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ANN Basics : Brief Review

ANN Basics : Brief Review . N. Saoulidou, Fermilab & G. Tzanakos, Univ. of Athens. -Methods: A rtificial N eural N etworks-. ANN can be trained by MC generated events

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ANN Basics : Brief Review

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  1. ANN Basics : Brief Review N. Saoulidou, Fermilab & G. Tzanakos, Univ. of Athens N. Saoulidou & G. Tzanakos

  2. -Methods:ArtificialNeural Networks- • ANN can be trained by MC generated events • A trained ANN provides multidimensional cuts for data that are difficult to deduce in the usual manner from 1-d or 2-d histogram plots. • ANN has been used in HEP • HEP Packages: • JETNET • SNNS • MLP fit N. Saoulidou & G. Tzanakos

  3. a1x+b1y+c1 a3x+b3y+c3 Y Y Signal Signal Background Background a2x+b2y+c2 X -ANN BASICS- X • Event sample characterized by two variables X and Y (left figure) • A linear combination of cuts can separate “signal” from “background” (right fig.) • Define “step function” • Separate “signal” from “background” with the following function: “Signal (x, y)” OUT “Signal (x, y)” IN N. Saoulidou & G. Tzanakos

  4. -ANN BASICS- Visualization of function C(x,y) • The diagram resembles a feed forwardneural network with two input neurons, three neurons in thefirst hidden layer and one output neuron. • Threshold produces thedesired offset. • Constantsai, bi are theweightswi,j(i and j are the neuron indices). X Y Thres. b1 a3 b3 a1 a2 b2 1 1 1 -2 c2 c1 c3 Output N. Saoulidou & G. Tzanakos

  5. -ANN basics : Schematic- HIDDEN LAYER Biological Neuron INPUT LAYER X1 WEIGHTS . . . OUTPUT LAYER neuron k . . . Bayesian Probability wik . . . Xi neuron i wkj neuron j INPUT PARAMETERS N. Saoulidou & G. Tzanakos Bias

  6. -ANN BASICS- • Output of tjeach neuron in the first hidden layer : • Transfer function is the sigmoid function : • For the standard backpropagation training procedure of neural networks, the derivative of the neuron transfer functions must exist in order to be able to minimize the network error (cost) function E. • Theorem 1 : Any continuous function of any number of variables on a compact set can be approximated to any accuracy by a linear combination of sigmoids • Theorem 2 : Trained with desired output 1 for signal and 0 for background the neural network function (output function tj) approximates the Bayesian Probability of an event being a signal. N. Saoulidou & G. Tzanakos

  7. -ANN Probability (review)- ANN analysis : Minimization of an Error (Cost) Function The ANN output is the Bayes a posteriori probability & in the proof no special assumption has been made on the a priori P(S) and P(B) probabilities (absolute normalization)…..TRUE BUT THEIR VALUES DO MATTER ………(They should be what nature gave us) N. Saoulidou & G. Tzanakos

  8. -ANN probability (review)- • Bayesian a posteriori probability : • ANN output : P(S/x) • ANN training examples : P(x/S) & P(x/B) • ANN number of Signal Training Examples P(S) • ANN number of Background Training Examples P(B) The MLP (ann) analysis and the Maximum Likelihood Method ( Bayes Classifier ) are equivalent. (c11 c22 = cost for making the correct decision & c12 c21 = cost for making the wrong decision ) N. Saoulidou & G. Tzanakos

  9. -ANN Probability cont.- • Worse hypothetical case 1: • One variable characterizing the populations, which is identical for S and B, therefore : • P(S)=0.1 & P(B)=0.9 • If we train with equal numbers for signal and background the ANN will wrongly computeP(S/x)=0.5 • If we train with the correct ratio for signal and background the ANN will correctly computeP(S/x)=0.1,which is exactly what Bayes a posteriori probability would give also. P(S/x)=0.5 ANN output P(S/x)=0.1 N. Saoulidou & G. Tzanakos

  10. -ANN Probability cont.- • Best hypothetical case : • One variable characterizing the populations, which is completely separated (different) for S and B. • P(S)=0.1 & P(B)=0.9 • If we train with equal numbers for signal and background the ANN will computeP(S/x)=1. • If we train with the correct ratio for signal and background the ANN will again computeP(S/x)=1. • In this case it does not matter if we use the correct a priori probabilities or not. P(S/x) =1 ANN output P(S/x) =1 N. Saoulidou & G. Tzanakos

  11. ANN Probability (final...) • The MLP output approximates the Bayesian a posteriori probability and the a priori class probabilities P(S) and P(B) should be considered correctly. • The more similar the characteristics of the populations are, the more important the a priori probabilities are, in calculation of the final a posteriori probability by the MLP. • In addition the more close to the boundary surface (between the two populations) an event is , the more sensitive it’s a posteriori probability is to changes in the a priori probabilities. N. Saoulidou & G. Tzanakos

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