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E. Fatemizadeh

Statistical Pattern Recognition. E. Fatemizadeh. PATTERN RECOGNITION. Typical application areas Machine vision Character recognition (OCR) Computer aided diagnosis Speech recognition Face recognition Biometrics Image Data Base retrieval Data mining Bionformatics

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E. Fatemizadeh

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  1. Statistical Pattern Recognition E. Fatemizadeh

  2. PATTERN RECOGNITION • Typical application areas • Machine vision • Character recognition (OCR) • Computer aided diagnosis • Speech recognition • Face recognition • Biometrics • Image Data Base retrieval • Data mining • Bionformatics • The task: Assign unknown objects – patterns – into the correct class. This is known as classification.

  3. Features:These are measurable quantities obtained from the patterns, and the classification task is based on their respective values. • Feature vectors:A number of features constitute the feature vector Feature vectors are treated as random vectors.

  4. An example:

  5. Patterns sensor feature generation feature selection classifier design system evaluation • The classifier consists of a set of functions, whose values, computed at , determine the class to which the corresponding pattern belongs • Classification system overview

  6. Supervised – unsupervised pattern recognition: The two major directions • Supervised: Patterns whose class is known a-priori are used for training. • Unsupervised: The number of classes is (in general) unknown and no training patterns are available.

  7. CLASSIFIERS BASED ON BAYES DECISION THEORY • Statistical nature of feature vectors • Assign the pattern represented by feature vector to the most probable of the available classes That is maximum

  8. Computation of a-posteriori probabilities • Assume known • a-priori probabilities This is also known as the likelihood of

  9. The Bayes rule (Μ=2) where

  10. The Bayes classification rule (for two classes M=2) • Given classify it according to the rule • Equivalently: classify according to the rule • For equiprobable classes the test becomes

  11. Equivalently in words: Divide space in two regions • Probability of error • Total shaded area • Bayesian classifier is OPTIMAL with respect to minimising the classification error probability!!!!

  12. Indeed: Moving the threshold the total shaded area INCREASES by the extra “grey” area.

  13. Minimizing Classification Error Probability • Our Claim: Bayesian Classifier is Optimal w.r to minimizing the classification error Probability.

  14. Minimizing Classification Error Probability R1 and R2 are Union of space:

  15. The Bayes classification rule for many (M>2) classes: • Given classify it to if: • Such a choice also minimizes the classification error probability • Minimizing the average risk • For each wrong decision, a penalty term is assigned since some decisions are more sensitive than others

  16. For M=2 • Define the loss matrix • penalty term for deciding class ,although the pattern belongs to , etc. • Risk with respect to

  17. Risk with respect to • RED lighted: • Average risk Probabilities of wrong decisions, weighted by the penalty terms

  18. Choose and so that ris minimized • Then assign to if • Equivalently:assign x in if : likelihood ratio

  19. If

  20. An example:

  21. Then the threshold value is: • Threshold for minimum r

  22. Thus moves to the left of (WHY?)

  23. MinMax Criteria • Goal: • Minimizing Maximum Possible Overall Risk (No Priori Knowledge)

  24. MinMax Criteria • With Some Simplification:

  25. MinMax Criteria • A Simple Case (λ11= λ22=0, λ12= λ21=1)

  26. DISCRIMINANT FUNCTIONS DECISION SURFACES • If are contiguous: is the surface separating the regions. On one side is positive (+), on the other is negative (-). It is known as Decision Surface + -

  27. For monotonic increasing f(.), the rule remains the same if we use: • is a discriminant function • In general, discriminant functions can be defined independent ofthe Bayesian rule. They lead to suboptimal solutions, yet if chosen appropriately, can be computationally more tractable.

  28. BAYESIAN CLASSIFIER FOR NORMAL DISTRIBUTIONS • Multivariate Gaussian pdf • The last one Called Covariance Matrix

  29. is monotonically increasing. Define: • Example:

  30. That is, is quadratic and the surfaces Quadrics, Ellipsoids, Parabolas, Hyperbolas, Pairs of lines. For example:

  31. Decision Hyperplanes • Quadratic terms come from: If ALL (the same) the quadratic terms are not of interest. They are not involved in comparisons. Then, equivalently, we can write: Discriminant functions are LINEAR

  32. Let in addition:

  33. Non Diagonal: • Decision hyperplane

  34. Minimum Distance Classifiers • equiprobable Euclidean Distance: smaller Mahalanobis Distance: smaller

  35. More • Geometry Analysis

  36. Example:

  37. Statistical Error Analysis • Minimum Attainable Classification Error (Bayes):

  38. ESTIMATION OF UNKNOWN PROBABILITY DENSITY FUNCTIONS • Maximum Likelihood

  39. Example:

  40. Maximum Aposteriori Probability Estimation • In ML method, θ was considered as a parameter • Here we shall look at θ as a random vector described by a pdf p(θ), assumed to be known • Given Compute the maximum of • From Bayes theorem

  41. The method:

  42. Example:

  43. Bayesian Inference

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