Image Modeling &

SegmentationAly Farag and Asem Ali

Lecture #3

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Parametric methods

These methods are useful when the underlying distribution is known in advance or is simple enough to be modeled by a simple distribution function or a mixture of such functions

The parametric model is very compact (low memory and CPU usage) where only few parameters need to fit.

The model’s parameters are estimated using these methods such as maximum likelihood estimation, Bayesian estimation and expectation maximization.

o A location parameter simply shifts the graph left or right on the horizontal

axis. o A scale parameter (>1) stretches or (<1 ) compress the pdf,

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Parametric methods (1- Maximum Likelihood Estimator: MLE)

Suppose that n samples x1, x2, … , xn are drawn independently and identically distributed (i.i.d.) ~ distribution φ(θ) with vector of parameters θ=(θ1,…. , θr)

Know: data samples, the distribution type Unknown : θ ?????

MLE Method estimates θ by maximizing the log likelihood of the data

By i.i.d.

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Parametric methods (1- Maximum Likelihood Estimator: MLE)Let

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Find θ by letting

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Example:

Suppose that n samples x1, x2, … , xn are drawn independently and identically distributed (i.i.d.)~ 1D N(μ,σ) find the MLE of μ and σ

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Matlab Demo

In some cases we can find a closed form for θ

Coin Example ………………..

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An estimator of a parameter is unbiased if the expected value of the estimate is the same as the true value of the parameters.

Parametric methods (1- Maximum Likelihood Estimator: MLE)

Example:

An estimator of a parameter is biased if the expected value of the estimate is different from the true value of the parameters.

Example:

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What if there are distinct subpopulations in observed data?

Parametric methods

Example Pearson in 1894, tried to model the distribution of the ratio

between measurements of forehead and body length on crabs.

He used a two-component mixture.

It was hypothesized that the two-component structure was related to the possibility of this particular population of crabs evolving into two new subspecies

Mixture Model

The underlying density is assumed to have the form

The weights, Constrained

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What is the difference between Mixture Model and the kernel-based estimator?

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Parametric methods

Example Given that {xi, Ci1, Ci2} n samples (complete data) drawn i.i.d. two

normal distributions

xi observed value of ith instance

Ci1 and Ci2 indicate which of two normal distributions was used

to generate xi

Cij=1 if Cij was used to generate xi, 0 otherwise

MLE

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How can we estimate the parameters given incomplete data (don’t know Ci1 and Ci2 )?

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Parametric methods (2- Expectation Maximization: EM)

The EM algorithm is a general method of finding the maximum-likelihood estimate of the parameters of an underlying distribution from a given data set when the data is incomplete or has missing values.

EM Algorithm:

Given initial parameters Θ0

Repeatedly

o Re-estimating expected values of hidden binary variables Cij

o Then recalculate the MLE of Θ using these expected values for the hidden variables Note:

EM unsupervised method, but MLE supervised

To use EM you must to know:

Number of classes K,

Parametric form of the distribution.

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Illustrative example

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Illustrative example

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Parametric methods (2- Expectation Maximization: EM)

Assume that a joint density function for complete data set

The EM algorithm first finds the expected value of the complete-data log-likelihood with respect to the unknown data C given the observed data x and the current parameter estimates Θi-1.

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The current parameters estimates that we used to evaluate the expectation

The new parameters that we optimize to maximize Q

The evaluation of this expectation is called the E-step of the algorithm

The second step (the M-step) of the EM algorithm is to maximize the expectation we computed in the first step.

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These two steps are repeated as necessary. Each iteration is guaranteed to increase the log likelihood and the algorithm is guaranteed to converge to a local maximum of the likelihood function.

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Parametric methods (2- Expectation Maximization: EM)

The mixture-density parameter estimation problem

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Parametric methods (2- Expectation Maximization: EM)

For some distributions, it is possible to get an analytical expressions for For example, if we assume d-dimensional Gaussian component distributions

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Parametric methods (2- Expectation Maximization: EM)

Example:

MATLAB demo