1b
This commit is contained in:
parent
9eee507d73
commit
9da52cfbfa
5 changed files with 143 additions and 136 deletions
|
|
@ -11,54 +11,56 @@
|
|||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
function [h, m, Q] = EMG(x, k, epochs, flag)
|
||||
|
||||
% variables
|
||||
num_clusters = k; % number of clusters
|
||||
eps = 1e-15; % small value that can be used to avoid obtaining 0's
|
||||
lambda = 1e-3; % value for improved version of EM
|
||||
[num_data, dim] = size(x);
|
||||
h = zeros(num_data, num_clusters); % expectation of data point being part of a cluster
|
||||
S = zeros(dim, dim, num_clusters); % covariance matrix for each cluster
|
||||
b = zeros(num_data,num_clusters); % cluster assignments, only used for intialization of pi and S
|
||||
Q = zeros(epochs*2,1); % vector that can hold complete data log-likelihood after each E and M step
|
||||
% variables
|
||||
num_clusters = k; % number of clusters
|
||||
eps = 1e-15; % small value that can be used to avoid obtaining 0's
|
||||
lambda = 1e-3; % value for improved version of EM
|
||||
[num_data, dim] = size(x);
|
||||
h = zeros(num_data, num_clusters); % expectation of data point being part of a cluster
|
||||
S = zeros(dim, dim, num_clusters); % covariance matrix for each cluster
|
||||
b = zeros(num_data,num_clusters); % cluster assignments, only used for intialization of pi and S
|
||||
Q = zeros(epochs*2,1); % vector that can hold complete data log-likelihood after each E and M step
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: Initialise cluster means using k-means
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: Initialise cluster means using k-means
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
means = kmeans(x, k);
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: Determine the b values for all data points
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: Initialize pi's (mixing coefficients)
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: Initialize the covariance matrix estimate
|
||||
% further modifications will need to be made when doing 2(d)
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
% Main EM loop
|
||||
for n=1:epochs
|
||||
%%%%%%%%%%%%%%%%
|
||||
% E-step
|
||||
%%%%%%%%%%%%%%%%
|
||||
fprintf('E-step, epoch #%d\n', n);
|
||||
[Q, h] = E_step(x, Q, h, pi, m, S, k);
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: Determine the b values for all data points
|
||||
% TODO: Store the value of the complete log-likelihood function
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%
|
||||
% M-step
|
||||
%%%%%%%%%%%%%%%%
|
||||
fprintf('M-step, epoch #%d\n', n);
|
||||
[Q, S, m] = M_step(x, Q, h, S, k);
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: Initialize pi's (mixing coefficients)
|
||||
% TODO: Store the value of the complete log-likelihood function
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: Initialize the covariance matrix estimate
|
||||
% further modifications will need to be made when doing 2(d)
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
% Main EM loop
|
||||
for n=1:epochs
|
||||
%%%%%%%%%%%%%%%%
|
||||
% E-step
|
||||
%%%%%%%%%%%%%%%%
|
||||
fprintf('E-step, epoch #%d\n', n);
|
||||
[Q, h] = E_step(x, Q, h, pi, m, S, k);
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: Store the value of the complete log-likelihood function
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%
|
||||
% M-step
|
||||
%%%%%%%%%%%%%%%%
|
||||
fprintf('M-step, epoch #%d\n', n);
|
||||
[Q, S, m] = M_step(x, Q, h, S, k);
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: Store the value of the complete log-likelihood function
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
end
|
||||
|
||||
|
||||
end
|
||||
|
|
@ -12,10 +12,11 @@
|
|||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
function [Q, h] = E_step(x, Q, h, pi, m, S, k)
|
||||
|
||||
[num_data, ~] = size(x);
|
||||
[num_data, ~] = size(x);
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: perform E-step of EM algorithm
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: perform E-step of EM algorithm
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
z = 1 + 1
|
||||
|
||||
end
|
||||
|
|
@ -11,24 +11,24 @@
|
|||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
function [Q, S, m] = M_step(x, Q, h, S, k)
|
||||
|
||||
% get size of data
|
||||
[num_data, dim] = size(x);
|
||||
eps = 1e-15;
|
||||
lambda = 1e-3; % value for improved version of EM
|
||||
% get size of data
|
||||
[num_data, dim] = size(x);
|
||||
eps = 1e-15;
|
||||
lambda = 1e-3; % value for improved version of EM
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: update mixing coefficients
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: update mixing coefficients
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: update cluster means
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: update cluster means
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: Calculate the covariance matrix estimate
|
||||
% further modifications will need to be made when doing 2(d)
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: Calculate the covariance matrix estimate
|
||||
% further modifications will need to be made when doing 2(d)
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
|
||||
end
|
||||
|
|
@ -1,87 +1,87 @@
|
|||
function [] = Problem2()
|
||||
|
||||
% file names
|
||||
stadium_fn = "stadium.jpg";
|
||||
goldy_fn = "goldy.jpg";
|
||||
% file names
|
||||
stadium_fn = "stadium.jpg";
|
||||
goldy_fn = "goldy.jpg";
|
||||
|
||||
% load image and preprocess it
|
||||
goldy_img = double(imread(goldy_fn))/255;
|
||||
stadium_img = double(imread(stadium_fn))/255;
|
||||
% load image and preprocess it
|
||||
goldy_img = double(imread(goldy_fn))/255;
|
||||
stadium_img = double(imread(stadium_fn))/255;
|
||||
|
||||
% convert RGB images
|
||||
goldy_x = reshape(permute(goldy_img, [2 1 3]), [], 3); % convert img from NxMx3 to N*Mx3
|
||||
stadium_x = reshape(permute(stadium_img, [2 1 3]), [], 3);
|
||||
% convert RGB images
|
||||
goldy_x = reshape(permute(goldy_img, [2 1 3]), [], 3); % convert img from NxMx3 to N*Mx3
|
||||
stadium_x = reshape(permute(stadium_img, [2 1 3]), [], 3);
|
||||
|
||||
% get dimensionality of stadium image
|
||||
[height, width, depth] = size(stadium_img);
|
||||
% get dimensionality of stadium image
|
||||
[height, width, depth] = size(stadium_img);
|
||||
|
||||
% set epochs (number of iterations to run algorithm for)
|
||||
epochs = 10;
|
||||
% set epochs (number of iterations to run algorithm for)
|
||||
epochs = 10;
|
||||
|
||||
%%%%%%%%%%
|
||||
% 2(a,b) %
|
||||
%%%%%%%%%%
|
||||
index = 1;
|
||||
figure();
|
||||
for k = 4:4:12
|
||||
fprintf("k=%d\n", k);
|
||||
%%%%%%%%%%
|
||||
% 2(a,b) %
|
||||
%%%%%%%%%%
|
||||
index = 1;
|
||||
figure();
|
||||
for k = 4:4:12
|
||||
fprintf("k=%d\n", k);
|
||||
|
||||
% call EM on data
|
||||
[h, m, Q] = EMG(stadium_x, k, epochs, false);
|
||||
% call EM on data
|
||||
[h, m, Q] = EMG(stadium_x, k, epochs, false);
|
||||
|
||||
% get compressed version of image
|
||||
[~,class_index] = max(h,[],2);
|
||||
compress = m(class_index,:);
|
||||
|
||||
% 2(a), plot compressed image
|
||||
subplot(3,2,index)
|
||||
imagesc(permute(reshape(compress, [width, height, depth]),[2 1 3]))
|
||||
index = index + 1;
|
||||
|
||||
% 2(b), plot complete data likelihood curves
|
||||
subplot(3,2,index)
|
||||
x = 1:size(Q);
|
||||
c = repmat([1 0 0; 0 1 0], length(x)/2, 1);
|
||||
scatter(x,Q,20,c);
|
||||
index = index + 1;
|
||||
end
|
||||
shg
|
||||
|
||||
%%%%%%%%
|
||||
% 2(c) %
|
||||
%%%%%%%%
|
||||
% get dimensionality of goldy image, and set k=7
|
||||
[height, width, depth] = size(goldy_img);
|
||||
k = 7;
|
||||
|
||||
% run EM on goldy image
|
||||
[h, m, Q] = EMG(goldy_x, k, epochs, false);
|
||||
|
||||
% plot goldy image using clusters from EM
|
||||
% get compressed version of image
|
||||
[~,class_index] = max(h,[],2);
|
||||
compress = m(class_index,:);
|
||||
figure();
|
||||
subplot(2,1,1)
|
||||
|
||||
% 2(a), plot compressed image
|
||||
subplot(3,2,index)
|
||||
imagesc(permute(reshape(compress, [width, height, depth]),[2 1 3]))
|
||||
index = index + 1;
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: plot goldy image after using clusters from k-means
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% begin code here
|
||||
% 2(b), plot complete data likelihood curves
|
||||
subplot(3,2,index)
|
||||
x = 1:size(Q);
|
||||
c = repmat([1 0 0; 0 1 0], length(x)/2, 1);
|
||||
scatter(x,Q,20,c);
|
||||
index = index + 1;
|
||||
end
|
||||
shg
|
||||
|
||||
% end code here
|
||||
shg
|
||||
%%%%%%%%
|
||||
% 2(c) %
|
||||
%%%%%%%%
|
||||
% get dimensionality of goldy image, and set k=7
|
||||
[height, width, depth] = size(goldy_img);
|
||||
k = 7;
|
||||
|
||||
%%%%%%%%
|
||||
% 2(e) %
|
||||
%%%%%%%%
|
||||
% run improved version of EM on goldy image
|
||||
[h, m, Q] = EMG(goldy_x, k, epochs, true);
|
||||
% run EM on goldy image
|
||||
[h, m, Q] = EMG(goldy_x, k, epochs, false);
|
||||
|
||||
% plot goldy image using clusters from improved EM
|
||||
[~,class_index] = max(h,[],2);
|
||||
compress = m(class_index,:);
|
||||
figure();
|
||||
imagesc(permute(reshape(compress, [width, height, depth]),[2 1 3]))
|
||||
shg
|
||||
% plot goldy image using clusters from EM
|
||||
[~,class_index] = max(h,[],2);
|
||||
compress = m(class_index,:);
|
||||
figure();
|
||||
subplot(2,1,1)
|
||||
imagesc(permute(reshape(compress, [width, height, depth]),[2 1 3]))
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% TODO: plot goldy image after using clusters from k-means
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% begin code here
|
||||
|
||||
% end code here
|
||||
shg
|
||||
|
||||
%%%%%%%%
|
||||
% 2(e) %
|
||||
%%%%%%%%
|
||||
% run improved version of EM on goldy image
|
||||
[h, m, Q] = EMG(goldy_x, k, epochs, true);
|
||||
|
||||
% plot goldy image using clusters from improved EM
|
||||
[~,class_index] = max(h,[],2);
|
||||
compress = m(class_index,:);
|
||||
figure();
|
||||
imagesc(permute(reshape(compress, [width, height, depth]),[2 1 3]))
|
||||
shg
|
||||
end
|
||||
|
|
@ -47,7 +47,7 @@ Updates:
|
|||
= Problem 1b
|
||||
|
||||
- $E(ww,vv|XX) = - sum_t r^t log y^t + (1 - r^t) log (1 - y^t)$
|
||||
- $y^t = "sigmoid"(v_2 z_2 + v_1 z_1 + v_0)$
|
||||
- $y^t = "sigmoid"(v_2 z^t_2 + v_1 z^t_1 + v_0)$
|
||||
- $z^t_1 = "ReLU"(w_2 x^t_2 + w_1 x^t_1 + w_0)$
|
||||
- $z^t_2 = tanh(w_2 x^t_2 + w_1 x^t_1 + w_0)$
|
||||
|
||||
|
|
@ -62,4 +62,8 @@ Updates:
|
|||
&= - sum_t frac(diff E, diff y^t) (frac(diff y^t, diff z^t_1) frac(diff z^t_1, diff w_j) + frac(diff y^t, diff z^t_2) frac(diff z^t_2, diff w_j)) \
|
||||
&= - sum_t (frac(r^t, y^t) - frac(1-r^t, 1-y^t)) (frac(diff y^t, diff z^t_1) frac(diff z^t_1, diff w_j) + frac(diff y^t, diff z^t_2) frac(diff z^t_2, diff w_j)) \
|
||||
&= - sum_t (frac(r^t-y^t, y^t (1-y^t))) (frac(diff y^t, diff z^t_1) frac(diff z^t_1, diff w_j) + frac(diff y^t, diff z^t_2) frac(diff z^t_2, diff w_j)) \
|
||||
&= - sum_t (frac(r^t-y^t, y^t (1-y^t))) (y^t (1-y^t) v_1 frac(diff z^t_1, diff w_j) + y^t (1-y^t) v_2 frac(diff z^t_2, diff w_j)) \
|
||||
&= - sum_t (r^t-y^t) (v_1 frac(diff z^t_1, diff w_j) + v_2 frac(diff z^t_2, diff w_j)) \
|
||||
&= - sum_t (r^t-y^t) (x^t_j v_1 cases(0 "if" ww dot xx < 0, 1 "otherwise") + x^t_j v_2 (1 - tanh^2 (ww dot xx))) \
|
||||
&= - sum_t (r^t-y^t) x^t_j (v_1 cases(0 "if" ww dot xx < 0, 1 "otherwise") + v_2 (1 - tanh^2 (ww dot xx))) \
|
||||
$
|
||||
Loading…
Reference in a new issue