Given a signal x(t), we need to find the symmetrical with respect to the Y-axis signal, x(-t)
If it's helpful to you, here's how my code works thus far:
t = [-5:0.01:5];
wt = (t>=0)&(t<=1);
r = #(t) t/5;
x = r(t).*wt;
%reflection - HERE IS WHERE I AM STUCK, basically looking for v(t) = x(-t)
%Shift by 2
y = v(t-2);
%The rest of the program - printing plots basically
I have tried using these:
v = x(t(1:end));
v = x(t(end:-1:1));
v = x(fliplr(t));
But it's not correct, since I get the error Array indices must be positive integers or logical values. as expected. Any ideas?
One solution is that you consider something like this:
x=signal; % with length 2*N+1 and symmetric
t= -N:N;
now, consider you want the value of index -2.
x(find(t==-2))
For ramp signal as an instance:
signal=[r(end:-1:1) 0 r]
with this assumption that r is a row-base vector and N length.
You should first define the signal function and then sampling it, not the other way round.
For instance, I here define a signal s(t) which is a windowed ramp:
s = #(t) t.*((t>=0)&(t<=1))
Then, I can find the samples for the signal and its symmetrical:
t = -5:0.01:5;
plot(t,s(t),t,s(-t))
What worked for me was defining a reflect function as follows:
function val = reflect(t)
val = -t;
end
And then using it with the shifting function in order to achieve my goal.
Given any sampled signal in an arbitrary time interval:
t = [-5.003:0.01:10];
x = randn(size(t));
You can reflect x around t=0 with:
t = -flip(t);
x = flip(x);
Note that in the example above, t=0 is not sampled. This is not necessary for this method.
Related
I want to find the maximum value using the second derivative of the the expression when x is between 0 and 1. In other words I am taking the derivative of cox(x^2) twice to get the second derivative resulting in - 2*sin(x^2) - 4*x^2*cos(x^2), then I want to evaluate this second derivative at x = 0 to x = 1, and display the maximum value of the populated values.
I have:
syms x
f = cos(x^2);
secondD = diff(diff(f));
for i = 0:1
y = max(secondD(i))
end
Can someone help?
You can do it easily by subs and double:
syms x
f = cos(x^2);
secondD = diff(diff(f));
% instead of the for loop
epsilon = 0.01;
specified_range = 0:epsilon:1;
[max_val, max_ind] = max(double(subs(secondD, specified_range)));
Please note that it is a numerical approach to find the maximum and the returned answer is not completely correct all the time. However, by increasing the epsilon, you can expect a better result in general (again in some cases it is not completely correct).
I have this task to create a script that acts similarly to normcdf on matlab.
x=linspace(-5,5,1000); %values for x
p= 1/sqrt(2*pi) * exp((-x.^2)/2); % THE PDF for the standard normal
t=cumtrapz(x,p); % the CDF for the standard normal distribution
plot(x,t); %shows the graph of the CDF
The problem is when the t values are assigned to 1:1000 instead of -5:5 in increments. I want to know how to assign the correct x values, that is -5:5,1000 to the t values output? such as when I do t(n) I get the same result as normcdf(n).
Just to clarify: the problem is I cannot simply say t(-5) and get result =1 as I would in normcdf(1) because the cumtrapz calculated values are assigned to x=1:1000 instead of -5 to 5.
Updated answer
Ok, having read your comment; here is how to do what you want:
x = linspace(-5,5,1000);
p = 1/sqrt(2*pi) * exp((-x.^2)/2);
cdf = cumtrapz(x,p);
q = 3; % Query point
disp(normcdf(q)) % For reference
[~,I] = min(abs(x-q)); % Find closest index
disp(cdf(I)) % Show the value
Sadly, there is no matlab syntax which will do this nicely in one line, but if you abstract finding the closest index into a different function, you can do this:
cdf(findClosest(x,q))
function I = findClosest(x,q)
if q>max(x) || q<min(x)
warning('q outside the range of x');
end
[~,I] = min(abs(x-q));
end
Also; if you are certain that the exact value of the query point q exists in x, you can just do
cdf(x==q);
But beware of floating point errors though. You may think that a certain range outght to contain a certain value, but little did you know it was different by a tiny roundoff erorr. You can see that in action for example here:
x1 = linspace(0,1,1000); % Range
x2 = asin(sin(x1)); % Ought to be the same thing
plot((x1-x2)/eps); grid on; % But they differ by rougly 1 unit of machine precision
Old answer
As far as I can tell, running your code does reproduce the result of normcdf(x) well... If you want to do exactly what normcdf does them use erfc.
close all; clear; clc;
x = linspace(-5,5,1000);
cdf = normcdf(x); % Result of normcdf for comparison
%% 1 Trapezoidal integration of normal pd
p = 1/sqrt(2*pi) * exp((-x.^2)/2);
cdf1 = cumtrapz(x,p);
%% 2 But error function IS the integral of the normal pd
cdf2 = (1+erf(x/sqrt(2)))/2;
%% 3 Or, even better, use the error function complement (works better for large negative x)
cdf3 = erfc(-x/sqrt(2))/2;
fprintf('1: Mean error = %.2d\n',mean(abs(cdf1-cdf)));
fprintf('2: Mean error = %.2d\n',mean(abs(cdf2-cdf)));
fprintf('3: Mean error = %.2d\n',mean(abs(cdf3-cdf)));
plot(x,cdf1,x,cdf2,x,cdf3,x,cdf,'k--');
This gives me
1: Mean error = 7.83e-07
2: Mean error = 1.41e-17
3: Mean error = 00 <- Because that is literally what normcdf is doing
If your goal is not not to use predefined matlab funcitons, but instead to calculate the result numerically (i.e. calculate the error function) then it's an interesting challange which you can read about for example here or in this stats stackexchange post. Just as an example, the following piece of code calculates the error function by implementing eq. 2 form the first link:
nerf = #(x,n) (-1)^n*2/sqrt(pi)*x.^(2*n+1)./factorial(n)/(2*n+1);
figure(1); hold on;
temp = zeros(size(x)); p =[];
for n = 0:20
temp = temp + nerf(x/sqrt(2),n);
if~mod(n,3)
p(end+1) = plot(x,(1+temp)/2);
end
end
ylim([-1,2]);
title('\Sigma_{n=0}^{inf} ( 2/sqrt(pi) ) \times ( (-1)^n x^{2*n+1} ) \div ( n! (2*n+1) )');
p(end+1) = plot(x,cdf,'k--');
legend(p,'n = 0','\Sigma_{n} 0->3','\Sigma_{n} 0->6','\Sigma_{n} 0->9',...
'\Sigma_{n} 0->12','\Sigma_{n} 0->15','\Sigma_{n} 0->18','normcdf(x)',...
'location','southeast');
grid on; box on;
xlabel('x'); ylabel('norm. cdf approximations');
Marcin's answer suggests a way to find the nearest sample point. It is easier, IMO, to interpolate. Given x and t as defined in the question,
interp1(x,t,n)
returns the estimated value of the CDF at x==n, for whatever value of n. But note that, for values outside the computed range, it will extrapolate and produce unreliable values.
You can define an anonymous function that works like normcdf:
my_normcdf = #(n)interp1(x,t,n);
my_normcdf(-5)
Try replacing x with 0.01 when you call cumtrapz. You can either use a vector or a scalar spacing for cumtrapz (https://www.mathworks.com/help/matlab/ref/cumtrapz.html), and this might solve your problem. Also, have you checked the original x-values? Is the problem with linspace (i.e. you are not getting the correct x vector), or with cumtrapz?
I need to produce a signal x=-2*cos(100*pi*n)+2*cos(140*pi*n)+cos(200*pi*n)
So I put it like this :
N=1024;
for n=1:N
x=-2*cos(100*pi*n)+2*cos(140*pi*n)+cos(200*pi*n);
end
But What I get is that the result keeps giving out 1
I tried to test each values according to each n, and I get the same results for any n
For example -2*cos(100*pi*n) with n=1 has to be -1.393310473. Instead of that, Matlab gave the result -2 for it and it always gave -2 for any n
I don't know how to fix it, so I hope someone could help me out! Thank you!
Not sure where you get the idea that -2*cos(100*pi) should be anything other than -2. Maybe you are not aware that Matlab works in radians?
Look at your expression. Each term can be factored to contain 2*pi*(an integer). And you should know that cos(2*pi*(an integer)) = 1.
So the results are exactly as expected.
What you are seeing is basically what happens when you under-sample a waveform. You may know that the Nyquist criterion says that you need to have a sampling rate that is at least two times greater than the highest frequency component present; but in your case, you are sampling one point every 50, 70, 100 complete cycles. So you are "far beyond Nyquist". And that can only be solved by sampling more closely.
For example, you could do:
t = linspace(0, 1, 1024); % sample the waveform 1024 times between 0 and 1
f1 = 50;
f2 = 70;
f3 = 100;
signal = -2*cos(2*pi*f1*t) + 2*cos(2*pi*f2*t) + cos(2*pi*f3*t);
figure; plot(t, signal)
I think you are using degrees when you are doing your calculations, so do this:
n = 1:1024
x=-2*cosd(100*pi*n)+2*cosd(140*pi*n)+cosd(200*pi*n);
cosd uses degrees instead of radians. Radians is the default for cos so matlab has a separate function when degree input is used. For me this gave:
-2*cosd(100*pi*1) = -1.3933
The first term that I got using:
x=-2*cosd(100*pi*1)+2*cosd(140*pi*1)+cosd(200*pi*1)
x = -1.0693
Also notice that I defined n as n = 1:1024; this will give all integers from 1,2,...,1024,
there is no need to use a for loop since many of Matlab's built in functions are vectorized. Meaning you can just input a vector and it will calculate the function for every element in the vector.
I’m currently a Physics student and for several weeks have been compiling data related to ‘Quantum Entanglement’. I’ve now got to a point where I have to plot my data (which should resemble a cos² graph - and does) to a sort of “best fit” cos² graph. The lab script says the following:
A more precise determination of the visibility V (this is basically how 'clean' the data is) follows from the best fit to the measured data using the function:
f(b) = A/2[1-Vsin(b-b(center)/P)]
Granted this probably doesn’t mean much out of context, but essentially A is the amplitude, b is an angle and P is the periodicity. Hence this is also a “wave” like the experimental data I have found.
From this I understand, as previously mentioned, I am making a “best fit” curve. However, I have been told that this isn’t possible with Excel and that the best approach is Matlab.
I know intermediate JavaScript but do not know Matlab and was hoping for some direction.
Is there a tutorial I can read for this? Is it possible for someone to go through it with me? I really have no idea what it entails, so any feed back would be greatly appreciated.
Thanks a lot!
Initial steps
I guess we should begin by getting a representation in Matlab of the function that you're trying to model. A direct translation of your formula looks like this:
function y = targetfunction(A,V,P,bc,b)
y = (A/2) * (1 - V * sin((b-bc) / P));
end
Getting hold of the data
My next step is going to be to generate some data to work with (you'll use your own data, naturally). So here's a function that generates some noisy data. Notice that I've supplied some values for the parameters.
function [y b] = generateData(npoints,noise)
A = 2;
V = 1;
P = 0.7;
bc = 0;
b = 2 * pi * rand(npoints,1);
y = targetfunction(A,V,P,bc,b) + noise * randn(npoints,1);
end
The function rand generates random points on the interval [0,1], and I multiplied those by 2*pi to get points randomly on the interval [0, 2*pi]. I then applied the target function at those points, and added a bit of noise (the function randn generates normally distributed random variables).
Fitting parameters
The most complicated function is the one that fits a model to your data. For this I use the function fminunc, which does unconstrained minimization. The routine looks like this:
function [A V P bc] = bestfit(y,b)
x0(1) = 1; %# A
x0(2) = 1; %# V
x0(3) = 0.5; %# P
x0(4) = 0; %# bc
f = #(x) norm(y - targetfunction(x(1),x(2),x(3),x(4),b));
x = fminunc(f,x0);
A = x(1);
V = x(2);
P = x(3);
bc = x(4);
end
Let's go through line by line. First, I define the function f that I want to minimize. This isn't too hard. To minimize a function in Matlab, it needs to take a single vector as a parameter. Therefore we have to pack our four parameters into a vector, which I do in the first four lines. I used values that are close, but not the same, as the ones that I used to generate the data.
Then I define the function I want to minimize. It takes a single argument x, which it unpacks and feeds to the targetfunction, along with the points b in our dataset. Hopefully these are close to y. We measure how far they are from y by subtracting from y and applying the function norm, which squares every component, adds them up and takes the square root (i.e. it computes the root mean square error).
Then I call fminunc with our function to be minimized, and the initial guess for the parameters. This uses an internal routine to find the closest match for each of the parameters, and returns them in the vector x.
Finally, I unpack the parameters from the vector x.
Putting it all together
We now have all the components we need, so we just want one final function to tie them together. Here it is:
function master
%# Generate some data (you should read in your own data here)
[f b] = generateData(1000,1);
%# Find the best fitting parameters
[A V P bc] = bestfit(f,b);
%# Print them to the screen
fprintf('A = %f\n',A)
fprintf('V = %f\n',V)
fprintf('P = %f\n',P)
fprintf('bc = %f\n',bc)
%# Make plots of the data and the function we have fitted
plot(b,f,'.');
hold on
plot(sort(b),targetfunction(A,V,P,bc,sort(b)),'r','LineWidth',2)
end
If I run this function, I see this being printed to the screen:
>> master
Local minimum found.
Optimization completed because the size of the gradient is less than
the default value of the function tolerance.
A = 1.991727
V = 0.979819
P = 0.695265
bc = 0.067431
And the following plot appears:
That fit looks good enough to me. Let me know if you have any questions about anything I've done here.
I am a bit surprised as you mention f(a) and your function does not contain an a, but in general, suppose you want to plot f(x) = cos(x)^2
First determine for which values of x you want to make a plot, for example
xmin = 0;
stepsize = 1/100;
xmax = 6.5;
x = xmin:stepsize:xmax;
y = cos(x).^2;
plot(x,y)
However, note that this approach works just as well in excel, you just have to do some work to get your x values and function in the right cells.
I have a MATLAB routine with one rather obvious bottleneck. I've profiled the function, with the result that 2/3 of the computing time is used in the function levels:
The function levels takes a matrix of floats and splits each column into nLevels buckets, returning a matrix of the same size as the input, with each entry replaced by the number of the bucket it falls into.
To do this I use the quantile function to get the bucket limits, and a loop to assign the entries to buckets. Here's my implementation:
function [Y q] = levels(X,nLevels)
% "Assign each of the elements of X to an integer-valued level"
p = linspace(0, 1.0, nLevels+1);
q = quantile(X,p);
if isvector(q)
q=transpose(q);
end
Y = zeros(size(X));
for i = 1:nLevels
% "The variables g and l indicate the entries that are respectively greater than
% or less than the relevant bucket limits. The line Y(g & l) = i is assigning the
% value i to any element that falls in this bucket."
if i ~= nLevels % "The default; doesnt include upper bound"
g = bsxfun(#ge,X,q(i,:));
l = bsxfun(#lt,X,q(i+1,:));
else % "For the final level we include the upper bound"
g = bsxfun(#ge,X,q(i,:));
l = bsxfun(#le,X,q(i+1,:));
end
Y(g & l) = i;
end
Is there anything I can do to speed this up? Can the code be vectorized?
If I understand correctly, you want to know how many items fell in each bucket.
Use:
n = hist(Y,nbins)
Though I am not sure that it will help in the speedup. It is just cleaner this way.
Edit : Following the comment:
You can use the second output parameter of histc
[n,bin] = histc(...) also returns an index matrix bin. If x is a vector, n(k) = >sum(bin==k). bin is zero for out of range values. If x is an M-by-N matrix, then
How About this
function [Y q] = levels(X,nLevels)
p = linspace(0, 1.0, nLevels+1);
q = quantile(X,p);
Y = zeros(size(X));
for i = 1:numel(q)-1
Y = Y+ X>=q(i);
end
This results in the following:
>>X = [3 1 4 6 7 2];
>>[Y, q] = levels(X,2)
Y =
1 1 2 2 2 1
q =
1 3.5 7
You could also modify the logic line to ensure values are less than the start of the next bin. However, I don't think it is necessary.
I think you shoud use histc
[~,Y] = histc(X,q)
As you can see in matlab's doc:
Description
n = histc(x,edges) counts the number of values in vector x that fall
between the elements in the edges vector (which must contain
monotonically nondecreasing values). n is a length(edges) vector
containing these counts. No elements of x can be complex.
I made a couple of refinements (including one inspired by Aero Engy in another answer) that have resulted in some improvements. To test them out, I created a random matrix of a million rows and 100 columns to run the improved functions on:
>> x = randn(1000000,100);
First, I ran my unmodified code, with the following results:
Note that of the 40 seconds, around 14 of them are spent computing the quantiles - I can't expect to improve this part of the routine (I assume that Mathworks have already optimized it, though I guess that to assume makes an...)
Next, I modified the routine to the following, which should be faster and has the advantage of being fewer lines as well!
function [Y q] = levels(X,nLevels)
p = linspace(0, 1.0, nLevels+1);
q = quantile(X,p);
if isvector(q), q = transpose(q); end
Y = ones(size(X));
for i = 2:nLevels
Y = Y + bsxfun(#ge,X,q(i,:));
end
The profiling results with this code are:
So it is 15 seconds faster, which represents a 150% speedup of the portion of code that is mine, rather than MathWorks.
Finally, following a suggestion of Andrey (again in another answer) I modified the code to use the second output of the histc function, which assigns entries to bins. It doesn't treat the columns independently, so I had to loop over the columns manually, but it seems to be performing really well. Here's the code:
function [Y q] = levels(X,nLevels)
p = linspace(0,1,nLevels+1);
q = quantile(X,p);
if isvector(q), q = transpose(q); end
q(end,:) = 2 * q(end,:);
Y = zeros(size(X));
for k = 1:size(X,2)
[junk Y(:,k)] = histc(X(:,k),q(:,k));
end
And the profiling results:
We now spend only 4.3 seconds in codes outside the quantile function, which is around a 500% speedup over what I wrote originally. I've spent a bit of time writing this answer because I think it's turned into a nice example of how you can use the MATLAB profiler and StackExchange in combination to get much better performance from your code.
I'm happy with this result, although of course I'll continue to be pleased to hear other answers. At this stage the main performance increase will come from increasing the performance of the part of the code that currently calls quantile. I can't see how to do this immediately, but maybe someone else here can. Thanks again!
You can sort the columns and divide+round the inverse indexes:
function Y = levels(X,nLevels)
% "Assign each of the elements of X to an integer-valued level"
[S,IX]=sort(X);
[grid1,grid2]=ndgrid(1:size(IX,1),1:size(IX,2));
invIX=zeros(size(X));
invIX(sub2ind(size(X),IX(:),grid2(:)))=grid1;
Y=ceil(invIX/size(X,1)*nLevels);
Or you can use tiedrank:
function Y = levels(X,nLevels)
% "Assign each of the elements of X to an integer-valued level"
R=tiedrank(X);
Y=ceil(R/size(X,1)*nLevels);
Surprisingly, both these solutions are slightly slower than the quantile+histc solution.