Is there a toolbox or available MATLAB function that will allow me to solve the following approximation of stochastic integrals, where z is a Wiener process:
%Lets say n is 100 and dt is 1/252 and k = .1
n = 100;
dt = 1/252;
k = 0.1;
dz = randn(n,1); %get random increments: normal
%dz = 2*(randi(2,n,1)-1.5); % or plus/minus ones : bernoulli
fnt = exp(-k*(n*dt - [0:n-1]*dt))*sqrt(dt)*dz;
Related
I am trying to find log Maximum likelihood estimation for Gaussian distribution, in order to estimate parameters.
I know that Matlab has a built-in function that does this by fitting a Gaussian distribution, but I need to do this with logMLE in order to expand this method later for other distributions.
So here is the log-likelihood function for gaussian dist :
Gaussian Log MLE
And I used this code to estimate the parameters for a set of variables (r) with fminsearch. but my search does not coverage and I don't fully understand where is the problem:
clear
clc
close all
%make random numbers with gaussian dist
r=[2.39587291079469
1.57478022109723
-0.442284350603745
4.39661178526569
7.94034385633171
7.52208574723178
5.80673144943155
-3.11338531920164
6.64267230284774
-2.02996003947964];
% mu=2 sigma=3
%introduce f
f=#(x,r)-(sum((-0.5.*log(2*3.14.*(x(2))))-(((r-(x(2))).^2)./(2.*(x(1))))))
fun = #(x)f(x,r);
% starting point
x0 = [0,0];
[y,fval,exitflag,output] = fminsearch(fun,x0)
f =
#(x,r)-(sum((-0.5.*log(2*3.14.*(x(2))))-(((r-(x(2))).^2)./(2.*(x(1))))))
Exiting: Maximum number of function evaluations has been exceeded
- increase MaxFunEvals option.
Current function value: 477814.233176
y = 1×2
1.0e+-3 *
0.2501 -0.0000
fval = 4.7781e+05 + 1.5708e+01i
exitflag = 0
output =
iterations: 183
funcCount: 400
algorithm: 'Nelder-Mead simplex direct search'
message: 'Exiting: Maximum number of function evaluations has been exceeded↵ - increase MaxFunEvals option.↵ Current function value: 477814.233176 ↵'
Rewrite f as follows:
function y = g(x, r)
n = length(r);
log_part = 0.5.*n.*log(x(2).^2);
sum_part = ((sum(r-x(1))).^2)./(2.*x(2).^2);
y = log_part + sum_part;
end
Use fmincon instead of fminsearch because standard deviation is
always a positif number.
Set standard deviation lower bound to zero 0
The entire code is as follows:
%make random numbers with gaussian dist
r=[2.39587291079469
1.57478022109723
-0.442284350603745
4.39661178526569
7.94034385633171
7.52208574723178
5.80673144943155
-3.11338531920164
6.64267230284774
-2.02996003947964];
% mu=2 sigma=3
fun = #(x)g(x, r);
% starting point
x0 = [0,0];
% borns
lb = [-inf, 0];
ub = [inf, inf];
[y, fval] = fmincon(fun,x0,[],[],[],[],lb,ub, []);
function y = g(x, r)
n = length(r);
log_part = 0.5.*n.*log(x(2).^2);
sum_part = ((sum(r-x(1))).^2)./(2.*x(2).^2);
y = log_part + sum_part;
end
Solution
y = [3.0693 0.0000]
For better estimation use mle() directly
The code is quiet simple:
y = mle(r,'distribution','normal')
Solution
y = [3.0693 3.8056]
The formula for the discrete double Fourier series that I'm attempting to code in MATLAB is:
The coefficient in front of the trigonometric sum (Fourier amplitude) is what I'm trying to extract from the fitting of the data through the double Fourier series seen above. Using my current code, the original function is not reconstructed, therefore my coefficients cannot be correct. I'm not certain if this is of any significance or insight, but the second term for the A coefficients (Akn(1))) is 13 orders of magnitude larger than any other coefficient.
Any suggestions, modifications, or comments about my program would be greatly appreciated.
%data = csvread('digitized_plot_data.csv',1);
%xdata = data(:,1);
%ydata = data(:,2);
%x0 = xdata(1);
lambda = 20; %km
tau = 20; %s
vs = 7.6; %k/s (velocity of CHAMP satellite)
L = 4; %S
% Number of terms to use:
N = 100;
% set up matrices:
M = zeros(length(xdata),1+2*N);
M(:,1) = 1;
for k=1:N
for n=1:N %error using *, inner matrix dimensions must agree...
M(:,2*n) = cos(2*pi/lambda*k*vs*xdata).*cos(2*pi/tau*n*xdata);
M(:,2*n+1) = sin(2*pi/lambda*k*vs*xdata).*sin(2*pi/tau*n*xdata);
end
end
C = M\ydata;
%least squares coefficients:
A0 = C(1);
Akn = C(2:2:end);
Bkn = C(3:2:end);
% reconstruct original function values (verification check):
y = A0;
for k=1:length(Akn)
y = y + Akn(k)*cos(2*pi/lambda*k*vs*xdata).*cos(2*pi/tau*n*xdata) + Bkn(k)*sin(2*pi/lambda*k*vs*xdata).*sin(2*pi/tau*n*xdata);
end
% plotting
hold on
plot(xdata,ydata,'ko')
plot(xdata,yk,'b--')
legend('Data','Least Squares','location','northeast')
xlabel('Centered Time Event [s]'); ylabel('J[\muA/m^2]'); title('Single FAC Event (50 Hz)')
I'm new with Matlab and Machine Learning and I tried to make a gradient descent function without using matrix.
m is the number of example on my training set
n is the number of feature for each example
The function gradientDescentMulti takes 5 arguments:
X mxn Matrix
y m-dimensional vector
theta : n-dimensional vector
alpha : a real number
nb_iters : a real number
I already have a solution using matrix multiplication
function theta = gradientDescentMulti(X, y, theta, alpha, num_iters)
for iter = 1:num_iters
gradJ = 1/m * (X'*X*theta - X'*y);
theta = theta - alpha * gradJ;
end
end
The result after iterations:
theta =
1.0e+05 *
3.3430
1.0009
0.0367
But now, I tried to do the same without matrix multiplication, this is the function:
function theta = gradientDescentMulti(X, y, theta, alpha, num_iters)
m = length(y); % number of training examples
n = size(X, 2); % number of features
for iter = 1:num_iters
new_theta = zeros(1, n);
%// for each feature, found the new theta
for t = 1:n
S = 0;
for example = 1:m
h = 0;
for example_feature = 1:n
h = h + (theta(example_feature) * X(example, example_feature));
end
S = S + ((h - y(example)) * X(example, n)); %// Sum each feature for this example
end
new_theta(t) = theta(t) - alpha * (1/m) * S; %// Calculate new theta for this example
end
%// only at the end of the function, update all theta simultaneously
theta = new_theta'; %// Transpose new_theta (horizontal vector) to theta (vertical vector)
end
end
The result, all the theta are the same :/
theta =
1.0e+04 *
3.5374
3.5374
3.5374
If you look at the gradient update rule, it may be more efficient to actually compute the hypothesis of all of your training examples first, then subtract this with the ground truth value of each training example and store these into an array or vector. Once you do this, you can then compute the update rule very easily. To me, it doesn't appear that you're doing this in your code.
As such, I rewrote the code, but I have a separate array that stores the difference in the hypothesis of each training example and ground truth value. Once I do this, I compute the update rule for each feature separately:
for iter = 1 : num_iters
%// Compute hypothesis differences with ground truth first
h = zeros(1, m);
for t = 1 : m
%// Compute hypothesis
for tt = 1 : n
h(t) = h(t) + theta(tt)*X(t,tt);
end
%// Compute difference between hypothesis and ground truth
h(t) = h(t) - y(t);
end
%// Now update parameters
new_theta = zeros(1, n);
%// for each feature, find the new theta
for tt = 1 : n
S = 0;
%// For each sample, compute products of hypothesis difference
%// and the right feature of the sample and accumulate
for t = 1 : m
S = S + h(t)*X(t,tt);
end
%// Compute gradient descent step
new_theta(tt) = theta(tt) - (alpha/m)*S;
end
theta = new_theta'; %// Transpose new_theta (horizontal vector) to theta (vertical vector)
end
When I do this, I get the same answers as using the matrix formulation.
How can I calculate the mean and median of a Gaussian Mixture Model with three components like the following parameters in MATLAB:
Priors[0.4,0.25,0.34]
Centers [0.44;0.74;0.05]
Co-variance [0.03,0.18,0.03]
Thanks
Here is the MATLAB code for calculating mean and median of a Gaussian Mixture Model (GMM):
Mean Calculation for N GMMs:
for i = 1:N
mu = center{i};
p = prior{i};
mean_mix(i) = mu(1)*p(1) + mu(2)*p(2) + mu(3)*p(3);
end
Median Calculation for N GMMs:
median = zeros(N,1);
for i = 1:N
for j = 2:N
if (fix(trapz(x(1:j), gmm_pdfs(1:j,i))*100) == 50);
median(i) = x(j);
end
end
end
Note: gmm_pdfs are the evaluated pdfs against x.
How to make a certain pulse function with Matlab, like this function on the graph:
I have the function which generates a pulse, but I want to generate as the graph above
clear all; clc; close all
% Step 1: Calculate Fourier Transform Coefficients
N = 101; % Number of coefficients
D = [];
% Negative Coefficients
for n = -(N-1)/2 : -1
D = [D (1/(n*pi))*sin(n*pi/2)];
end
% Append D0
D0 = 0.5; D = [D D0];
% Positive Coefficients
for n = 1 : (N-1)/2
D = [D (1/(n*pi))*sin(n*pi/2)];
end
% Step 2: Calculate the series terms
series = [];
f0 = 1; % Frequency in Hz
T0 = 1/f0; % Period in seconds
dt = T0/100; % Sample interval
t = -T0 : dt : T0;
for n = -(N-1)/2 : (N-1)/2
r = n + (N+1)/2;
temp = D(r)*exp(j*n*2*pi*f0*t);
series = [series; temp];
end
% Step 3: Summation of the series terms
w = 8*sum(series);
% Step 4: Plot the approximated signal
plot(t,w); grid
xlabel('Time, seconds'); ylabel('signal, s(t)')
title('An approximation of a rectangular singal using Fourier series')'
You need to calculate the continuous Fourier series for your waveform, then modify the coefficients in D to match.
You can find information on general Fourier series elsewhere on the internet.