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CSE190a, Winter 2010

Pattern classification

Biometrics CSE 190-a Lecture 3

CSE190a, Winter 2010

Biometrics: A Pattern Recognition System

•  False accept rate (FAR): Proportion of imposters accepted •  False reject rate (FRR): Proportion of genuine users rejected • Failure to enroll rate (FTE): portion of population that cannot be enrolled • Failure to acquire rate (FTA): portion of population that cannot be verified

Authentication Enrollment

Yes/No

CSE190a, Winter 2010

Error Rates

False Match (False Accept): Mistaking biometric measurements from two different persons to be from the same person; False Non-match (False reject): Mistaking two biometric measurements from the same person to be from two different persons

(c) Jain 2004 CSE190a, Winter 2010

Error vs Threshold

(c) Jain 2004

FAR: False accept rate FRR: False reject rate

CSE190a, Winter 2010

ROC Curve

Accuracy requirements of a biometric system are application dependent (c) Jain 2004

Pattern Classification

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An Example

•  “Sorting incoming Fish on a conveyor according to species using optical sensing”

Sea bass Species Salmon

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•  Adopt the lightness and add the width of the fish

Fish xT = [x1, x2]

Lightness Width

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•  However, our satisfaction is premature because the central aim of designing a classifier is to correctly classify novel input

Issue of generalization!

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Bayesian Decision Theory Continuous Features

(Sections 2.1-2.2)

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Introduction

•  The sea bass/salmon example

•  State of nature, prior

• State of nature is a random variable

• The catch of salmon and sea bass is equiprobable

•  P(ω1), P(ω2) Prior probabilities

•  P(ω1) = P(ω2) (uniform priors)

•  P(ω1) + P( ω2) = 1 (exclusivity and exhaustivity)

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•  Decision rule with only the prior information • Decide ω1 if P(ω1) > P(ω2) otherwise decide ω2

•  Use of the class–conditional information

•  P(x | ω1) and P(x | ω2) describe the difference in lightness between populations of sea-bass and salmon

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•  Posterior, likelihood, evidence

•  P(ωj | x) = (P(x | ωj) * P (ωj)) / P(x) (BAYES RULE)

•  In words, this can be said as: Posterior = (Likelihood * Prior) / Evidence

• Where in case of two categories

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•  Intuitive decision rule given the posterior probabilities: Given x: if P(ω1 | x) > P(ω2 | x) True state of nature = ω1 if P(ω1 | x) < P(ω2 | x) True state of nature = ω2

Why do this?: Whenever we observe a particular x, the probability of error is : P(error | x) = P(ω1 | x) if we decide ω2 P(error | x) = P(ω2 | x) if we decide ω1

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•  Since decision rule is optimal for each feature value X, there is not better rule for all x.

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Bayesian Decision Theory – Continuous Features

Generalization of the preceding ideas

•  Use of more than one feature

•  Use more than two states of nature

•  Allowing actions and not only decide on the state of nature

•  Introduce a loss of function (more general than the probability of error)

•  Allowing actions other than classification primarily allows the possibility of rejection

•  Refusing to make a decision in close or bad cases!

•  Letting loss function state how costly each action taken is

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•  Let X be a vector of features.

•  Let {ω1, ω2,…, ωc} be the set of c states of nature (or “classes”)

•  Let {α1, α2,…, αa} be the set of possible actions

•  Let λ(αi | ωj) be the loss for action αi when the state of nature is ωj

Bayesian Decision Theory – Continuous Features

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What is the Expected Loss for action αi ?

R(αi | x) is called the Conditional Risk (or Expected Loss)

For any given x the expected loss is

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Overall risk R = Sum of all R(αi | x) for i = 1,…,a

Minimizing R Minimizing R(αi | x) for i = 1,…, a

for i = 1,…,a

Conditional risk

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Given a measured feature vector x, which actiion should we take?

Select the action αi for which R(αi | x) is minimum

R is minimum and R in this case is called the Bayes risk = best performance that can be achieved!

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Two-Category Classification

α1 : deciding ω1

α2 : deciding ω2

λij = λ(αi | ωj)

loss incurred for deciding ωi when the true state of nature is ωj

Conditional risk:

R(α1 | x) = λ11P(ω1 | x) + λ12P(ω2 | x) R(α2 | x) = λ21P(ω1 | x) + λ22P(ω2 | x)

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Our rule is the following: if R(α1 | x) < R(α2 | x)

λ11P(ω1 | x) + λ12P(ω2 | x) < λ21P(ω1 | x) + λ22P(ω2 | x) action α1: “decide ω1” is taken

This results in the equivalent rule : decide ω1 if:

(λ21- λ11) P(x | ω1) P(ω1) > (λ12- λ22) P(x | ω2) P(ω2)

and decide ω2 otherwise

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x (λ21- λ11)

x (λ12- λ22)

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Two-Category Decision Theory: Chopping Machine

α1 = chop

α2 = DO NOT chop ω1 = NO hand in machine

ω2 = hand in machine

λ11 = λ(α1 | ω1) = $ 0.00

λ12 = λ(α1 | ω2) = $ 100.00

λ21 = λ(α2 | ω1) = $ 0.01

λ22 = λ(α1 | ω1) = $ 0.01

Therefore our rule becomes

(λ21- λ11) P(x | ω1) P(ω1) > (λ12- λ22) P(x | ω2) P(ω2)

0.01 P(x | ω1) P(ω1) > 99.99 P(x | ω2) P(ω2)

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x 0.01

x 99.99

α1 = chop α2 = DO NOT chop

ω1 = NO hand in machine ω2 = hand in machine

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Exercise to do at home!!

Select the optimal decision where: W = {ω1, ω2}

P(x | ω1) N(2, 0.5) (Normal distribution) P(x | ω2) N(1.5, 0.2)

P(ω1) = 2/3 P(ω2) = 1/3

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revisited

•  Actions are decisions on classes If action αi is taken and the true state of nature is ωj then the decision is correct if i = j and in error if i ≠ j

•  Seek a decision rule that minimizes the probability of error which is the error rate

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•  Introduction of the zero-one loss function:

Therefore, the conditional risk for each action is:

“The risk corresponding to this loss function is the average (or expected) probability error”

Average Prob. of Error

Since

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• Minimize the risk requires maximize P(ωi | x) (since R(αi | x) = 1 – P(ωi | x))

•  For Minimum error rate

• Decide ωi if P (ωi | x) > P(ωj | x) ∀j ≠ i

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Likelihood ratio:

The preceding rule is equivalent to the following rule:

Then take action α1 (decide ω1) Otherwise take action α2 (decide ω2)