I c | I θ | θ | θ ∈ Θ 0 � � k ∈ I c k ∈ I θ θ - Tail probabilities � � � y 2 | I c θ || I θ | P H 0 ( T W ≥ y ) ≤ − 2 exp 4 m 3 σ 2 | W | θ |≤ σ 2 | W | θ ∈ Θ 0 : | I c yH � � � σ 2 | W | y + 2 exp − 2 Hm 3 | I θ | + θ || I θ | . 4 H 2 m 3 | I c θ | > σ 2 | W | θ ∈ Θ 0 : | I c yH

Seite 14 Change-point detection in random fields | Change-point methods for anomaly detection | 10.10.2019 Tail probabilities - If ξ k ’s are Gaussian, then H = σ. If | ξ k | ≤ M then H = M . σ 2 - Particularly, if y < H ( 1 − γ 0 ) then � � y 2 P H 0 ( T W ≥ y ) ≤ 2 | Θ 0 | exp − 4 m 3 σ 2 | W | γ 0 ( 1 − γ 0 ) , σ 2 and if y > H ( 1 − γ 1 ) then � � y P H 0 ( T W ≥ y ) ≤ 2 | Θ 0 | exp − 4 Hm 3 γ 0 | W | .

Seite 15 Applications | Change-point methods for anomaly detection | 10.10.2019 Simulated data Homogeneous RSA data: Attr. | Θ 0 | Var. Test stat. p − value ˜ x 39395 0.04360 0.0344 1.00 ˜ y 39395 0.03743 0.0130 1.00 ˜ z 39395 0.03749 0.0146 1.00 � 16536 0.08984 0.0942 1.00 E Layered RSA data: Attr. | Θ 0 | Var. Test stat. p − value 4 . 6 × 10 − 30 ˜ 39395 0.10592 0.44036 x 2 . 8 × 10 − 23 ˜ y 39395 0.10948 0.43163 ˜ z 39395 0.06151 0.18764 0.301 � E 16536 0.3583 1.07030 0.00

Seite 16 Applications | Change-point methods for anomaly detection | 10.10.2019 Classification by spatial SAEM algorithm: layered RSA image. Spatial SAEM classification by 2000 × 2000 × 2100 voxels entropy and mean local directions

Seite 17 Applications | Change-point methods for anomaly detection | 10.10.2019 Real glass fibre reinforced polymer - The images are provided by the Institute for Composite Materials (IVW) in Kaiserslautern: 970 × 1469 × 1217 voxels, the estimated radius of 3 voxels. We obtain 64 × 97 × 80 small windows � W i with 15 × 15 × 15 voxels. - Change point testing: σ 2 Attr. H m | Θ 0 | Var. Test stat. p − value ˜ 0.5 0.2 7 33004 0.04589 0.15995 1.00 x 2 . 1 × 10 − 10 ˜ y 0.5 0.2 7 33004 0.06795 0.44733 1 . 3 × 10 − 6 ˜ z 0.5 0.2 7 33004 0.07982 0.43383 3 . 96 × 10 − 8 E 0.7071 0.5 1 12366 0.30126 0.46811

Seite 18 Applications | Change-point methods for anomaly detection | 10.10.2019 Classification by spatial SAEM algorithm: real data. Spatial SAEM classification by 970 × 1469 × 1217 voxels entropy and mean local directions

Seite 19 Acknowledgments | Change-point methods for anomaly detection | 10.10.2019 Preprint: Detecting anomalies in fibre systems using 3-dimensional image data. arXiv:1907.06988. Support: ◮ German Ministry of Education and Research (BMBF): the project “AniS” ◮ DFG Research Training Group GRK 1932; ◮ DFG grant SP 971/10-1; ◮ DAAD scientific exchange program “Strategic Partnerships”. Thank you for your attention!