Statistical Methods in Particle Physics

Transcription

1 Statistical Methods in Particle Physics 8. Multivariate Analysis Prof. Dr. Klaus Reygers (lectures) Dr. Sebastian Neubert (tutorials) Heidelberg University WS 2017/18

2 Multi-Variate Classification Consider events which can be either signal or background events. Each event is characterized by n observables: ~x =(x 1,...,x n ) "feature vector" Goal: classify events as signal or background in an optimal way. This is usually done by mapping the feature vector to a single variable, i.e., to scalar test statistic: R n! R : y(~x) A cut y > c to classify events as signal corresponds to selecting a potentially complicated hyper-surface in feature space. In general superior to classical "rectangular" cuts on the xi. Problem closely related to machine learning (pattern recognition, data mining, ) 2

3 Classification: Different Approaches rectangular cuts non linear linear H 0 k-nearest-neighbor, Boosted Decision Trees, Multi-Layer Perceptrons, Support Vector Machines G. Cowan': 3

4 Signal Probability Instead of Hard Decisions Example: test statistic y for signal and background from a Multi-Layer Perceptron (MLP): Normalized TMVA output for classifier: MLP Signal Background Instead of a hard yes/no decision one can also define the probability of an event to be a signal event: P s (y) P(S y) = p(y B) TMVA manual p(y S) MLP U/O-flow (S,B): (0.0, 0.0)% / (0.0, 0.0)% Normalized TMVA output for classifier: BDT Signal Background Figure 4: Example plots for classifier output distributions for signal and background event test sample. Shown are likelihood (upper left), PDE range search (upper right), Multilay lower left) and boosted decision trees. p(y S) f s p(y S) f s + p(y B) (1 f s ), f s = n s n s + n b TMVA tutorial: An up-to-date reference of all configuration options for the TMVA Factory, 4 the MVA methods:

5 ROC Curve Quality of the classification can be characterized by the receiver operating characteristic (ROC curve) Background rejection versus Signal efficiency 1 εb εb: background efficiency Background rejection good MVA Method: Fisher MLP BDT PDERS Likelihood better Figure 5: Example for the background rejection versus signal e on the classifier outputs for the events of the test sample. Signal efficiency ciency ( ROC curve ) obtained by cutting 5

6 Different Approaches to Classification Neyman-Pearson lemma states that likelihood ratio provides an optimal test statistic for classification: y(~x) = p(~x S) p(~x B) Problem: the underlying pdf's are almost never known explicitly. Two approaches: 1. Estimate signal and background pdf's and construct test statistic based on Neyman-Pearson lemma, e.g. Naïve Bayes classifier (= Likelihood classifier) 2. Decision boundaries determined directly without approximating the pdf's (linear discriminants, decision trees, neural networks, ) 6

7 Supervised Machine Learning (I) Supervised Machine Learning requires labeled training data, i.e., a training sample where for each event it is known whether it is a signal or background event Decision boundary defined by minimizing a loss function ("training") Bias-variance tradeoff Classifiers with a small number of degrees of freedom are less prone to statistical fluctuations: different training samples would result in a similar classification boundaries ("small variance") However, if the data contain features that a model with few degrees of freedom cannot describe, a bias is introduced. In this case a classifier with more degrees of freedom would be better. User has to find a good balance 7

8 Supervised Machine Learning (II) Training, validation, and test sample Decision boundary fixed with training sample Performance on training sample becomes better with more iterations Danger of overtraining: Statistical fluctuations of the training sample will be learnt Validation sample = independent labeled data set not used for training check for overtraining Sign of overtraining: performance on validation sample becomes worse Stop training when signs of overtraining are observed ("early stopping") Performance: apply classifier to independent test sample Often: test sample = validation sample (only small bias) 8

9 Supervised Machine Learning (III) Rule of thumb if training data not expensive Training sample: 50% Validation sample: 25% Test sample: 25% often test sample = validation sample, i.e., training : validation/test = 50:50 Cross validation (efficient use of scarce training data) Split training sample in k independent subset Tk of the full sample T Train on T \ Tk resulting in k different classifiers For each training event there is one classifier that didn't use this event for training Validation results are then combined 9

10 Estimating PDFs from Histograms? Consider 2d example: G. Cowan': signal background Approximate approximate pdfs PDF using byn(x,y s), y S) N(x,y b) andn(x, in y B) corresponding cells. M bins per variable in d dimensions: M d cells hard to generate enough training data (often not practical for d > 1) In general in machine learning, problems related to a large number of dimensions of the feature space are referred to as the "curse of dimensionality" 10

11 k-nearest Neighbor Method (I) k-nn classifier Estimates probability density around the input vector p(~x S) and p(~x B) are approximated by the number of signal and background events in the training sample that lie in a small volume around the point ~x Algorithms finds k nearest neighbors: k = k s + k b Probability for the event to be of signal type: p s (~x) = k s (~x) k s (~x)+k b (~x) 11

12 k-nearest Neighbor Method (II) Simplest choice for distance measure in feature space is the Euclidean distance: 1.5 TMVA manual R = ~x ~y 1 Better: take correlations between variables into account: R = q (~x ~y) T V 1 (~x ~y) x V = covariance matrix "Mahalanobis distance" x 0 Figure 14: Example for the k-nearest The k-nn classifier has best performance when the boundary that separates signal and background events has irregular discriminating features that input cannot variables). be easily The thre approximated by parametric learning methods. The full (open) circles are the signal (bac in the nearest neighborhood (circle) of th signal and 7 background points so that 12 q

13 Kernel Density Estimator (KDE) [Just mentioned briefly here] Idea: Smear training data to get an estimate of the PDFs ˆf h (~x) = 1 n nx i=1 K h (~x ~x i )= 1 nh nx i=1 K( ~x ~x i h ) K = Kernel h = bandwidth (smoothing parameter) Use, e.g., Gaussian kernel: K(~x) = 1 ~x e 2 /2 d = dimension of feature space (2 ) d/2 x 2 Lecture Niklas Berger x 1 13

14 Gaussian KDE in 1 Dimension G. Cowan': Adaptive KDE: adjust width of kernel according to PDF (wide where pdf is low) Advantage of KDE: no training! Disadvantage of KDE: need to sum of all training events to evaluate PDF, i.e., method can be slow 14

15 Fisher Linear Discriminant Linear discriminant is simple. Can still be optimal if amount of training data is limited. n nx Ansatz for test statistic: yx= y(~x) w= i x i = ~w T i x i = w T x ~x i=1 i=1 f y s, f y b Choose parameters wi so that separation between signal and background distribution is maximum. Need to define "separation". f (y) t s t b Fisher: maximize J(~w) = ( s b ) 2 2 s + 2 b s b J w= s b 2 G. Cowan': s 2 b 2 y 15

16 Fisher Linear Discriminant: Variable Definitions Mean and covariance for signal and background: µ s,b i = V s,b ij = Z Z x i f (~x H s,b )d~x (x i µ s,b i )(x j µ s,b j ) f (~x H s,b )d~x Mean and covariance of y(~x) for signal and background: Z s,b = y(~x)f (~x H s,b )d~x = ~w T ~µ s,b Z 2 s,b = (y(~x) s,b ) 2 f (~x H s,b )d~x = ~w T V s,b ~w G. Cowan': 16

17 Fisher Linear Discriminant: Determining the Coefficients wi Numerator of J(~w) : ( s b ) 2 = J(~w) Denominator of : nx i=1 2 s + 2 b = w i (µ s i µ b i )! 2 = nx i,j=1 nx i,j=1 nx i,j=1 w i w j (µ s i µ b i )(µ s j µ b j ) w i w j B ij = ~w T B ~w w i w j (V s + V b ) ij ~w T W ~w Maximize: J(~w) = ~w T B ~w ~w T W ~w = separation between classes separation within classes G. Cowan': 17

18 Fisher Linear Discriminant: Determining the Coefficients i =0 gives: y(~x) =~w T ~x with ~w / W 1 (~µ s ~µ b ) We obtain linear decision boundaries. linear decision boundary Weight vector ~w can be interpreted as a direction in feature space on which the events are projected. G. Cowan': 18

19 Fisher Linear Discriminant: Remarks In case the signal and background pdfs f (~x H s ) and f (~x H b ) are both multivariate Gaussian with the same covariance but different means, the Fisher discriminant is y(~x) / ln f (~x H s) f (~x H b ) That is, in this case the Fisher discriminant is an optimal classifier according to the Neyman-Pearson lemma (as y(~x) is a monotonic function of the likelihood ratio) Test statistic can be written as y(~x) =w 0 + nx i=1 w i x i where events with y > 0 are classified as signal. Same functional form as for the perceptron (prototype of neural networks). 19

20 Perceptron Discriminant: y(~x) =h w 0 + nx i=1 w i x i! The nonlinear, monotonic function h is called activation function. Typical choices for h: 1 1+e x ( sigmoid ), tanhx x 1 h(x) y(~x) x n x 20

21 The Biological Inspiration: the Neuron Human brain: neurons, each with on average 7000 synaptic connections y l 1 1 y l 1 2. y l 1 n Input w l 1 1j w l 1 2j w l 1 nj ρ Output Σ y l j 21

22 Feedforward Neural Network with One Hidden Layer superscripts indicates layer number 1(~x) y(~x) i(~x) =h (1) i0 + nx j=1 1 w (1) ij x j A m(~x) y(~x) =h (2) 10 + m X j=1 w (2) 1j 1 j(~x) A Straightforward to generalize to multiple hidden layers 22

23 Network Training ~x a : training event, a = 1,..., N t a : correct label for training event a e.g., ta = 1, 0 for signal and background, respectively ~w : vector containing all weights Error function: E(~w) = 1 2 NX (y(~x a, ~w) t a ) 2 = NX E a (~w) a=1 a=1 Weights are determined by minimizing the error function. 23

24 Backpropagation ~w (0) Start with an initial guess for the weights an then update weights after each training event: ~w ( +1) = ~w ( ) re a (~w ( ) ) learning rate Let's write network output as follows: mx y(~x) =h(u(~x)) with u(~x) = w (2) 1j j(~x), j(~x) =h Here we defined φ0 = x0 = 1 and the sums start from 0 to include the offsets. Weights from hidden layer to output: E a = 1 2 (y a t a ) (2) 1j j=0 =(y a t a )h 0 (u(~x (2) 1j =(y a t a )h 0 (u(~x a )) j (~x a ) Weights from input layer to hidden layer ( further application of chain rule): nx k=0 w (1) jk x k! h (v j (1) jk =(y a t a )h 0 (u(~x a ))w (2) 1j h 0 (v j (~x a )) x a,k ~x a (x a,1,...,x a,n ) 24

25 Neural Network Output and Decision Boundaries P. Bhat, Multivariate Analysis Methods in Particle Physics, inspirehep.net/record/ a Input layer Hidden layer Output layer 800 b x 1 x 2 x 3 Number of events Signal Background output of neural network 200 x d x i h j 0(x) = f(x,w) NN output 25 c d 20 Signal Background NN contours decision boundaries for different cuts on NN output Variable x Variable x p(s x 1, x 2 ) Variable x Variable x signal probability p(s x1, x2) 25

26 Example of Overtraining Too many neurons/layers make a neural network too flexible overtraining training sample G. Cowan: test sample Network "learns" features that are merely statistical fluctuations in the training sample 26

27 Monitoring Overtraining G. Cowan: Monitor fraction of misclassified events (or error function:) error rate optimum = minimum of error rate for test sample overtraining = increase of error rate test sample training sample flexibility (e.g., number of nodes/layers) 27

28 Example: Identification of Events with Top-Quarks a t t! W + bw b! l bq q b Shaded = signal b likelihood method Background Signal Events (arbitrary units) x 1 = missing E T x 2 = aplanarity Events (arbitrary units) D LB Background Signal x 3 x 4 D0 experiment, plot from inspirehep.net/record/ neural net D NN 28

29 Deep Neural Networks Deep networks: many hidden layers with large number of neurons Challenges Hard too train ("vanishing gradient problem") Training slow Risk of overtraining Big progress in recent years Interest in NN waned before ca Milestone: paper by G. Hinton (2006): "learning for deep belief nets" Image recognition, AlphaGo, Soon: self-driving cars, 29

30 Fun with Neural Nets in the Browser 30

31 Decision Trees (I) arxiv:physics/ v1 root node B 4/37 S 7/1 S/B 52/48 < PMT Hits? S/B 9/10 S/B 48/11 < 0.2 GeV 0.2 GeV Energy? < 500 cm 500 cm Radius? B 2/9 branch node (node with further branching) S 39/1 MiniBooNE: 1520 photomultiplier signals, goal: separation of νe from νμ events leaf node (no further branching) Leaf nodes classify events as either signal or background 31

32 Decision Trees (II) Ann.Rev.Nucl.Part.Sci. 61 (2011) a Root node {x 1 5, x 2 } b x 1 θ 1 x 1 > θ 1 x 2 R 3 θ 3 x 2 θ 2 x 2 > θ 2 x 2 θ 3 x 2 > θ 3 R 2 R 1 R 2 R 3 θ 2 R 4 R 5 R 1 x 1 θ 4 x 1 > θ 4 θ 1 θ 4 x 1 R 4 R 5 Easy to interpret and visualize: Space of feature vectors split up into rectangular volumes (attributed to either signal or background) How to build a decision tree in an optimal way? 32

33 Finding Optimal Cuts Separation btw. signal and background is often measured with the Gini index: G = p(1 p) Here p is the purity: p = P signal w i P signal w i + P background w i w i = weight of event i [usefulness of weights will become apparent soon] Improvement in signal/background separation after splitting a set A into two sets B and C: = W A G A W B G B W C G C where W X = X X w i 33

34 Separation Measures arbitrary unit Split criterion Misclas. error Entropy Gini signal purity Cross entropy: p ln p (1 p)ln(1 p) p Gini index: p(1 p) [after Corrado Gini, used to measure income and wealth inequalities, 1912] Misclassification rate: 1 max(p,1 p) 34

35 Decision Tree Pruning When to stop growing a tree? When all nodes are essentially pure? Well, that's overfitting! Pruning Cut back fully grown tree to avoid overtraining 35

36 Boosted Decision Trees: Idea Drawback of decisions trees: very sensitive to statistical fluctuations in training sample Solution: boosting One tree several trees ("forrest") Trees are derived from the same training ensemble by reweighting events Individual trees are then combined: weighted average of individual trees Boosting is a general method of combining a set of classifiers (not necessarily decisions trees) into a new, more stable classifier with smaller error. Popular example: AdaBoost (Freund, Schapire, 1997) 36

37 AdaBoost (short for Adaptive Boosting) Initial training sample with equal weights normalized as nx Train first classifier f1: ~x 1,...,~x n : multivariate event data y 1,...,y n : true class labels, +1 or 1 w (1) (1) 1,...,w n event weights i=1 w (1) i =1 f 1 (~x i ) > 0 f 1 (~x i ) < 0 classify as signal classify as background 37

38 Updating Events Weights Define training sample k+1 from training sample k by updating weights: w (k+1) i = w (k) i e k f k (~x i )y i /2 Z k i = event index normalization factor so that nx i=1 w (k) i =1 Weight is increased if event was misclassified by the previous classifier "Next classifier should pay more attention to misclassified events" At each step the classifier fk minimizes error rate " k = nx i=1 w (k) i I (y i f k (~x i ) apple 0), I (X )=1ifX is true, 0 otherwise 38

39 Assigning the Classifier Score Assign score to each classifier according to its error rate: k =ln 1 " k " k Combined classifier (weighted average): f (~x) = KX k=1 k f k (~x) It can be shown that the error rate of the combined classifier satisfies " apple KY k=1 2 p " k (1 " k ) 39

40 General Remarks on Multi-Variate Analyses MVA Methods More effective than classic cut-based analyses Take correlations of input variables into account Important: find good input variables for MVA methods Good separation power between S and B Little correlations among variables No correlation with the parameters you try to measure in your signal sample! Pre-processing Apply obvious variable transformations and let MVA method do the rest Make use of obvious symmetries: if e.g. a particle production process is symmetric in polar angle θ use cos θ and not cos θ as input variable It is generally useful to bring all input variables to a similar numerical range H. Voss, Multivariate Data Analysis and Machine Learning in High Energy Physics 40

41 Classifiers and Their Properties H. Voss, Multivariate Data Analysis and Machine Learning in High Energy Physics Classifiers Criteria Cuts Likelihood PDERS / k-nn H-Matrix Fisher MLP BDT RuleFit SVM Performance no / linear correlations nonlinear correlations Speed Training Response / Robust -ness Overtraining Weak input variables Curse of dimensionality Transparency 41