I'm trying to apply the Karhunen Loeve procedure to a translation-invariant data set. I understand the KL procedure, and how to create a mask to smooth out missing data. However, I'm having a hard time creating a program to model my Translation invariant data set.
The data set that I need to plot in matlab is:
Translation-Invariant Data Set
And here's the matlab code that I tried to use to model it:
function [fmu] = kLProcedure(N, P, M)
for k = 1:N;
for m = 1:M;
for n = 1:P;
x(m) = ((m-1)*2.*(pi))/M;
t(n) = ((n - 1)*2.*(pi))/P;
k = 1:3;
fmu(x(m),t(n)) = (1/N).*symsum((1/k).*sin(k(x(m)-t(n))),k);
end
end
end
With N=3, P=64, M=64;
I'm trying to use a nested for-loop to calculate each iteration of m, n, and t. and keep getting the error:
Error using /
Matrix dimensions must agree.
Error in kLProcedure (line 28)
fmu(x(m),t(n)) = (1/N).*symsum((1/k).*sin(k(x(m)-t(n))),k);
And advice would be greatly appreciated. Thank you.
Honestly, I don't have an in-depth knowledge of the algorithms you are using, but looking at the formulation of the one you are trying to reproduce, I think the following code is what you are looking for:
fmu = kLProcedure(3,64,64);
plot(fmu);
function [fmu] = kLProcedure(N,M,P)
k = 1:N;
ki = 1 ./ k;
Ni = 1 / N;
m = 1:M;
x = ((m - 1) .* 2 .* pi()) ./ M;
n = 1:P;
t = ((n - 1) .* 2 .* pi()) ./ P;
fmu = zeros(M,P);
for i = m
for j = n
fmu(i,j) = Ni .* sum(ki .* sin(ki .* (x(i) - t(j))));
end
end
end
Output:
Translational-Invariant Data
function [fmu] = kLProcedure(N,M,P)
m = 1:M;
x = ((m - 1) .* 2 .* pi()) ./ M;
n = 1:P;
t = ((n - 1) .* 2 .* pi()) ./ P;
fmu = zeros(M,P);
for i = m
for j = n
fmu(i,j) = 0;
for k = 1:3
fmu(i,j) = fmu(i,j) + (1/3).*sum((1/k).*(sin((k) .* (x(i) - t(j)))));
end
end
end
Related
Construct an experiment to study the performance of the Cramer rule (with two implementations
determinants) in relation to Gauss's algorithm.
In each iteration 10 random arrays A (NxN), and vectors b (Nx1) will be created.
The 10 linear systems will be solved using the Cramer rule ("cramer.m") using
of rec_det (A) and using det (A), and the Gaussian algorithm
(“GaussianElimination.m”), and the time for each technique will be the average of 10 values.
Repeat the above for N = 2 to 10 and make a graph of the average time
in relation to the dimension N.
This is my task. I dont know if the way that I calculate the average time is correct and the graphic is not displayed.
T1=0;
T2=0;
T3=0;
for N=2:10
for i=1:10
A=rand(N,N);
b=rand(N,1);
t1=[1,i];
t2=[1,i];
t3=[1,i];
tic;
crammer(A,b);
t1(i)=toc;
tic
crammer_rec(A,b);
t2(i)=toc;
tic
gaussianElimination(A,b);
t3(i)=toc;
T1=T1+t1(i);
T2=T2+t2(i);
T3=T3+t3(i);
end
avT1=T1/10;
avT2=T2/10;
avT3=T3/10;
end
plot(2:10 , avT1 , 2:10 , avT2 , 2:10 , avT3);
function x = cramer(A, b)
n = length(b);
d = det(A);
% d = rec_det(A);
x = zeros(n, 1);
for j = 1:n
x(j) = det([A(:,1:j-1) b A(:,j+1:end)]) / d;
% x(j) = rec_det([A(:,1:j-1) b A(:,j+1:end)]) / d;
end
end
function x = cramer(A, b)
n = length(b);
d = rec_det(A);
x = zeros(n, 1);
for j = 1:n
x(j) = rec_det([A(:,1:j-1) b A(:,j+1:end)]) / d;
end
end
function deta = rec_det(R)
if size(R,1)~=size(R,2)
error('Error.Matrix must be square.')
else
n = size(R,1);
if ( n == 2 )
deta=(R(1,1)*R(2,2))-(R(1,2)*R(2,1));
else
for i=1:n
deta_temp=R;
deta_temp(1,:)=[ ];
deta_temp(:,i)=[ ];
if i==1
deta=(R(1,i)*((-1)^(i+1))*rec_det(deta_temp));
else
deta=deta+(R(1,i)*((-1)^(i+1))*rec_det(deta_temp));
end
end
end
end
end
function x = gaussianElimination(A, b)
[m, n] = size(A);
if m ~= n
error('Matrix A must be square!');
end
n1 = length(b);
if n1 ~= n
error('Vector b should be equal to the number of rows and columns of A!');
end
Aug = [A b]; % build the augmented matrix
C = zeros(1, n + 1);
% elimination phase
for k = 1:n - 1
% ensure that the pivoting point is the largest in its column
[pivot, j] = max(abs(Aug(k:n, k)));
C = Aug(k, :);
Aug(k, :) = Aug(j + k - 1, :);
Aug(j + k - 1, :) = C;
if Aug(k, k) == 0
error('Matrix A is singular');
end
for i = k + 1:n
r = Aug(i, k) / Aug(k, k);
Aug(i, k:n + 1) = Aug(i, k:n + 1) - r * Aug(k, k: n + 1);
end
end
% back substitution phase
x = zeros(n, 1);
x(n) = Aug(n, n + 1) / Aug(n, n);
for k = n - 1:-1:1
x(k) = (Aug(k, n + 1) - Aug(k, k + 1:n) * x(k + 1:n)) / Aug(k, k);
end
end
I think the easiest way to do this is by creating a 9 * 3 dimensional matrix to contain all the total times, and then take the average at the end.
allTimes = zeros(9, 3);
for N=2:10
for ii=1:10
A=rand(N,N);
b=rand(N,1);
tic;
crammer(A,b);
temp = toc;
allTimes(N-1,1) = allTimes(N-1,1) + temp;
tic
crammer_rec(A,b);
temp = toc;
allTimes(N-1,2) = allTimes(N-1,2) + temp;
tic
gaussianElimination(A,b);
temp = toc;
allTimes(N-1,3) = allTimes(N-1,3) + temp;
end
end
allTimes = allTimes/10;
figure; plot(2:10, allTimes);
You can use this approach because the numbers are quite straightforward and simple. If you had a more complicated setup, the way to store the times/calculate the averages would have to be tweaked.
If you had more functions you could also use function handles and create a third inner loop, but this is a little more advanced.
What would be the best way to simplify a function by getting rid of a loop?
function Q = gs(f, a, b)
X(4) = sqrt((3+2*sqrt(6/5))/7);
X(3) = sqrt((3-2*sqrt(6/5))/7);
X(2) = -sqrt((3-2*sqrt(6/5))/7);
X(1) = -sqrt((3+2*sqrt(6/5))/7);
W(4) = (18-sqrt(30))/36;
W(3) = (18+sqrt(30))/36;
W(2) = (18+sqrt(30))/36;
W(1) = (18-sqrt(30))/36;
Q = 0;
for i = 1:4
W(i) = (W(i)*(b-a))/2;
X(i) = ((b-a)*X(i)+(b+a))/2;
Q = Q + W(i) * f(X(i));
end
end
Is there any way to use any vector-like solution instead of a for loop?
sum is your best friend here. Also, declaring some constants and creating vectors is useful:
function Q = gs(f, a, b)
c = sqrt((3+2*sqrt(6/5))/7);
d = sqrt((3-2*sqrt(6/5))/7);
e = (18-sqrt(30))/36;
g = (18+sqrt(30))/36;
X = [-c -d d c];
W = [e g g e];
W = ((b - a) / 2) * W;
X = ((b - a)*X + (b + a)) / 2;
Q = sum(W .* f(X));
end
Note that MATLAB loves to handle element-wise operations, so the key is to replace the for loop at the end with scaling all of the elements in W and X with those scaling factors seen in your loop. In addition, using the element-wise multiplication (.*) is key. This of course assumes that f can handle things in an element-wise fashion. If it doesn't, then there's no way to avoid the for loop.
I would highly recommend you consult the MATLAB tutorial on element-wise operations before you venture onwards on your MATLAB journey: https://www.mathworks.com/help/matlab/matlab_prog/array-vs-matrix-operations.html
this is my first time here so I hope that someone can help me.
I'm trying to implementing the Gauss-Seidel method and the power method using a matrix with the storage CSR or called Morse storage. Unfortunately I can't manage to do better then the following codes:
GS-MORSE:
function [y] = gs_morse(aa, diag, col, row, nmax, tol)
[n, n] = size(A);
y = [1, 1, 1, 1];
m = 1;
while m < nmax,
for i = 1: n,
k1 = row(i);
k2 = row(i + 1) - 1;
for k = k1: k2,
y(i) = y(i) + aa(k) * x(col(k));
y(col(k)) = y(col(k)) + aa(k) * diag(i);
end
k2 = k2 + 1;
y(i) = y(i) + aa(k) * diag(i);
end
if (norm(y - x)) < tol
disp(y);
end
m = m + 1;
for i = 1: n,
x(i) = y(i);
end
end
POWER-MORSE:
I was able only to implement the power method but I don't understand how to use the former matrix... so my code for power method is:
function [y, l] = potencia_iterada(A, v)
numiter=100;
eps=1e-10;
x = v(:);
y = x/norm(x);
l = 0;
for k = 1: numiter,
x = A * y;
y = x / norm(x);
l0 = x.' * y;
if abs(l0) < eps
return
end
l = l0;
end
Please anyone can help me for completing these codes or can explain me how can I do that? I really don't understand how to do. Thank you very much
here is my code where i have one error regarding array index out out of bound. plzz help me to rectify it
I = imread('E:\degraded images\village.jpg');
imshow(I)
I = im2double(I);
I = log(1 + I);
M = 2*size(I,1) + 1;
N = 2*size(I,2) + 1;
sigma = 10;
[X, Y] = meshgrid(1:N,1:M);
centerX = ceil(N/2);
centerY = ceil(M/2);
gaussianNumerator = (X - centerX).^2 + (Y - centerY).^2;
H = exp(-gaussianNumerator./(2*sigma.^2));
H = 1 - H;
imshow(H,'InitialMagnification',25)
H = fftshift(H);
If = fft2(I, M, N);
Iout = real(ifft2(H.*If)); ** here the code has error . ??? Error using ==> times Number of array dimensions must match for binary array op.**
H is 2-D while If is 3-D. You can use repmat with H or subset If. I don't know which one is correct for your situation. For instance,
rempat( H, [1, 1, 3 ] ) .* If;
or
H .* If(:,:,ind); % ind is the index of the 2-D array you want to subset
I'm trying my hand at regularized LR, simple with this formulas in matlab:
The cost function:
J(theta) = 1/m*sum((-y_i)*log(h(x_i)-(1-y_i)*log(1-h(x_i))))+(lambda/2*m)*sum(theta_j)
The gradient:
∂J(theta)/∂theta_0 = [(1/m)*(sum((h(x_i)-y_i)*x_j)] if j=0
∂j(theta)/∂theta_n = [(1/m)*(sum((h(x_i)-y_i)*x_j)]+(lambda/m)*(theta_j) if j>1
This is not matlab code is just the formula.
So far I've done this:
function [J, grad] = costFunctionReg(theta, X, y, lambda)
J = 0;
grad = zeros(size(theta));
temp_theta = [];
%cost function
%get the regularization term
for jj = 2:length(theta)
temp_theta(jj) = theta(jj)^2;
end
theta_reg = lambda/(2*m)*sum(temp_theta);
temp_sum =[];
%for the sum in the cost function
for ii =1:m
temp_sum(ii) = -y(ii)*log(sigmoid(theta'*X(ii,:)'))-(1-y(ii))*log(1-sigmoid(theta'*X(ii,:)'));
end
tempo = sum(temp_sum);
J = (1/m)*tempo+theta_reg;
%regulatization
%theta 0
reg_theta0 = 0;
for jj=1:m
reg_theta0(jj) = (sigmoid(theta'*X(m,:)') -y(jj))*X(jj,1)
end
reg_theta0 = (1/m)*sum(reg_theta0)
grad_temp(1) = reg_theta0
%for the rest of thetas
reg_theta = [];
thetas_sum = 0;
for ii=2:size(theta)
for kk =1:m
reg_theta(kk) = (sigmoid(theta'*X(m,:)') - y(kk))*X(kk,ii)
end
thetas_sum(ii) = (1/m)*sum(reg_theta)+(lambda/m)*theta(ii)
reg_theta = []
end
for i=1:size(theta)
if i == 1
grad(i) = grad_temp(i)
else
grad(i) = thetas_sum(i)
end
end
end
And the cost function is giving correct results, but I have no idea why the gradient (one step) is not, the cost gives J = 0.6931 which is correct and the gradient grad = 0.3603 -0.1476 0.0320, which is not, the cost starts from 2 because the parameter theta(1) does not have to be regularized, any help? I guess there is something wrong with the code, but after 4 days I can't see it.Thanks
Vectorized:
function [J, grad] = costFunctionReg(theta, X, y, lambda)
hx = sigmoid(X * theta);
m = length(X);
J = (sum(-y' * log(hx) - (1 - y')*log(1 - hx)) / m) + lambda * sum(theta(2:end).^2) / (2*m);
grad =((hx - y)' * X / m)' + lambda .* theta .* [0; ones(length(theta)-1, 1)] ./ m ;
end
I used more variables, so you could see clearly what comes from the regular formula, and what comes from "the regularization cost added". Additionally, It is a good practice to use "vectorization" instead of loops in Matlab/Octave. By doing this, you guarantee a more optimized solution.
function [J, grad] = costFunctionReg(theta, X, y, lambda)
%Hypotheses
hx = sigmoid(X * theta);
%%The cost without regularization
J_partial = (-y' * log(hx) - (1 - y)' * log(1 - hx)) ./ m;
%%Regularization Cost Added
J_regularization = (lambda/(2*m)) * sum(theta(2:end).^2);
%%Cost when we add regularization
J = J_partial + J_regularization;
%Grad without regularization
grad_partial = (1/m) * (X' * (hx -y));
%%Grad Cost Added
grad_regularization = (lambda/m) .* theta(2:end);
grad_regularization = [0; grad_regularization];
grad = grad_partial + grad_regularization;
Finally got it, after rewriting it again like for the 4th time, this is the correct code:
function [J, grad] = costFunctionReg(theta, X, y, lambda)
J = 0;
grad = zeros(size(theta));
temp_theta = [];
for jj = 2:length(theta)
temp_theta(jj) = theta(jj)^2;
end
theta_reg = lambda/(2*m)*sum(temp_theta);
temp_sum =[];
for ii =1:m
temp_sum(ii) = -y(ii)*log(sigmoid(theta'*X(ii,:)'))-(1-y(ii))*log(1-sigmoid(theta'*X(ii,:)'));
end
tempo = sum(temp_sum);
J = (1/m)*tempo+theta_reg;
%regulatization
%theta 0
reg_theta0 = 0;
for i=1:m
reg_theta0(i) = ((sigmoid(theta'*X(i,:)'))-y(i))*X(i,1)
end
theta_temp(1) = (1/m)*sum(reg_theta0)
grad(1) = theta_temp
sum_thetas = []
thetas_sum = []
for j = 2:size(theta)
for i = 1:m
sum_thetas(i) = ((sigmoid(theta'*X(i,:)'))-y(i))*X(i,j)
end
thetas_sum(j) = (1/m)*sum(sum_thetas)+((lambda/m)*theta(j))
sum_thetas = []
end
for z=2:size(theta)
grad(z) = thetas_sum(z)
end
% =============================================================
end
If its helps anyone, or anyone has any comments on how can I do it better. :)
Here is an answer that eliminates the loops
m = length(y); % number of training examples
predictions = sigmoid(X*theta);
reg_term = (lambda/(2*m)) * sum(theta(2:end).^2);
calcErrors = -y.*log(predictions) - (1 -y).*log(1-predictions);
J = (1/m)*sum(calcErrors)+reg_term;
% prepend a 0 column to our reg_term matrix so we can use simple matrix addition
reg_term = [0 (lambda*theta(2:end)/m)'];
grad = sum(X.*(predictions - y)) / m + reg_term;