Pattern recognition

Pattern recognition Lecture 15 – Non Parametric Methods Professor Aly A. Farag Computer Vision and Image Processing Laboratory University of Louisville URL: www.cvip.uofl.edu ; E-mail: aly.farag@louisville.edu Planned for ECE 620 and ECE 655 - Summer 2011 TA/Grader: MelihAslan; CVIP Lab Rm 6, msaslan01@lousiville.edu

Introduction • All Parametric densities are unimodal (have a single local maximum), whereas many practical problems involve multi-modal densities • Nonparametric procedures can be used with arbitrary distributions and without the assumption that the forms of the underlying densities are known • There are two types of nonparametric methods: • Estimating P(x | j ) • Bypass probability and go directly to a-posteriori probability estimation

Density Estimation • Basic idea: • Probability that a vector x will fall in region R is: • P is a smoothed (or averaged) version of the density function p(x) if we have a sample of size n; therefore, the probability that k points fall in R is then: and the expected value for k is: E(k) = nP (3) Pattern Classification, Ch4 (Part 1)

ML estimation of P =  is reached for Therefore, the ratio k/n is a good estimate for the probability P and hence for the density function p. p(x) is continuous and that the region R is so small that p does not vary significantly within it, we can write: where is a point within R and V the volume enclosed by R. Pattern Classification, Ch4 (Part 1)

Combining equation (1) , (3) and (4) yields: Pattern Classification, Ch4 (Part 1)

Density Estimation (cont.) • Justification of equation (4) We assume that p(x) is continuous and that region R is so small that p does not vary significantly within R. Since p(x) = constant, it is not a part of the sum. Pattern Classification, Ch4 (Part 1)

Where: (R) is: a surface in the Euclidean space R2 a volume in the Euclidean space R3 a hypervolume in the Euclidean space Rn Since p(x)  p(x’) = constant, therefore in the Euclidean space R3: Pattern Classification, Ch4 (Part 1)

Condition for convergence The fraction k/(nV) is a space averaged value of p(x). p(x) is obtained only if V approaches zero. This is the case where no samples are included in R: it is an uninteresting case! In this case, the estimate diverges: it is an uninteresting case! Pattern Classification, Ch4 (Part 1)

The volume V needs to approach 0 anyway if we want to use this estimation • Practically, V cannot be allowed to become small since the number of samples is always limited • One will have to accept a certain amount of variance in the ratio k/n • Theoretically, if an unlimited number of samples is available, we can circumvent this difficulty To estimate the density of x, we form a sequence of regions R1, R2,…containing x: the first region contains one sample, the second two samples and so on. Let Vnbe the volume of Rn, kn the number of samples falling in Rn and pn(x) be the nth estimate for p(x): pn(x) = (kn/n)/Vn(7) Pattern Classification, Ch4 (Part 1)

Three necessary conditions should apply if we want pn(x) to converge to p(x): There are two different ways of obtaining sequences of regions that satisfy these conditions: (a) Shrink an initial region where Vn = 1/n and show that This is called “the Parzen-window estimation method” (b) Specify kn as some function of n, such as kn = n; the volume Vn is grown until it encloses kn neighbors of x. This is called “the kn-nearest neighbor estimation method” Pattern Classification, Ch4 (Part 1)

Pattern Classification, Ch4 (Part 1)

Parzen Windows • Parzen-window approach to estimate densities assume that the region Rn is a d-dimensional hypercube • ((x-xi)/hn) is equal to unity if xi falls within the hypercube of volume Vn centered at x and equal to zero otherwise. Pattern Classification, Ch4 (Part 1)

The number of samples in this hypercube is: By substituting kn in equation (7), we obtain the following estimate: Pn(x) estimates p(x) as an average of functions of x and the samples (xi) (i = 1,… ,n). These functions  can be general! Pattern Classification, Ch4 (Part 1)

Illustration • The behavior of the Parzen-window method • Case where p(x) N(0,1) Let (u) = (1/(2) exp(-u2/2) and hn = h1/n (n>1) (h1: known parameter) Thus: is an average of normal densities centered at the samples xi. Pattern Classification, Ch4 (Part 1)

Numerical results: For n = 1 and h1=1 For n = 10 and h = 0.1, the contributions of the individual samples are clearly observable ! Pattern Classification, Ch4 (Part 1)

Analogous results are also obtained in two dimensions as illustrated: Pattern Classification, Ch4 (Part 1)

Case where p(x) = 1.U(a,b) + 2.T(c,d) (unknown density) (mixture of a uniform and a triangle density) Pattern Classification, Ch4 (Part 1)

Classification example In classifiers based on Parzen-window estimation: • We estimate the densities for each category and classify a test point by the label corresponding to the maximum posterior • The decision region for a Parzen-window classifier depends upon the choice of window function as illustrated in the following figure. Pattern Classification, Ch4 (Part 1)

Chapter 4 (part 2):Non-Parametric Classification (Sections 4.3-4.5) Parzen Window (cont.) Kn –Nearest Neighbor Estimation The Nearest-Neighbor Rule

x1 x2 . . . xd Parzen Windows (cont.) • Parzen Windows – Probabilistic Neural Networks • Compute a Parzen estimate based on n patterns • Patterns with d features sampled from c classes • The input unit is connected to n patterns . . . . . W11 p1 p2 . . . Input patterns Input unit . . Wd2 Wdn pn Modifiable weights (trained)

. . . p1 pn . p2 Input patterns . . . 1 . 2 . . . Category units . pk . . . . c pn Activations (Emission of nonlinear functions)

Training the network • Algorithm • Normalize each pattern x of the training set to 1 • Place the first training pattern on the input units • Set the weights linking the input units and the first pattern units such that: w1 = x1 • Make a single connection from the first pattern unit to the category unit corresponding to the known class of that pattern • Repeat the process for all remaining training patterns by setting the weights such that wk = xk (k = 1, 2, …, n) We finally obtain the following network

Testing the network • Algorithm • Normalize the test pattern x and place it at the input units • Each pattern unit computes the inner product in order to yield the net activation and emit a nonlinear function • Each output unit sums the contributions from all pattern units connected to it • Classify by selecting the maximum value of Pn(x | j) (j = 1, …, c)

Kn - Nearest neighbor estimation • Goal: a solution for the problem of the unknown “best” window function • Let the cell volume be a function of the training data • Center a cell about x and let it grows until it captures knsamples (kn = f(n)) • knare called the knnearest-neighbors of x 2 possibilities can occur: • Density is high near x; therefore the cell will be small which provides a good resolution • Density is low; therefore the cell will grow large and stop until higher density regions are reached We can obtain a family of estimates by setting kn=k1/n and choosing different values for k1

Illustration For kn = n = 1 ; the estimate becomes: Pn(x) = kn / n.Vn = 1 / V1 =1 / 2|x-x1|

Estimation of a-posteriori probabilities • Goal: estimate P(i | x) from a set of n labeled samples • Let’s place a cell of volume V around x and capture k samples • kisamples amongst k turned out to be labeled ithen: pn(x, i) = ki /n.V An estimate for pn(i| x) is:

ki/k is the fraction of the samples within the cell that are labeled i • For minimum error rate, the most frequently represented category within the cell is selected • If k is large and the cell sufficiently small, the performance will approach the best possible

The nearest –neighbor rule • Let Dn = {x1, x2, …, xn} be a set of n labeled prototypes • Let x’  Dn be the closest prototype to a test point xthen the nearest-neighbor rule for classifying x is to assign it the label associated with x’ • The nearest-neighbor rule leads to an error rate greater than the minimum possible: the Bayes rate • If the number of prototype is large (unlimited), the error rate of the nearest-neighbor classifier is never worse than twice the Bayes rate (it can be demonstrated!) • If n  , it is always possible to find x’ sufficiently close so that: P(i | x’)  P(i | x)

Example: x = (0.68, 0.60)t Decision:5is the label assigned to x

If P(m | x)  1, then the nearest neighbor selection is almost always the same as the Bayes selection

The k – nearest-neighbor rule • Goal: Classify x by assigning it the label most frequently represented among the k nearest samples and use a voting scheme

Example: k = 3 (odd value) and x = (0.10, 0.25)t Closest vectors to x with their labels are: {(0.10, 0.28, 2); (0.12, 0.20, 2); (0.15, 0.35,1)} One voting scheme assigns the label 2 to x since 2 is the most frequently represented

Pattern recognition