Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion The z -Vertex Trigger for Belle II Sebastian Skambraks Technische Universit¨ at M¨ unchen July 3, 2015 1 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion Outline Introduction Motivation Signal Flow Multi Layer Perceptron - MLP Theory Setup Preprocessing Least Square fit - LS Results MLP and LS results Conclusion Neuro Team F. Abudinen (LMU), Y. Chen (TUM), M. Feindt (KIT), R. Fr¨ uhwirth (HEPHY), M. Heck (KIT), C. Kiesling (MPI), A. Knoll (TUM), S. Neuhaus (TUM), S. Paul (TUM), T. R¨ oder (TUM), J. Schieck (HEPHY), S. Skambraks (TUM) 2 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion Introduction Goals ◮ build a z -vertex track trigger Z distribution ◮ achieve high precision # of events / 5 mm (spatial resolution ∆ z ≈ 2 cm) 1000 ◮ get a fast decision ( < 1 µ s) 800 600 Method 400 ◮ Input: ◮ CDC Track Segment data 200 [IDs & clock cycle (2 ns timing)] 0 -40 -30 -20 -10 0 10 20 30 40 ◮ Algorithms: z (cm) Offline z distribution in the Belle 1. Bayes Classifier / Hough Transformation Experiment a . (pattern recognition) 2. LS - Least Square fit (linear estimation) a) T. Abe et al., Belle II Technical Design Report , 3. MLP - Multi Layer Perceptron KEK-REPORT-2010-1, arXiv:1011.0352v1 [physics.ins-det] (2010). (nonlinear correction) 3 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion Main approach Pattern recognition - Bayes Classifier / Hough Transform sectorize Input in the track parameters ( p T , ϕ, ϑ ) P ( Sector | Hits ) = P ( Hits | Sector ) · P ( Sector ) (1) P ( Hits ) Fitter - Least Squares n = ( X T X ) − 1 X T � � y (2) X and � y contain the hits, � n defines a track parameter sector Neural Network - MLP z ( Hits , p T , ϕ, ϑ ) = NN ( f ( Hits , p T , ϕ, ϑ )) (3) ◮ output float value interpreted as scaled z -position ◮ NN input transformation (function f ) requires preprocessing 4 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion Signal flow in the CDC Trigger 2D Finder 3D Trigger TS axial Axial Pattern 1. 2D Fit TSF CDC Recognition 2. 3D Fit Digitizer Global Merger Decision Logic TS axial , p T , ϕ NeuroTrigger 1. Preprocessing: TS stereo Stereo z HHimprove ( p T , ϕ ) & TSF HHsectorize phase space 2. MLP HHoutput: improved z t < 5 µ s → The neural network trigger will be implemented on a Virtex 7 FPGA 5 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion Signal flow in the CDC Trigger 2D Finder 3D Trigger TS axial Axial Pattern 1. 2D Fit TSF CDC Recognition 2. 3D Fit Digitizer Global Merger Decision Logic TS axial , p T , ϕ NeuroTrigger 1. Preprocessing: TS stereo Stereo z HHimprove ( p T , ϕ ) & TSF HHsectorize phase space 2. MLP HHoutput: improved z t < 5 µ s → The neural network trigger will be implemented on a Virtex 7 FPGA 5 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion MLP - Multi Layer Perceptron input hidden output Properties layer layer layer ◮ supervised machine learning w ji ◮ function approximation w kj t i t i t i t i t i ◮ short deterministic runtime ◮ one neuron: ϕ rel i ϕ rel i ϕ rel i ϕ rel i ϕ rel i z � y = tanh( w i · x i + w 0 ) µ i µ i µ i µ i µ i i =1 Input . . . 3 nodes per SL ( t , ϕ rel , µ ) . with t : drift time, ϕ rel : relative . . wire position, µ : 2D arc length 6 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion MLP - Setup Sectorization ◮ the track parameter space is sectorized in ( p T , ϕ, ϑ ) ◮ for each sector an expert MLP is trained ◮ asymmetry in ϑ , and p T can be taken into account ◮ preprocessing selects the proper MLP Two different sectors in ( p T , ϕ ) (left) and in ϑ (right). 7 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion “Expert” MLP - Capabilities a) b) Figure: z -vertex prediction with an “expert” MLP for a small sector in two p T regions with φ ∈ [0 , 360] ◦ , θ ∈ [56 , 62] ◦ and z ∈ [ − 10 , 10] cm. a) p T ∈ [0 . 3 , 0 . 317] GeV. b) p T ∈ [3 . 5 , 9 . 625] GeV. ! high accuracy on the z -vertex within a small sector 8 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion Preprocessing TS axial , TS stereo TSF NN input Preprocessing z MLP 2D p T , ϕ Finder Figure: Information flow in the NeuroTrigger Tasks 1. match Track Segments to tracks 2. improve ( p T , ϕ ) estimate (2D fit) → Least Squares fit including drift times 3. provide 3D estimate ( ϑ, z ) 4. prepare Neural Net Input (& choose sector) 9 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion Preprocessing - Least Square fit n = ( X T X ) − 1 X T � solve the linear equation � y = X · � n by: � y 2D fit y circle fit; center at � c ; track from origin. p T , ϕ ← c x , c y x i , y i : cartesian coordinates of axial hits in the ( r , ϕ ) plane. c � ( x 2 i + y 2 i ) = 2 c x · x i + 2 c y · y i ϕ x 3D fit z line fit in the ( µ, z ) plane. µ i , z i : stereo hits transformed by z -vertex 2D fit result. z i = cot ( ϑ ) · µ i + z 0 µ 10 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion Results - 2D LS Fit ( ϕ, p T ) 90% RMS ∆ p T [%] ∆ ϕ [ ◦ ] p T 9 nonlinear xt nonlinear xt 4 inner detector 8 inner detector no background no background 7 3 6 5 2 4 3 1 2 1 1 2 3 4 5 1 2 3 4 5 p T [ GeV ] p T [ GeV ] p T ∈ [0 . 3 , 5] GeV ϕ ∈ [0 , 90] ◦ ϑ ∈ [35 , 123] ◦ z ∈ [ − 50 , 50] cm 11 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion Results - 3D LS Fit ( ϑ, z ) 90% RMS ∆ θ [ ◦ ] ∆ z [ cm ] 13 13 12 12 11 11 10 10 9 9 nonlinear xt nonlinear xt 8 inner detector 8 inner detector no background no background 7 7 6 6 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5 p T [ GeV ] 1 2 3 4 5 p T [ GeV ] p T ∈ [0 . 3 , 5] GeV ϕ ∈ [0 , 90] ◦ ϑ ∈ [35 , 123] ◦ z ∈ [ − 50 , 50] cm 12 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion LS Fitter efficiency Cuts lead to efficiency decrease ◮ min 3 axial hits in different layers ◮ max 10 axial hits total ◮ min 2 stereo hits in different layers ◮ max 8 stereo hits total ◮ min 5 hits total ε [%] 100 90 80 nonlinear xt inner detector 70 no background 60 50 1 2 3 4 5 p T [ GeV ] 13 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion z -90% RMS with LS fit and MLP ∆ z [ cm ] 9 MLP, inner detector, nonlinear xt 8 LS Fit, inner detector, nonlinear xt 7 MLP, small ( p T , ϑ ) sectors 6 no inner detector, nonlinear xt 5 4 3 2 sector 1 1 sector 2 1 2 3 4 5 p T [ GeV ] p T ∈ [0 . 3 , 5] GeV ϕ ∈ [0 , 90] ◦ ϑ ∈ [35 , 123] ◦ z ∈ [ − 50 , 50] cm 14 / 15
Introduction Multi Layer Perceptron - MLP Preprocessing Results Conclusion Conclusion MLP ◮ MLP requires preprocessing ◮ MLP can improve z -RMS in low pt region ◮ Sectorization improves MLP prediction Preprocessing ◮ LS fit useful for preprocessing ◮ LS fit achieves good z -RMS for high p T tracks Outlook ◮ MLP optimization for low p T tracks ◮ further preprocessing studies ◮ hardware implementation on Virtex 7 FPGA 15 / 15
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