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I am struggling to plot the PDF and CDF graphs of where
Sn=X1+X2+X3+....+Xn
using central limit theorem where n = 1; 2; 3; 4; 5; 10; 20; 40
I am taking Xi to be a uniform continuous random variable for values between (0,3).
Here is what i have done so far -
close all
%different sizes of input X
%N=[1 5 10 50];
N = [1 2 3 4 5 10 20 40];
%interval (1,6) for random variables
a=0;
b=3;
%to store sum of differnet sizes of input
for i=1:length(N)
%generates uniform random numbers in the interval
X = a + (b-a).*rand(N(i),1);
S=zeros(1,length(X));
S=cumsum(X);
cd=cdf('Uniform',S,0,3);
plot(cd);
hold on;
end
legend('n=1','n=2','n=3','n=4','n=5','n=10','n=20','n=40');
title('CDF PLOT')
figure;
for i=1:length(N)
%generates uniform random numbers in the interval
X = a + (b-a).*rand(N(i),1);
S=zeros(1,length(X));
S=cumsum(X);
cd=pdf('Uniform',S,0,3);
plot(cd);
hold on;
end
legend('n=1','n=2','n=3','n=4','n=5','n=10','n=20','n=40');
title('PDF PLOT')
My output is nowhere near what I am expecting any help is much appreciated.
This can be done with vectorization using rand() and cumsum().
For example, the code below generates 40 replications of 10000 samples of a Uniform(0,3) distribution and stores in X. To meet the Central Limit Theorem (CLT) assumptions, they are independent and identically distributed (i.i.d.). Then cumsum() transforms this into 10000 copies of the Sn = X1 + X2 + ... where the first row is n = 10000copies of Sn = X1, the 5th row is n copies of S_5 = X1 + X2 + X3 + X4 + X5. The last row is n copies of S_40.
% MATLAB R2019a
% Setup
N = [1:5 10 20 40]; % values of n we are interested in
LB = 0; % lowerbound for X ~ Uniform(LB,UB)
UB = 3; % upperbound for X ~ Uniform(LB,UB)
n = 10000; % Number of copies (samples) for each random variable
% Generate random variates
X = LB + (UB - LB)*rand(max(N),n); % X ~ Uniform(LB,UB) (i.i.d.)
Sn = cumsum(X);
You can see from the image that the n = 2 case, the sum is indeed a Triangular(0,3,6) distribution. For the n = 40 case, the sum is approximately Normally distributed (Gaussian) with mean 60 (40*mean(X) = 40*1.5 = 60). This shows the convergence in distribution for both the probability density function (PDF) and the cumulative distribution function (CDF).
Note: The CLT is often stated with convergence in distribution to a Normal distribution with zero mean as it has been shifted. Shifting the results by subtracting mean(Sn) = n*mean(X) = n*0.5*(LB+UB) from Sn gets this done.
Code below isn't the gold standard but it produced the image.
figure
s(11) = subplot(6,2,1) % n = 1
histogram(Sn(1,:),'Normalization','pdf')
title(s(11),'n = 1')
s(12) = subplot(6,2,2)
cdfplot(Sn(1,:))
title(s(12),'n = 1')
s(21) = subplot(6,2,3) % n = 2
histogram(Sn(2,:),'Normalization','pdf')
title(s(21),'n = 2')
s(22) = subplot(6,2,4)
cdfplot(Sn(2,:))
title(s(22),'n = 2')
s(31) = subplot(6,2,5) % n = 5
histogram(Sn(5,:),'Normalization','pdf')
title(s(31),'n = 5')
s(32) = subplot(6,2,6)
cdfplot(Sn(5,:))
title(s(32),'n = 5')
s(41) = subplot(6,2,7) % n = 10
histogram(Sn(10,:),'Normalization','pdf')
title(s(41),'n = 10')
s(42) = subplot(6,2,8)
cdfplot(Sn(10,:))
title(s(42),'n = 10')
s(51) = subplot(6,2,9) % n = 20
histogram(Sn(20,:),'Normalization','pdf')
title(s(51),'n = 20')
s(52) = subplot(6,2,10)
cdfplot(Sn(20,:))
title(s(52),'n = 20')
s(61) = subplot(6,2,11) % n = 40
histogram(Sn(40,:),'Normalization','pdf')
title(s(61),'n = 40')
s(62) = subplot(6,2,12)
cdfplot(Sn(40,:))
title(s(62),'n = 40')
sgtitle({'PDF (left) and CDF (right) for Sn with n \in \{1, 2, 5, 10, 20, 40\}';'note different axis scales'})
for tgt = [11:10:61 12:10:62]
xlabel(s(tgt),'Sn')
if rem(tgt,2) == 1
ylabel(s(tgt),'pdf')
else % rem(tgt,2) == 0
ylabel(s(tgt),'cdf')
end
end
Key functions used for plot: histogram() from base MATLAB and cdfplot() from the Statistics toolbox. Note this could be done manually without requiring the Statistics toolbox with a few lines to obtain the cdf and then just calling plot().
There was some concern in comments over the variance of Sn.
Note the variance of Sn is given by (n/12)*(UB-LB)^2 (derivation below). Monte Carlo simulation shows our samples of Sn do have the correct variance; indeed, it converges to this as n gets larger. Simply call var(Sn(40,:)).
% with n = 10000
var(Sn(40,:)) % var(S_40) = 30 (will vary slightly depending on random seed)
(40/12)*((UB-LB)^2) % 29.9505
You can see the convergence is very good by S_40:
step = 0.01;
Domain = 40:step:80;
mu = 40*(LB+UB)/2;
sigma = sqrt((40/12)*((UB-LB)^2));
figure, hold on
histogram(Sn(40,:),'Normalization','pdf')
plot(Domain,normpdf(Domain,mu,sigma),'r-','LineWidth',1.4)
ylabel('pdf')
xlabel('S_n')
Derivation of mean and variance for Sn:
For the expectation (mean), the second equality holds by linearity of expectation. The third equality holds since X_i are identically distributed.
The discrete version of this is posted here.
I have a vector containining the speed of 200 walks:
a = 50;
b = 100;
speed = (b-a).*rand(200,1) + a;
and another vector contaning the number of steps for each walk:
a = 8;
b = 100;
steps = (b-a).*rand(200,1) + a;
I would like to create a histogram plot with on the x axis the speeds and on the y axes the sum of the steps of each speed.
What I do is the following but I guess there is a more elegant way to do this:
unique_speed = unique(speed);
y_unique_speed = zeros(size(unique_speed));
for i = 1 : numel(unique_speed)
speed_idx = unique_speed(i);
idx = speed==speed_idx ;
y_unique_speed (i) = sum(steps (idx));
end
First you need to discretize your speed variable. Unlike in the other answer, the following allows you to choose arbitrary step size, e.g. I've chosen 1.5. Note that the last bin edge should be strictly more than the maximum data point, hence I used max(speed)+binstep. You can then use histc to determine which bin each pairs falls into, and accumarray to determine total number of steps in each bin. Finally, use bar to plot.
binstep = 1.5;
binranges = (min(speed):binstep:max(speed)+binstep)';
[~, ind] = histc(speed, binranges);
bincounts = accumarray(ind, steps, size(binranges));
hFig = figure(); axh = axes('Parent', hFig); hold(axh, 'all'); grid(axh, 'on');
bar(axh, binranges, bincounts); axis(axh, 'tight');
First, bin the speed data to discrete values:
sspeed = ceil(speed);
and then accumulate the various step sizes to each bin:
numsteps = accumarray(sspeed, steps);
plot(numsteps)
I'm trying to get volume-time graph of .wav file. First, I recorded sound (patient exhalations) via android as .wav file, but when I read this .wav file in MATLAB it has negative values. What is the meaning of negative values? Second, MATLAB experts could you please check if the code below does the same as written in my comments? Also another question. Y = fft(WindowArray);
p = abs(Y).^2;
I took the power of values returned from fft...is that correct and what is the goal of this step??
[data, fs] = wavread('newF2');
% read exhalation audio wav file (1 channel, mono)
% frequency is 44100 HZ
% windows of 0.1 s and overlap of 0.05 seconds
WINDOW_SIZE = fs*0.1; %4410 = fs*0.1
array_size = length(data); % array size of data
numOfPeaks = (array_size/(WINDOW_SIZE/2)) - 1;
step = floor(WINDOW_SIZE/2); %step size used in loop
transformed = data;
start =1;
k = 1;
t = 1;
g = 1;
o = 1;
% performing fft on each window and finding the peak of windows
while(((start+WINDOW_SIZE)-1)<=array_size)
j=1;
i =start;
while(j<=WINDOW_SIZE)
WindowArray(j) = transformed(i);
j = j+1;
i = i +1;
end
Y = fft(WindowArray);
p = abs(Y).^2; %power
[a, b] = max(abs(Y)); % find max a and its indices b
[m, i] = max(p); %the maximum of the power m and its indices i
maximum(g) = m;
index(t) = i;
power(o) = a;
indexP(g) = b;
start = start + step;
k = k+1;
t = t+1;
g = g+1;
o=o+1;
end
% low pass filter
% filtering noise: ignor frequencies that are less than 5% of maximum frequency
for u=1:length(maximum)
M = max(maximum); %highest value in the array
Accept = 0.05* M;
if(maximum(u) > Accept)
maximum = maximum(u:length(maximum));
break;
end
end
% preparing the time of the graph,
% Location of the Peak flow rates are estimated
TotalTime = (numOfPeaks * 0.1);
time1 = [0:0.1:TotalTime];
if(length(maximum) > ceil(numOfPeaks));
maximum = maximum(1:ceil(numOfPeaks));
end
time = time1(1:length(maximum));
% plotting frequency-time graph
figure(1);
plot(time, maximum);
ylabel('Frequency');
xlabel('Time (in seconds)');
% plotting volume-time graph
figure(2);
plot(time, cumsum(maximum)); % integration over time to get volume
ylabel('Volume');
xlabel('Time (in seconds)');
(I only answer the part of the question which I understood)
Per default Matlab normalizes the audio wave to - 1...1 range. Use the native option if you want the integer data.
First, in your code it should be p = abs(Y)**2, this is the proper way to square the values returned from the FFT. The reason why you take the absolute value of the FFT return values is because those number are complex numbers with a Real and Imaginary part, therefore the absolute value (or modulus) of an imaginary number is the magnitude of that number. The goal of taking the power could be for potentially obtaining an RMS value (root mean squared) of your overall amplitude values, but you could also have something else in mind. When you say volume-time I assume you want decibels, so try something like this:
def plot_signal(file_name):
sampFreq, snd = wavfile.read(file_name)
snd = snd / (2.**15) #Convert sound array to floating point values
#Floating point values range from -1 to 1
s1 = snd[:,0] #left channel
s2 = snd[:,1] #right channel
timeArray = arange(0, len(snd), 1)
timeArray = timeArray / sampFreq
timeArray = timeArray * 1000 #scale to milliseconds
timeArray2 = arange(0, len(snd), 1)
timeArray2 = timeArray2 / sampFreq
timeArray2 = timeArray2 * 1000 #scale to milliseconds
n = len(s1)
p = fft(s1) # take the fourier transform
m = len(s2)
p2 = fft(s2)
nUniquePts = ceil((n+1)/2.0)
p = p[0:nUniquePts]
p = abs(p)
mUniquePts = ceil((m+1)/2.0)
p2 = p2[0:mUniquePts]
p2 = abs(p2)
'''
Left Channel
'''
p = p / float(n) # scale by the number of points so that
# the magnitude does not depend on the length
# of the signal or on its sampling frequency
p = p**2 # square it to get the power
# multiply by two (see technical document for details)
# odd nfft excludes Nyquist point
if n % 2 > 0: # we've got odd number of points fft
p[1:len(p)] = p[1:len(p)] * 2
else:
p[1:len(p) -1] = p[1:len(p) - 1] * 2 # we've got even number of points fft
plt.plot(timeArray, 10*log10(p), color='k')
plt.xlabel('Time (ms)')
plt.ylabel('LeftChannel_Power (dB)')
plt.show()
'''
Right Channel
'''
p2 = p2 / float(m) # scale by the number of points so that
# the magnitude does not depend on the length
# of the signal or on its sampling frequency
p2 = p2**2 # square it to get the power
# multiply by two (see technical document for details)
# odd nfft excludes Nyquist point
if m % 2 > 0: # we've got odd number of points fft
p2[1:len(p2)] = p2[1:len(p2)] * 2
else:
p2[1:len(p2) -1] = p2[1:len(p2) - 1] * 2 # we've got even number of points fft
plt.plot(timeArray2, 10*log10(p2), color='k')
plt.xlabel('Time (ms)')
plt.ylabel('RightChannel_Power (dB)')
plt.show()
I hope this helps.
I've studied "Simulating Ocean Water" article by Jerry Tessendorf and tried to program the Statistical Wave Model but I didn't get correct result and I don't understand why.
In my program I tried only to create a wave height field at time t = 0 without any further changes in time. After execution of my program I got not what I was expecting:
Here's my source code:
clear all; close all; clc;
rng(11); % setting seed for random numbers
meshSize = 64; % field size
windDir = [1, 0]; % ||windDir|| = 1
patchSize = 64;
A = 1e+4;
g = 9.81; % gravitational constant
windSpeed = 1e+2;
x1 = linspace(-10, 10, meshSize+1); x = x1(1:meshSize);
y1 = linspace(-10, 10, meshSize+1); y = y1(1:meshSize);
[X,Y] = meshgrid(x, y);
H0 = zeros(size(X)); % height field at time t = 0
for i = 1:meshSize
for j = 1:meshSize
kx = 2.0 * pi / patchSize * (-meshSize / 2.0 + x(i)); % = 2*pi*n / Lx
ky = 2.0 * pi / patchSize * (-meshSize / 2.0 + y(j)); % = 2*pi*m / Ly
P = phillips(kx, ky, windDir, windSpeed, A, g); % phillips spectrum
H0(i,j) = 1/sqrt(2) * (randn(1) + 1i * randn(1)) * sqrt(P);
end
end
H0 = H0 + conj(H0);
surf(X,Y,abs(ifft(H0)));
axis([-10 10 -10 10 -10 10]);
And the phillips function:
function P = phillips(kx, ky, windDir, windSpeed, A, g)
k_sq = kx^2 + ky^2;
L = windSpeed^2 / g;
k = [kx, ky] / sqrt(k_sq);
wk = k(1) * windDir(1) + k(2) * windDir(2);
P = A / k_sq^2 * exp(-1.0 / (k_sq * L^2)) * wk^2;
end
Is there any matlab ocean simulation source code which could help me to understand my mistakes? Fast google search didn't get any results.
Here's a "correct" result I got from "CUDA FFT Ocean Simulation". I didn't achieve this behavior in Matlab yet but I've ploted "surf" in matlab using data from "CUDA FFT Ocean Simulation". Here's what it looks like:
I've made an experiment and got an interesting result:
I've taken generated h0 from "CUDA FFT Ocean Simulation". So I have to do ifft to transform from frequency domain to spatial domain to plot the graph. I've done it for the same h0 using matlab ifft and using cufftExecC2C from CUDA library. Here's the result:
CUDA ifft:
Matlab ifft:
Either I don't understand some aspects of realization of cufftExecC2C or cufftExecC2C and matlab ifft are different algorithms with different results.
By the way parameters for generating such surface are:
meshSize = 32
A = 1e-7
patchSize = 80
windSpeed = 10
Well that was definitely a funny exercise. This is a completely rewritten answer since you found the issues you were asking about by yourself.
Instead of deleting my answer, there is still merit in posting to help you vectorize and/or explain a few bits of code.
I completely rewrote the GUI I gave in my former answer in order to incorporate your changes and add a couple of options. It started to grew arms and legs so I won't put the listing here but you can find the full file there:
ocean_simulator.m.
This is completely self contained and it includes all the calculating functions I vectorized and list separately below.
The GUI will allow you to play with the parameters, animate the waves, export GIF file (and a few other options like the "preset", but they are not too ironed out yet). A few examples of what you can achieve:
Basic
This is what you get with the quick default settings, and a couple of rendering options. This uses a small grid size and a fast time step, so it runs pretty quickly on any machine.
I am quite limited at home (Pentium E2200 32bit), so I could only practice with limited settings. The gui will run even with the settings maxed but it will become to slow to really enjoy.
However, with a quick run of ocean_simulator at work (I7 64 bit, 8 cores, 16GB ram, 2xSSD in Raid), it makes it much more fun! Here are a few examples:
Although done on a much better machine, I didn't use any parallel functionality nor any GPU calculations, so Matlab was only using a portion of these specs, which means it could probably run just as good on any 64bit system with decent RAM
Windy lake
This is a rather flat water surface like a lake. Even high winds do not produce high amplitude waves (but still a lot of mini wavelets). If you're a wind surfer looking at that from your window on top of the hill, your heart is going to skip a beat and your next move is to call Dave "Man! gear up. Meet you in five on the water!"
Swell
This is you looking from the bridge of your boat on the morning, after having battled with the storm all night. The storm has dissipated and the long large waves are the last witness of what was definitely a shaky night (people with sailing experience will know ...).
T-Storm
And this was what you were up to the night before...
second gif done at home, hence the lack of detail ... sorry
To the bottom:
Finally, the gui will let you add a patch around the water domain. In the gui it is transparent so you could add objects underwater or a nice ocean bottom. Unfortunately, the GIF format cannot include an alpha channel so no transparency here (but if you export in a video then you should be ok).
Moreover, the export to GIF degrade the image, the joint between the domain border and the water surface is flawless if you run that in Matlab. In some case it also make Matlab degrade the rendering of the lighting, so this is definitely not the best option for export, but it allows more things to play within matlab.
Now onto the code:
Instead of listing the full GUI, which would be super long (this post is long enough already), I will just list here the re-written version of your code, and explain the changes.
You should notice a massive increase of speed execution (orders of magnitude), thanks to the remaining vectorization, but mostly for two reasons:
(i) A lot of calculations were repeated. Caching values and reusing them is much faster than recalculating full matrices in loops (during the animation part).
(ii) Note how I defined the surface graphic object. It is defined only once (empty even), then all the further calls (in the loop) only update the underlying ZData of the surface object (instead of re-creating a surface object at each iteration.
Here goes:
%% // clear workspace
clear all; close all; clc;
%% // Default parameters
param.meshsize = 128 ; %// main grid size
param.patchsize = 200 ;
param.windSpeed = 100 ; %// what unit ? [m/s] ??
param.winddir = 90 ; %// Azimuth
param.rng = 13 ; %// setting seed for random numbers
param.A = 1e-7 ; %// Scaling factor
param.g = 9.81 ; %// gravitational constant
param.xLim = [-10 10] ; %// domain limits X
param.yLim = [-10 10] ; %// domain limits Y
param.zLim = [-1e-4 1e-4]*2 ;
gridSize = param.meshsize * [1 1] ;
%% // Define the grid X-Y domain
x = linspace( param.xLim(1) , param.xLim(2) , param.meshsize ) ;
y = linspace( param.yLim(1) , param.yLim(2) , param.meshsize ) ;
[X,Y] = meshgrid(x, y);
%% // get the grid parameters which remain constants (not time dependent)
[H0, W, Grid_Sign] = initialize_wave( param ) ;
%% // calculate wave at t0
t0 = 0 ;
Z = calc_wave( H0 , W , t0 , Grid_Sign ) ;
%% // populate the display panel
h.fig = figure('Color','w') ;
h.ax = handle(axes) ; %// create an empty axes that fills the figure
h.surf = handle( surf( NaN(2) ) ) ; %// create an empty "surface" object
%% // Display the initial wave surface
set( h.surf , 'XData',X , 'YData',Y , 'ZData',Z )
set( h.ax , 'XLim',param.xLim , 'YLim',param.yLim , 'ZLim',param.zLim )
%% // Change some rendering options
axis off %// make the axis grid and border invisible
shading interp %// improve shading (remove "faceted" effect)
blue = linspace(0.4, 1.0, 25).' ; cmap = [blue*0, blue*0, blue]; %'// create blue colormap
colormap(cmap)
%// configure lighting
h.light_handle = lightangle(-45,30) ; %// add a light source
set(h.surf,'FaceLighting','phong','AmbientStrength',.3,'DiffuseStrength',.8,'SpecularStrength',.9,'SpecularExponent',25,'BackFaceLighting','unlit')
%% // Animate
view(75,55) %// no need to reset the view inside the loop ;)
timeStep = 1./25 ;
nSteps = 2000 ;
for time = (1:nSteps)*timeStep
%// update wave surface
Z = calc_wave( H0,W,time,Grid_Sign ) ;
h.surf.ZData = Z ;
pause(0.001);
end
%% // This block of code is only if you want to generate a GIF file
%// be carefull on how many frames you put there, the size of the GIF can
%// quickly grow out of proportion ;)
nFrame = 55 ;
gifFileName = 'MyDancingWaves.gif' ;
view(-70,40)
clear im
f = getframe;
[im,map] = rgb2ind(f.cdata,256,'nodither');
im(1,1,1,20) = 0;
iframe = 0 ;
for time = (1:nFrame)*.5
%// update wave surface
Z = calc_wave( H0,W,time,Grid_Sign ) ;
h.surf.ZData = Z ;
pause(0.001);
f = getframe;
iframe= iframe+1 ;
im(:,:,1,iframe) = rgb2ind(f.cdata,map,'nodither');
end
imwrite(im,map,gifFileName,'DelayTime',0,'LoopCount',inf)
disp([num2str(nFrame) ' frames written in file: ' gifFileName])
You'll notice that I changed a few things, but I can assure you the calculations are exactly the same. This code calls a few subfunctions but they are all vectorized so if you want you can just copy/paste them here and run everything inline.
The first function called is initialize_wave.m
Everything calculated here will be constant later (it does not vary with time when you later animate the waves), so it made sense to put that into a block on it's own.
function [H0, W, Grid_Sign] = initialize_wave( param )
% function [H0, W, Grid_Sign] = initialize_wave( param )
%
% This function return the wave height coefficients H0 and W for the
% parameters given in input. These coefficients are constants for a given
% set of input parameters.
% Third output parameter is optional (easy to recalculate anyway)
rng(param.rng); %// setting seed for random numbers
gridSize = param.meshsize * [1 1] ;
meshLim = pi * param.meshsize / param.patchsize ;
N = linspace(-meshLim , meshLim , param.meshsize ) ;
M = linspace(-meshLim , meshLim , param.meshsize ) ;
[Kx,Ky] = meshgrid(N,M) ;
K = sqrt(Kx.^2 + Ky.^2); %// ||K||
W = sqrt(K .* param.g); %// deep water frequencies (empirical parameter)
[windx , windy] = pol2cart( deg2rad(param.winddir) , 1) ;
P = phillips(Kx, Ky, [windx , windy], param.windSpeed, param.A, param.g) ;
H0 = 1/sqrt(2) .* (randn(gridSize) + 1i .* randn(gridSize)) .* sqrt(P); % height field at time t = 0
if nargout == 3
Grid_Sign = signGrid( param.meshsize ) ;
end
Note that the initial winDir parameter is now expressed with a single scalar value representing the "azimuth" (in degrees) of the wind (anything from 0 to 360). It is later translated to its X and Y components thanks to the function pol2cart.
[windx , windy] = pol2cart( deg2rad(param.winddir) , 1) ;
This insure that the norm is always 1.
The function calls your problematic phillips.m separately, but as said before it works even fully vectorized so you can copy it back inline if you like. (don't worry I checked the results against your versions => strictly identical). Note that this function does not output complex numbers so there was no need to compare the imaginary parts.
function P = phillips(Kx, Ky, windDir, windSpeed, A, g)
%// The function now accept scalar, vector or full 2D grid matrix as input
K_sq = Kx.^2 + Ky.^2;
L = windSpeed.^2 ./ g;
k_norm = sqrt(K_sq) ;
WK = Kx./k_norm * windDir(1) + Ky./k_norm * windDir(2);
P = A ./ K_sq.^2 .* exp(-1.0 ./ (K_sq * L^2)) .* WK.^2 ;
P( K_sq==0 | WK<0 ) = 0 ;
end
The next function called by the main program is calc_wave.m. This function finishes the calculations of the wave field for a given time. It is definitely worth having that on its own because this is the mimimun set of calculations which will have to be repeated for each given time when you want to animate the waves.
function Z = calc_wave( H0,W,time,Grid_Sign )
% Z = calc_wave( H0,W,time,Grid_Sign )
%
% This function calculate the wave height based on the wave coefficients H0
% and W, for a given "time". Default time=0 if not supplied.
% Fourth output parameter is optional (easy to recalculate anyway)
% recalculate the grid sign if not supplied in input
if nargin < 4
Grid_Sign = signGrid( param.meshsize ) ;
end
% Assign time=0 if not specified in input
if nargin < 3 ; time = 0 ; end
wt = exp(1i .* W .* time ) ;
Ht = H0 .* wt + conj(rot90(H0,2)) .* conj(wt) ;
Z = real( ifft2(Ht) .* Grid_Sign ) ;
end
The last 3 lines of calculations require a bit of explanation as they received the biggest changes (all for the same result but a much better speed).
Your original line:
Ht = H0 .* exp(1i .* W .* (t * timeStep)) + conj(flip(flip(H0,1),2)) .* exp(-1i .* W .* (t * timeStep));
recalculate the same thing too many times to be efficient:
(t * timeStep) is calculated twice on the line, at each loop, while it is easy to get the proper time value for each line when time is initialised at the beginning of the loop for time = (1:nSteps)*timeStep.
Also note that exp(-1i .* W .* time) is the same than conj(exp(1i .* W .* time)). Instead of doing 2*m*n multiplications to calculate them each, it is faster to calculate one once, then use the conj() operation which is much faster.
So your single line would become:
wt = exp(1i .* W .* time ) ;
Ht = H0 .* wt + conj(flip(flip(H0,1),2)) .* conj(wt) ;
Last minor touch, flip(flip(H0,1),2)) can be replaced by rot90(H0,2) (also marginally faster).
Note that because the function calc_wave is going to be repeated extensively, it is definitely worth reducing the number of calculations (as we did above), but also by sending it the Grid_Sign parameter (instead of letting the function recalculate it every iteration). This is why:
Your mysterious function signCor(ifft2(Ht),meshSize)), simply reverse the sign of every other element of Ht. There is a faster way of achieving that: simply multiply Ht by a matrix the same size (Grid_Sign) which is a matrix of alternated +1 -1 ... and so on.
so signCor(ifft2(Ht),meshSize) becomes ifft2(Ht) .* Grid_Sign.
Since Grid_Sign is only dependent on the matrix size, it does not change for each time in the loop, you only calculate it once (before the loop) then use it as it is for every other iteration. It is calculated as follow (vectorized, so you can also put it inline in your code):
function sgn = signGrid(n)
% return a matrix the size of n with alternate sign for every indice
% ex: sgn = signGrid(3) ;
% sgn =
% -1 1 -1
% 1 -1 1
% -1 1 -1
[x,y] = meshgrid(1:n,1:n) ;
sgn = ones( n ) ;
sgn(mod(x+y,2)==0) = -1 ;
end
Lastly, you will notice a difference in how the grids [Kx,Ky] are defined between your version and this one. They do produce slightly different result, it's just a matter of choice.
To explain with a simple example, let's consider a small meshsize=5. Your way of doing things will split that into 5 values, equally spaced, like so:
Kx(first line)=[-1.5 -0.5 0.5 1.5 2.5] * 2 * pi / patchSize
while my way of producing the grid will produce equally spaced values, but also centered on the domain limits, like so:
Kx(first line)=[-2.50 -1.25 0.0 1.25 2.50] * 2 * pi / patchSize
It seems to respect more your comment % = 2*pi*n / Lx, -N/2 <= n < N/2 on the line where you define it.
I tend to prefer symmetric solutions (plus it is also slightly faster but it is only calculated once so it is not a big deal), so I used my vectorized way, but it is purely a matter of choice, you can definitely keep your way, it only ever so slightly "offset" the whole result matrix, but it doesn't perturbate the calculations per se.
last remains of the first answer
Side programming notes:
I detect you come from the C/C++ world or family. In Matlab you do not need to define decimal number with a coma (like 2.0, you used that for most of your numbers). Unless specifically defined otherwise, Matlab by default cast any number to double, which is a 64 bit floating point type. So writing 2 * pi is enough to get the maximum precision (Matlab won't cast pi as an integer ;-)), you do not need to write 2.0 * pi. Although it will still work if you don't want to change your habits.
Also, (one of the great benefit of Matlab), adding . before an operator usually mean "element-wise" operation. You can add (.+), substract (.-), multiply (.*), divide (./) full matrix element wise this way. This is how I got rid of all the loops in your code. This also work for the power operator: A.^2 will return a matrix the same size as A with every element squared.
Here's the working program.
First of all - source code:
clear all; close all; clc;
rng(13); % setting seed for random numbers
meshSize = 128; % field size
windDir = [0.1,1];
patchSize = 200;
A = 1e-7;
g = 9.81; % gravitational constant
windSpeed = 100;
timeStep = 1/25;
x1 = linspace(-10, 10, meshSize+1); x = x1(1:meshSize);
y1 = linspace(-10, 10, meshSize+1); y = y1(1:meshSize);
[X,Y] = meshgrid(x,y); % wave field
i = 1:meshSize; j = 1:meshSize; % indecies
[I,J] = meshgrid(i,j); % field of indecies
Kx = 2.0 * pi / patchSize * (-meshSize / 2.0 + I); % = 2*pi*n / Lx, -N/2 <= n < N/2
Ky = 2.0 * pi / patchSize * (-meshSize / 2.0 + J); % = 2*pi*m / Ly, -M/2 <= m < M/2
K = sqrt(Kx.^2 + Ky.^2); % ||K||
W = sqrt(K .* g); % deep water frequencies (empirical parameter)
P = zeros(size(X)); % Cant compute P without loops
for i = 1:meshSize
for j = 1:meshSize
P(i,j) = phillips(Kx(i,j), Ky(i,j), windDir, windSpeed, A, g); % phillips spectrum
end
end
H0 = 1/sqrt(2) .* (randn(size(X)) + 1i .* randn(size(X))) .* sqrt(P); % height field at time t = 0
rotate3d on;
for t = 1:10000 % 10000 * timeStep (sec)
Ht = H0 .* exp(1i .* W .* (t * timeStep)) + ...
conj(flip(flip(H0,1),2)) .* exp(-1i .* W .* (t * timeStep));
[az,el] = view;
surf(X,Y,real(signCor(ifft2(Ht),meshSize)));
axis([-10 10 -10 10 -1e-4 1e-4]); view(az,el);
blue = linspace(0.4, 1.0, 25)'; map = [blue*0, blue*0, blue];
%shading interp; % improve shading (remove "faceted" effect)
colormap(map);
pause(1/60);
end
phillips.m: (I've tried to vectorize the computation of Phillips spectrum but I faced with a difficulty which I'll show further)
function P = phillips(kx, ky, windDir, windSpeed, A, g)
k_sq = kx^2 + ky^2;
if k_sq == 0
P = 0;
else
L = windSpeed^2 / g;
k = [kx, ky] / sqrt(k_sq);
wk = k(1) * windDir(1) + k(2) * windDir(2);
P = A / k_sq^2 * exp(-1.0 / (k_sq * L^2)) * wk^2;
if wk < 0
P = 0;
end
end
end
signCor.m: (This function is an absolutely mystery for me... I've copied it from "CUDA FFT Ocean Simulation" realization. Simulation works much worse without it. And again I don't know how to vectorize this function.)
function H = signCor(H1, meshSize)
H = H1;
for i = 1:meshSize
for j = 1:meshSize
if mod(i+j,2) == 0
sign = -1; % works fine if we change signs vice versa
else
sign = 1;
end
H(i,j) = H1(i,j) * sign;
end
end
end
The biggest mistake that I've done is that I used ifft instead of using ifft2, that's why CUDA ifft and Matlab ifft didn't match.
My second mistake was in this lines of code:
kx = 2.0 * pi / patchSize * (-meshSize / 2.0 + x(i)); % = 2*pi*n / Lx
ky = 2.0 * pi / patchSize * (-meshSize / 2.0 + y(j)); % = 2*pi*m / Ly
I should've write:
kx = 2.0 * pi / patchSize * (-meshSize / 2.0 + i); % = 2*pi*n / Lx
ky = 2.0 * pi / patchSize * (-meshSize / 2.0 + j); % = 2*pi*m / Ly
I've played a bit with parameters A, meshSize, patchSize and I came to the conclusion that:
Somehow plausible parameter of wave amplitude is A * (patchSize / meshSize), where A is nothing but a scaling factor.
For 'calm' patchSize / meshSize <= 0.5.
For 'tsunami' patchSize / meshSize >= 3.0.
Difficulty with a vectorization of Phillips spectrum:
I have 2 functions:
% non-vectorized spectrum
function P = phillips1(kx, ky, windDir, windSpeed, A, g)
k_sq = kx^2 + ky^2;
if k_sq == 0
P = 0;
else
L = windSpeed^2 / g;
k = [kx, ky] / sqrt(k_sq);
wk = k(1) * windDir(1) + k(2) * windDir(2);
P = A / k_sq^2 * exp(-1.0 / (k_sq * L^2)) * wk^2;
if wk < 0
P = 0;
end
end
end
% vectorized spectrum
function P = phillips2(Kx, Ky, windDir, windSpeed, A, g)
K_sq = Kx .^ 2 + Ky .^ 2;
L = -g^2 / windSpeed^4;
WK = (Kx ./ K_sq) .* windDir(1) + (Ky ./ K_sq) .* windDir(2);
P = (A ./ (K_sq .^ 2)) .* ( exp(L ./ K_sq) .* (WK .^ 2) );
P(K_sq == 0) = 0;
P(WK < 0) = 0;
P(isinf(P)) = 0;
end
After I compute P1 using phillips1 and P2 using phillips2 I plot their difference:
subplot(2,1,1); surf(X,Y,real(P2-P1)); title('Difference in real part');
subplot(2,1,2); surf(X,Y,imag(P2-P1)); title('Difference in imaginary part');
It perfectly illustrates that there's a huge difference between this 2 spectrums in real part.
I've found myself needing to do a least-squares (or similar matrix-based operation) for every pixel in an image. Every pixel has a set of numbers associated with it, and so it can be arranged as a 3D matrix.
(This next bit can be skipped)
Quick explanation of what I mean by least-squares estimation :
Let's say we have some quadratic system that is modeled by Y = Ax^2 + Bx + C and we're looking for those A,B,C coefficients. With a few samples (at least 3) of X and the corresponding Y, we can estimate them by:
Arrange the (lets say 10) X samples into a matrix like X = [x(:).^2 x(:) ones(10,1)];
Arrange the Y samples into a similar matrix: Y = y(:);
Estimate the coefficients A,B,C by solving: coeffs = (X'*X)^(-1)*X'*Y;
Try this on your own if you want:
A = 5; B = 2; C = 1;
x = 1:10;
y = A*x(:).^2 + B*x(:) + C + .25*randn(10,1); % added some noise here
X = [x(:).^2 x(:) ones(10,1)];
Y = y(:);
coeffs = (X'*X)^-1*X'*Y
coeffs =
5.0040
1.9818
0.9241
START PAYING ATTENTION AGAIN IF I LOST YOU THERE
*MAJOR REWRITE*I've modified to bring it as close to the real problem that I have and still make it a minimum working example.
Problem Setup
%// Setup
xdim = 500;
ydim = 500;
ncoils = 8;
nshots = 4;
%// matrix size for each pixel is ncoils x nshots (an overdetermined system)
%// each pixel has a matrix stored in the 3rd and 4rth dimensions
regressor = randn(xdim,ydim, ncoils,nshots);
regressand = randn(xdim, ydim,ncoils);
So my problem is that I have to do a (X'*X)^-1*X'*Y (least-squares or similar) operation for every pixel in an image. While that itself is vectorized/matrixized the only way that I have to do it for every pixel is in a for loop, like:
Original code style
%// Actual work
tic
estimate = zeros(xdim,ydim);
for col=1:size(regressor,2)
for row=1:size(regressor,1)
X = squeeze(regressor(row,col,:,:));
Y = squeeze(regressand(row,col,:));
B = X\Y;
% B = (X'*X)^(-1)*X'*Y; %// equivalently
estimate(row,col) = B(1);
end
end
toc
Elapsed time = 27.6 seconds
EDITS in reponse to comments and other ideas
I tried some things:
1. Reshaped into a long vector and removed the double for loop. This saved some time.
2. Removed the squeeze (and in-line transposing) by permute-ing the picture before hand: This save alot more time.
Current example:
%// Actual work
tic
estimate2 = zeros(xdim*ydim,1);
regressor_mod = permute(regressor,[3 4 1 2]);
regressor_mod = reshape(regressor_mod,[ncoils,nshots,xdim*ydim]);
regressand_mod = permute(regressand,[3 1 2]);
regressand_mod = reshape(regressand_mod,[ncoils,xdim*ydim]);
for ind=1:size(regressor_mod,3) % for every pixel
X = regressor_mod(:,:,ind);
Y = regressand_mod(:,ind);
B = X\Y;
estimate2(ind) = B(1);
end
estimate2 = reshape(estimate2,[xdim,ydim]);
toc
Elapsed time = 2.30 seconds (avg of 10)
isequal(estimate2,estimate) == 1;
Rody Oldenhuis's way
N = xdim*ydim*ncoils; %// number of columns
M = xdim*ydim*nshots; %// number of rows
ii = repmat(reshape(1:N,[ncoils,xdim*ydim]),[nshots 1]); %//column indicies
jj = repmat(1:M,[ncoils 1]); %//row indicies
X = sparse(ii(:),jj(:),regressor_mod(:));
Y = regressand_mod(:);
B = X\Y;
B = reshape(B(1:nshots:end),[xdim ydim]);
Elapsed time = 2.26 seconds (avg of 10)
or 2.18 seconds (if you don't include the definition of N,M,ii,jj)
SO THE QUESTION IS:
Is there an (even) faster way?
(I don't think so.)
You can achieve a ~factor of 2 speed up by precomputing the transposition of X. i.e.
for x=1:size(picture,2) % second dimension b/c already transposed
X = picture(:,x);
XX = X';
Y = randn(n_timepoints,1);
%B = (X'*X)^-1*X'*Y; ;
B = (XX*X)^-1*XX*Y;
est(x) = B(1);
end
Before: Elapsed time is 2.520944 seconds.
After: Elapsed time is 1.134081 seconds.
EDIT:
Your code, as it stands in your latest edit, can be replaced by the following
tic
xdim = 500;
ydim = 500;
n_timepoints = 10; % for example
% Actual work
picture = randn(xdim,ydim,n_timepoints);
picture = reshape(picture, [xdim*ydim,n_timepoints])'; % note transpose
YR = randn(n_timepoints,size(picture,2));
% (XX*X).^-1 = sum(picture.*picture).^-1;
% XX*Y = sum(picture.*YR);
est = sum(picture.*picture).^-1 .* sum(picture.*YR);
est = reshape(est,[xdim,ydim]);
toc
Elapsed time is 0.127014 seconds.
This is an order of magnitude speed up on the latest edit, and the results are all but identical to the previous method.
EDIT2:
Okay, so if X is a matrix, not a vector, things are a little more complicated. We basically want to precompute as much as possible outside of the for-loop to keep our costs down. We can also get a significant speed-up by computing XT*X manually - since the result will always be a symmetric matrix, we can cut a few corners to speed things up. First, the symmetric multiplication function:
function XTX = sym_mult(X) % X is a 3-d matrix
n = size(X,2);
XTX = zeros(n,n,size(X,3));
for i=1:n
for j=i:n
XTX(i,j,:) = sum(X(:,i,:).*X(:,j,:));
if i~=j
XTX(j,i,:) = XTX(i,j,:);
end
end
end
Now the actual computation script
xdim = 500;
ydim = 500;
n_timepoints = 10; % for example
Y = randn(10,xdim*ydim);
picture = randn(xdim,ydim,n_timepoints); % 500x500x10
% Actual work
tic % start timing
picture = reshape(picture, [xdim*ydim,n_timepoints])';
% Here we precompute the (XT*Y) calculation to speed things up later
picture_y = [sum(Y);sum(Y.*picture)];
% initialize
est = zeros(size(picture,2),1);
picture = permute(picture,[1,3,2]);
XTX = cat(2,ones(n_timepoints,1,size(picture,3)),picture);
XTX = sym_mult(XTX); % precompute (XT*X) for speed
X = zeros(2,2); % preallocate for speed
XY = zeros(2,1);
for x=1:size(picture,2) % second dimension b/c already transposed
%For some reason this is a lot faster than X = XTX(:,:,x);
X(1,1) = XTX(1,1,x);
X(2,1) = XTX(2,1,x);
X(1,2) = XTX(1,2,x);
X(2,2) = XTX(2,2,x);
XY(1) = picture_y(1,x);
XY(2) = picture_y(2,x);
% Here we utilise the fact that A\B is faster than inv(A)*B
% We also use the fact that (A*B)*C = A*(B*C) to speed things up
B = X\XY;
est(x) = B(1);
end
est = reshape(est,[xdim,ydim]);
toc % end timing
Before: Elapsed time is 4.56 seconds.
After: Elapsed time is 2.24 seconds.
This is a speed up of about a factor of 2. This code should be extensible to X being any dimensions you want. For instance, in the case where X = [1 x x^2], you would change picture_y to the following
picture_y = [sum(Y);sum(Y.*picture);sum(Y.*picture.^2)];
and change XTX to
XTX = cat(2,ones(n_timepoints,1,size(picture,3)),picture,picture.^2);
You would also change a lot of 2s to 3s in the code, and add XY(3) = picture_y(3,x) to the loop. It should be fairly straight-forward, I believe.
Results
I sped up your original version, since your edit 3 was actually not working (and also does something different).
So, on my PC:
Your (original) version: 8.428473 seconds.
My obfuscated one-liner given below: 0.964589 seconds.
First, for no other reason than to impress, I'll give it as I wrote it:
%%// Some example data
xdim = 500;
ydim = 500;
n_timepoints = 10; % for example
estimate = zeros(xdim,ydim); %// initialization with explicit size
picture = randn(xdim,ydim,n_timepoints);
%%// Your original solution
%// (slightly altered to make my version's results agree with yours)
tic
Y = randn(n_timepoints,xdim*ydim);
ii = 1;
for x = 1:xdim
for y = 1:ydim
X = squeeze(picture(x,y,:)); %// or similar creation of X matrix
B = (X'*X)^(-1)*X' * Y(:,ii);
ii = ii+1;
%// sometimes you keep everything and do
%// estimate(x,y,:) = B(:);
%// sometimes just the first element is important and you do
estimate(x,y) = B(1);
end
end
toc
%%// My version
tic
%// UNLEASH THE FURY!!
estimate2 = reshape(sparse(1:xdim*ydim*n_timepoints, ...
builtin('_paren', ones(n_timepoints,1)*(1:xdim*ydim),:), ...
builtin('_paren', permute(picture, [3 2 1]),:))\Y(:), ydim,xdim).'; %'
toc
%%// Check for equality
max(abs(estimate(:)-estimate2(:))) % (always less than ~1e-14)
Breakdown
First, here's the version that you should actually use:
%// Construct sparse block-diagonal matrix
%// (Type "help sparse" for more information)
N = xdim*ydim; %// number of columns
M = N*n_timepoints; %// number of rows
ii = 1:N;
jj = ones(n_timepoints,1)*(1:N);
s = permute(picture, [3 2 1]);
X = sparse(ii,jj(:), s(:));
%// Compute ALL the estimates at once
estimates = X\Y(:);
%// You loop through the *second* dimension first, so to make everything
%// agree, we have to extract elements in the "wrong" order, and transpose:
estimate2 = reshape(estimates, ydim,xdim).'; %'
Here's an example of what picture and the corresponding matrix X looks like for xdim = ydim = n_timepoints = 2:
>> clc, picture, full(X)
picture(:,:,1) =
-0.5643 -2.0504
-0.1656 0.4497
picture(:,:,2) =
0.6397 0.7782
0.5830 -0.3138
ans =
-0.5643 0 0 0
0.6397 0 0 0
0 -2.0504 0 0
0 0.7782 0 0
0 0 -0.1656 0
0 0 0.5830 0
0 0 0 0.4497
0 0 0 -0.3138
You can see why sparse is necessary -- it's mostly zeros, but will grow large quickly. The full matrix would quickly consume all your RAM, while the sparse one will not consume much more than the original picture matrix does.
With this matrix X, the new problem
X·b = Y
now contains all the problems
X1 · b1 = Y1
X2 · b2 = Y2
...
where
b = [b1; b2; b3; ...]
Y = [Y1; Y2; Y3; ...]
so, the single command
X\Y
will solve all your systems at once.
This offloads all the hard work to a set of highly specialized, compiled to machine-specific code, optimized-in-every-way algorithms, rather than the interpreted, generic, always-two-steps-away from the hardware loops in MATLAB.
It should be straightforward to convert this to a version where X is a matrix; you'll end up with something like what blkdiag does, which can also be used by mldivide in exactly the same way as above.
I had a wee play around with an idea, and I decided to stick it as a separate answer, as its a completely different approach to my other idea, and I don't actually condone what I'm about to do. I think this is the fastest approach so far:
Orignal (unoptimised): 13.507176 seconds.
Fast Cholesky-decomposition method: 0.424464 seconds
First, we've got a function to quickly do the X'*X multiplication. We can speed things up here because the result will always be symmetric.
function XX = sym_mult(X)
n = size(X,2);
XX = zeros(n,n,size(X,3));
for i=1:n
for j=i:n
XX(i,j,:) = sum(X(:,i,:).*X(:,j,:));
if i~=j
XX(j,i,:) = XX(i,j,:);
end
end
end
The we have a function to do LDL Cholesky decomposition of a 3D matrix (we can do this because the (X'*X) matrix will always be symmetric) and then do forward and backwards substitution to solve the LDL inversion equation
function Y = fast_chol(X,XY)
n=size(X,2);
L = zeros(n,n,size(X,3));
D = zeros(n,n,size(X,3));
B = zeros(n,1,size(X,3));
Y = zeros(n,1,size(X,3));
% These loops compute the LDL decomposition of the 3D matrix
for i=1:n
D(i,i,:) = X(i,i,:);
L(i,i,:) = 1;
for j=1:i-1
L(i,j,:) = X(i,j,:);
for k=1:(j-1)
L(i,j,:) = L(i,j,:) - L(i,k,:).*L(j,k,:).*D(k,k,:);
end
D(i,j,:) = L(i,j,:);
L(i,j,:) = L(i,j,:)./D(j,j,:);
if i~=j
D(i,i,:) = D(i,i,:) - L(i,j,:).^2.*D(j,j,:);
end
end
end
for i=1:n
B(i,1,:) = XY(i,:);
for j=1:(i-1)
B(i,1,:) = B(i,1,:)-D(i,j,:).*B(j,1,:);
end
B(i,1,:) = B(i,1,:)./D(i,i,:);
end
for i=n:-1:1
Y(i,1,:) = B(i,1,:);
for j=n:-1:(i+1)
Y(i,1,:) = Y(i,1,:)-L(j,i,:).*Y(j,1,:);
end
end
Finally, we have the main script which calls all of this
xdim = 500;
ydim = 500;
n_timepoints = 10; % for example
Y = randn(10,xdim*ydim);
picture = randn(xdim,ydim,n_timepoints); % 500x500x10
tic % start timing
picture = reshape(pr, [xdim*ydim,n_timepoints])';
% Here we precompute the (XT*Y) calculation
picture_y = [sum(Y);sum(Y.*picture)];
% initialize
est2 = zeros(size(picture,2),1);
picture = permute(picture,[1,3,2]);
% Now we calculate the X'*X matrix
XTX = cat(2,ones(n_timepoints,1,size(picture,3)),picture);
XTX = sym_mult(XTX);
% Call our fast Cholesky decomposition routine
B = fast_chol(XTX,picture_y);
est2 = B(1,:);
est2 = reshape(est2,[xdim,ydim]);
toc
Again, this should work equally well for a Nx3 X matrix, or however big you want.
I use octave, thus I can't say anything about the resulting performance in Matlab, but would expect this code to be slightly faster:
pictureT=picture'
est=arrayfun(#(x)( (pictureT(x,:)*picture(:,x))^-1*pictureT(x,:)*randn(n_ti
mepoints,1)),1:size(picture,2));