Pattern Recognition Assignments II

Sevdanur GENC

27 Haz 2014

7 dakikada okunur

Pattern Recognition – Oruntu Tanima alaninda calismaniz icin Maksimum olabilirlik kestirimi, Bayesian siniflayici, knn, kmeans, Gaussian parzen penceresi, RBF siniflayici konulari ornek niteliginde isinize yarayabilecek soru ve cevaplari bu makale iceriginde bulabilirsiniz.

Assignment I

QA1) Maksimum olabilirlik kestrimi cozumleri; Matlab Kodu; clear all; close all; clc; veri=1+randn(1,100); standartSapma=1; degerAraligi=-10:0.1:10; L=zeros(1,length(degerAraligi)); for i=1:length(degerAraligi) L(i) = sum(1/sqrt(2*pi*standartSapma^2)*exp(-(veri-degerAraligi(i)).^2/(2*standartSapma^2))); end

=max(L); display(degerAraligi(index)); plot(degerAraligi,L); hold on; stem(degerAraligi(index),L(index),'r');

Assignment II

QA2) w1 = <8;4>

; <6;4>

; <7;5>

; <7;3>

; w2 = <10;-2>

; <4;-2>

; <7;-1>

; <7;-3>

; Matlab Kodu; clear all; close all; clc; W1 = [ 8,4; 6,4; 7,5; 7,3; ] W2 = [ 10,-2; 4,-2; 7,-1; 7,-3;] %% A Sikki %% nokta_w1 = [ 8,6,7,7; 4,4,5,3; ]; figure; plot(nokta_w1(1,:),nokta_w1(2,:),'*r'); nokta_w2 = [ 10,4,7,7; -2,-2,-1,-3;] hold on; plot(nokta_w2(1,:),nokta_w2(2,:),'*b'); axis(<-15 15 -15 15 -15 15>

); grid on; %% B Sikki %% ortalamaW1 = mean(W1) ortalamaW2 = mean(W2) kovaryansW1 = cov(W1) kovaryansW2 = cov(W2) %% C Sikki %% E1 = <1/2 , 0; 0, 1/2>

; E2 = <4.5, 0; 0 0.5>

; Wb1 = - (1/2) * E1'; Wb2 = - (1/2) * E2'; wk1 = E1' * ortalamaW1(1,1); wk2 = E2' * ortalamaW1(1,1); wk10 = <- (1/2) * ortalamaw1(1,1)' * e1' * ortalamaw1(1,1)>

- [ (1/2) * (log(det(E1)) + log(exp(1))) ]; wk20 = <- (1/2) * ortalamaw2(1,1)' * e2' * ortalamaw2(1,1)>

- [ (1/2) * (log(det(E2)) + log(exp(1))) ]; for i=1:length(W1) G11 = W1(i,1)' * Wb1 * W1(i,1) + wk1' * W1(i,1) + wk10; G12 = W1(i,2)' * Wb1 * W1(i,2) + wk1' * W1(i,2) + wk10; end for i=1:length(W1) G21 = W2(i,1)' * Wb2 * W2(i,1) + wk2' * W2(i,1) + wk20; G22 = W2(i,2)' * Wb2 * W2(i,2) + wk2' * W2(i,2) + wk20; end %% E Sikki %% prototype = <9,1>

for i=1:length(W1) sonuc(i,:) = <(prototype(1,1) - w1(i,1))^2 + (prototype(1,2) - w1(i,2))^2>

; end NNMatrixW1 = sonuc(:,:) vektorAraligi1 = min(NNMatrixW1) for i=1:length(W2) sonuc(i,:) = <(prototype(1,1) - w2(i,1))^2 + (prototype(1,2) - w2(i,2))^2>

; end NNMatrixW2 = sonuc(:,:) vektorAraligi2 = min(NNMatrixW2) prototype = <0,1>

%% 2.satir'a ait. for i=1:length(W1) sonuc(i,:) = <(prototype(1,1) - w1(i,1))^2 + (prototype(1,2) - w1(i,2))^2>

; end NNMatrixW1 = sonuc(:,:) vektorAraligi1 = min(NNMatrixW1) for i=1:length(W2) sonuc(i,:) = <(prototype(1,1) - w2(i,1))^2 + (prototype(1,2) - w2(i,2))^2>

; end NNMatrixW2 = sonuc(:,:) vektorAraligi2 = min(NNMatrixW2) hold on; prototypeNokta1 = <9; 1>

; plot(prototypeNokta1(1,:),prototypeNokta1(2,:),'Or'); hold on; prototypeNokta2 = <0; 1>

; plot(prototypeNokta2(1,:),prototypeNokta2(2,:),'Ob');

Assignment III

QA3) w1 = <8;4>

; <6;4>

; <7;5>

; <7;3>

; w2 = <10;-2>

; <4;-2>

; <7;-1>

; <7;-3>

; A Sikki) Matlab Kodu; clear all; close all; clc; W1 = [ 8,4; 6,4; 7,5; 7,3; ] W2 = [ 10,-2; 4,-2; 7,-1; 7,-3;] x = <0,1>

; varyans = 1; for i=1:length(W1) solutionA(i,1) = 1 / sqrt(2*pi) * exp(-((W1(i,1)- x(1,1))^2) / 2 * varyans^2); solutionA(i,2) = 1 / sqrt(2*pi) * exp(-((W1(i,2)- x(1,2))^2) / 2 * varyans^2); end Cozum1 = solutionA(:,:) p=0; for i=1:length(solutionA) p = p + solutionA(i,:); end p / length(W1) for i=1:length(W2) solutionB(i,1) = 1 / sqrt(2*pi) * exp(-((W2(i,1)- x(1,1))^2) / 2 * varyans^2); solutionB(i,2) = 1 / sqrt(2*pi) * exp(-((W2(i,2)- x(1,2))^2) / 2 * varyans^2); end Cozum2 = solutionB(:,:) p=0; for i=1:length(solutionB) p = p + solutionB(i,:); end p / length(W2) B Sikki) Matlab Kodu; clear all; close all; clc; x = [ 8 4; 6 4; 7 5; 7 3; 10 -2; 4 -2; 7 -1; 7 -3;] knn = 3; c1 = <0,1>

axis(<-5 15 -10 10>

); axis equal; hold on; plot(c1(1,1),c1(1,2),'rX'); for i=1:8 plot(x(i,1),x(i,2),'rO'); distanceX(i)=norm(c1(1,:)- x(i,:)); end;

= sort(distanceX); for i=1:knn plot(

,'b:'); end; C Sikki) Matlab Kodu; clear all; close all; clc; x = [ 8 4; 6 4; 7 5; 7 3; 10 -2; 4 -2; 7 -1; 7 -3;] c1 = <7,0>

c2 = <0,1>

axis(<-5 15 -10 10>

); axis equal; hold on; plot(c1(1,1),c1(1,2),'X'); plot(c2(1,1),c2(1,2),'X'); for i=1:8; plot(x(i,1),x(i,2),'O'); end; for i=1:8; plot(

,'r:'); plot(

,'r:'); end;

Assignment IV

QA4) Matlab Kodu; clear all; close all; clc; x = [ 8 4; 6 4; 7 5; 7 3; 10 -2; 4 -2; 7 -1; 7 -3;] varyans = 1; c =

axis(<0 15 -10 10>

); hold on; for i=1:4; plot(x(i,1),x(i,2),'o'); y(i)=0; end; for i=5:8 ; plot(x(i,1),x(i,2),'x'); y(i)=1; end; for i=1:8; phi1(i) = exp(- ((x(i,1)-c(1,1))^2 + (x(i,2)-c(1,2))^2 )/2 * varyans^2 ); phi2(i) = exp(- ((x(i,1)-c(2,1))^2 + (x(i,2)-c(2,2))^2 )/2 * varyans^2 ); phi3(i) = exp(- ((x(i,1)-c(3,1))^2 + (x(i,2)-c(3,2))^2 )/2 * varyans^2 ); phi4(i) = exp(- ((x(i,1)-c(4,1))^2 + (x(i,2)-c(4,2))^2 )/2 * varyans^2 ); end; P =

w = inv(P'*P )*P'*y'; ty = sign(P*w); for i=1:64; for j=1:64; x1(i,j)=i/64; y1(i,j)=j/64; p1 = exp(- ((x1(i,j)-c(1,1))^2 + (y1(i,j)-c(1,2))^2)/2 * varyans^2); p2 = exp(- ((x1(i,j)-c(2,1))^2 + (y1(i,j)-c(2,2))^2)/2 * varyans^2); p3 = exp(- ((x1(i,j)-c(3,1))^2 + (y1(i,j)-c(3,2))^2)/2 * varyans^2); p4 = exp(- ((x1(i,j)-c(4,1))^2 + (y1(i,j)-c(4,2))^2)/2 * varyans^2); z(i,j)= w(1)*p1 + w(2)*p2 + w(3)*p3 + w(4)*p4; end; end; contour(x1,y1,z, <0 -10>

);

Assignment V

QA5) W1 = <-5.01, -8.12, -3.68; -5.43, -3.48, -3.54; 1.08, -5.52, 1.66; 0.86, -3.78, -4.11; -2.67, 0.63, 7.39; 4.94, 3.29, 2.08; -2.51, 2.09, -2.59; -2.25, -2.13, -6.94; 5.56, 2.86, -2.26; 1.03, -3.33, 4.33; >

W2 = <-0.91, -0.18, -0.05; 1.30, -2.06, -3.53; -7.75, -4.54, -0.95; -5.47, 0.50, 3.92; 6.14, 5.72, -4.85; 3.60, 1.26, 4.36; 5.37, -4.63, -3.65; 7.18, 1.46, -6.66; -7.39, 1.17, 6.30; -7.50, -6.32, -0.31; >

W3 = [ 5.35, 2.26, 8.13; 5.12, 3.22, -2.66; -1.34, -5.31, -9.87; 4.48, 3.42, 5.19; 7.11, 2.39, 9.21; 7.17, 4.33, -0.98; 5.75, 3.97, 6.65; 0.77, 0.27, 2.41; 0.90, -0.43, -8.71; 3.52, -0.36, 6.43 ] Matlab Kodu; clear all; close all; clc; %% E Sikki %% W1 = <-5.01, -8.12, -3.68; -5.43, -3.48, -3.54; 1.08, -5.52, 1.66; 0.86, -3.78, -4.11; -2.67, 0.63, 7.39; 4.94, 3.29, 2.08; -2.51, 2.09, -2.59; -2.25, -2.13, -6.94; 5.56, 2.86, -2.26; 1.03, -3.33, 4.33; >

nokta_w1 = [ -5.01,-5.43,1.08,0.86,-2.67,4.94,-2.51,-2.25,5.56,1.03; -8.12,-3.48,-5.52,-3.78,0.63,3.29,2.09,-2.13,2.86,-3.33; -3.68,-3.54,1.66,-4.11,7.39,2.08,-2.59,-6.94,-2.26,4.33; ]; figure; plot3(nokta_w1(1,:),nokta_w1(2,:),nokta_w1(3,:),'*r'); W2 = <-0.91, -0.18, -0.05; 1.30, -2.06, -3.53; -7.75, -4.54, -0.95; -5.47, 0.50, 3.92; 6.14, 5.72, -4.85; 3.60, 1.26, 4.36; 5.37, -4.63, -3.65; 7.18, 1.46, -6.66; -7.39, 1.17, 6.30; -7.50, -6.32, -0.31; >

nokta_w2 = [ -0.91,1.30,-7.75,-5.47,6.14,3.60,5.37,7.18,-7.39,-7.50; -0.18,-2.06,-4.54,0.50,5.72,1.26,-4.63,1.46,1.17,-6.32; -0.05,-3.53,-0.95,3.92,-4.85,4.36,-3.65,-6.66,6.30,-0.31; ] hold on; plot3(nokta_w2(1,:),nokta_w2(2,:),nokta_w2(3,:),'*b'); W3 = [ 5.35, 2.26, 8.13; 5.12, 3.22, -2.66; -1.34, -5.31, -9.87; 4.48, 3.42, 5.19; 7.11, 2.39, 9.21; 7.17, 4.33, -0.98; 5.75, 3.97, 6.65; 0.77, 0.27, 2.41; 0.90, -0.43, -8.71; 3.52, -0.36, 6.43 ] nokta_w3 = [ 5.35,5.12,-1.34,4.48,7.11,7.17,5.75,0.77,0.90,3.52; 2.26,3.22,-5.31,3.42,2.39,4.33,3.97,0.27,-0.43,-0.36; 8.13,-2.66,-9.87,5.19,9.21,-0.98,6.65,2.41,-8.71,6.43; ] hold on; plot3(nokta_w3(1,:),nokta_w3(2,:),nokta_w3(3,:),'*g'); axis(<-15 15 -15 15 -15 15>

); grid on; %% A Sikki %% mW1 = mean(W1) mW2 = mean(W2) mW3 = mean(W3) sigmaW1 = cov(W1) sigmaW2 = cov(W2) sigmaW3 = cov(W3) %% B ve D Sikki %% x1 = <1 2 1>

'; x2 = <5 3 2>

'; x3 = <0 0 0>

'; x4 = <1 0 0>

'; Wi1 = -0.5 * inv(sigmaW1) wi1 = inv(sigmaW1)* mW1' w01 = -0.5 * (sigmaW1') * inv(sigmaW1) * mW1' - 0.5*(log(det(sigmaW1))/log(exp(1))) g1x1 = x1' * Wi1 * x1 + wi1' * x1 + w01 Wi2 = -0.5 * inv(sigmaW2) wi2 = inv(sigmaW2)* mW2' w02 = -0.5 * (sigmaW2') * inv(sigmaW2) * mW2' - 0.5*(log(det(sigmaW2))/log(exp(1))) g2x1 = x1' * Wi2 * x1 + wi2' * x1 + w02 Wi3 = -0.5 * inv(sigmaW3) wi3 = inv(sigmaW3)* mW3' w03 = -0.5 * (sigmaW3') * inv(sigmaW3) * mW3' - 0.5*(log(det(sigmaW3))/log(exp(1))) g3x1 = x1' * Wi3 * x1 + wi3' * x1 + w03 %%%%%%%%%%%%%%%%%%%%%%% Wi1 = -0.5 * inv(sigmaW1) wi1 = inv(sigmaW1)* mW1' w01 = -0.5 * (sigmaW1') * inv(sigmaW1) * mW1' - 0.5*(log(det(sigmaW1))/log(exp(1))) g1x2 = x2' * Wi1 * x2 + wi1' * x2 + w01 Wi2 = -0.5 * inv(sigmaW2) wi2 = inv(sigmaW2)* mW2' w02 = -0.5 * (sigmaW2') * inv(sigmaW2) * mW2' - 0.5*(log(det(sigmaW2))/log(exp(1))) g2x2 = x2' * Wi2 * x2 + wi2' * x2 + w02 Wi3 = -0.5 * inv(sigmaW3) wi3 = inv(sigmaW3)* mW3' w03 = -0.5 * (sigmaW3') * inv(sigmaW3) * mW3' - 0.5*(log(det(sigmaW3))/log(exp(1))) g3x2 = x2' * Wi3 * x2 + wi3' * x2 + w03 %%%%%%%%%%%%%%%%%%%%%%% Wi1 = -0.5 * inv(sigmaW1) wi1 = inv(sigmaW1)* mW1' w01 = -0.5 * (sigmaW1') * inv(sigmaW1) * mW1' - 0.5*(log(det(sigmaW1))/log(exp(1))) g1x3 = x3' * Wi1 * x3 + wi1' * x3 + w01 Wi2 = -0.5 * inv(sigmaW2) wi2 = inv(sigmaW2)* mW2' w02 = -0.5 * (sigmaW2') * inv(sigmaW2) * mW2' - 0.5*(log(det(sigmaW2))/log(exp(1))) g2x3 = x3' * Wi2 * x3 + wi2' * x3 + w02 Wi3 = -0.5 * inv(sigmaW3) wi3 = inv(sigmaW3)* mW3' w03 = -0.5 * (sigmaW3') * inv(sigmaW3) * mW3' - 0.5*(log(det(sigmaW3))/log(exp(1))) g3x3 = x3' * Wi3 * x3 + wi3' * x3 + w03 %%%%%%%%%%%%%%%%%%%%%%% Wi1 = -0.5 * inv(sigmaW1) wi1 = inv(sigmaW1)* mW1' w01 = -0.5 * (sigmaW1') * inv(sigmaW1) * mW1' - 0.5*(log(det(sigmaW1))/log(exp(1))) g1x4 = x4' * Wi1 * x4 + wi1' * x4 + w01 Wi2 = -0.5 * inv(sigmaW2) wi2 = inv(sigmaW2)* mW2' w02 = -0.5 * (sigmaW2') * inv(sigmaW2) * mW2' - 0.5*(log(det(sigmaW2))/log(exp(1))) g2x4 = x4' * Wi2 * x4 + wi2' * x4 + w02 Wi3 = -0.5 * inv(sigmaW3) wi3 = inv(sigmaW3)* mW3' w03 = -0.5 * (sigmaW3') * inv(sigmaW3) * mW3' - 0.5*(log(det(sigmaW3))/log(exp(1))) g3x4 = x4' * Wi3 * x4 + wi3' * x4 + w03 %% C Sikki %% W = <-5.01, -8.12, -3.68; -5.43, -3.48, -3.54; 1.08, -5.52, 1.66; 0.86, -3.78, -4.11; -2.67, 0.63, 7.39; 4.94, 3.29, 2.08; -2.51, 2.09, -2.59; -2.25, -2.13, -6.94; 5.56, 2.86, -2.26; 1.03, -3.33, 4.33; -0.91, -0.18, -0.05; 1.30, -2.06, -3.53; -7.75, -4.54, -0.95; -5.47, 0.50, 3.92; 6.14, 5.72, -4.85; 3.60, 1.26, 4.36; 5.37, -4.63, -3.65; 7.18, 1.46, -6.66; -7.39, 1.17, 6.30; -7.50, -6.32, -0.31; 5.35, 2.26, 8.13; 5.12, 3.22, -2.66; -1.34, -5.31, -9.87; 4.48, 3.42, 5.19; 7.11, 2.39, 9.21; 7.17, 4.33, -0.98; 5.75, 3.97, 6.65; 0.77, 0.27, 2.41; 0.90, -0.43, -8.71; 3.52, -0.36, 6.43 >

knn = 3; c1 = <1 2 1>

; c2 = <5 3 2>

; c3 = <0 0 0>

; c4 = <1 0 0>

; axis(<-5 15 -10 10>

); axis equal; hold on; plot3(c1(1,1),c1(1,2),c1(1,3),'rX'); plot3(c2(1,1),c2(1,2),c2(1,3),'rX'); plot3(c3(1,1),c3(1,2),c3(1,3),'rX'); plot3(c4(1,1),c4(1,2),c4(1,3),'rX'); for i=1:30 plot3(W(i,1),W(i,2),W(i,3),'rO'); distanceX1(i)=norm(c1(1,:)- W(i,:)); distanceX2(i)=norm(c2(1,:)- W(i,:)); distanceX3(i)=norm(c3(1,:)- W(i,:)); distanceX4(i)=norm(c4(1,:)- W(i,:)); end;

= sort(distanceX1);

= sort(distanceX2);

= sort(distanceX3);

= sort(distanceX4); for i=1:knn plot3(

,'b:'); plot3(

,'b:'); end; Keyifli Calismalar Dilerim.