I'm using random points to determine the area below a curve (Monte-Carlo):
X: 1xn vector of x values for the function
Y: 1xn vector of y = f(x)
RP: mxn matrix of m random y for each x
I would like to split RP into RPA and RPB depending on it being above or below the curve. The idea is then to plot RPA and RPB against X, in different colors. This code doesn't work because RPA and RPB number of columns is not the same than X:
clf
f = #(x) sin(x/10) + cos(x/60); % Function
xMin = 1; xMax = 100; % x interval
X = [xMin:xMax];
Y = f(X);
plot(X,Y), hold on % Plot function
yMin = min(Y); yMax = max(Y); % Axes limits
set(gca, 'xlim', [xMin, xMax], 'ylim', [yMin, yMax])
m = 20; % Random points per x value
RP = rand(m, columns(X)) .* (yMax-yMin) .+ yMin;
% Split points (doesn't work)
RPA = RP(RP>Y);
RPB = RP(RP<=Y);
br = size(RPB) / size(RP) % Ratio of points below
a = (xMax - xMin) * (yMax - yMin) * br % Area below
% Plot points
plot(X, RPA, 'r.') % Above
plot(X, RPB, 'g.') % Below
Is there a possibility to build RPA and RPB so that they are the same size than RP, with the excluded points y being NaN or something similar, which can be counted and plotted?
You gave a good answer yourself. You can build RPA and RPB with strategic NaNs:
% Split points (works!)
RPA = RP;
RPA(RP<=Y) = NaN;
RPB = RP;
RPB(RPB > Y) = NaN;
And than calculating the ration as the not-NaN:
br = sum(~isnan(RPB)) / sum(~isnan(RP)) % Ratio of points below
I get this nice image:
I'm trying to plot the following equation (let's call it "Equation 1"):
This is the code I'm testing:
clear all;
xl=0; xr=1; % x domain [xl,xr]
J = 10; % J: number of division for x
dx = (xr-xl) / J; % dx: mesh size
tf = 0.1; % final simulation time
Nt = 60; % Nt: number of time steps
dt = tf/Nt/4;
x = xl : dx : xr; % generate the grid point
u_ex = zeros(J+1,Nt);
for n = 1:Nt
t = n*dt; % current time
for j=1:J+1
xj = xl + (j-1)*dx;
suma = zeros(100 , 1);
for k= 1:100
suma(k) = 4/(((2*k-1)^2) *pi*pi);
suma(k) = suma(k) * exp(-((2*k-1)^2) *pi*pi*t) * cos(2*k-1)*pi*xj;
end
m = sum(suma);
u_ex(j, n)= 0.5 - m;
end
end
tt = dt : dt : Nt*dt;
figure(1)
surf(x,tt, u_ex'); % 3-D surface plot
xlabel('x')
ylabel('t')
zlabel('u')
The problem is that all I get is a flat surface:
Equation 1 is suppossed to be the solution of the following parabolic partial differential equation with boundary values:
And after getting the numerical solution, it should look like this:
This plot gets the right values at the boundaries x = 0 and x = 1. The plot of Equation 1 doesn't have those values at the boundaries.
My complete .m code (that plots both the numerical solution and Equation 1) is:
clear all; % clear all variables in memory
xl=0; xr=1; % x domain [xl,xr]
J = 10; % J: number of division for x
dx = (xr-xl) / J; % dx: mesh size
tf = 0.1; % final simulation time
Nt = 60; % Nt: number of time steps
dt = tf/Nt/4;
mu = dt/(dx)^2;
if mu > 0.5 % make sure dt satisy stability condition
error('mu should < 0.5!')
end
% Evaluate the initial conditions
x = xl : dx : xr; % generate the grid point
% store the solution at all grid points for all time steps
u = zeros(J+1,Nt);
u_ex = zeros(J+1,Nt);
% Find the approximate solution at each time step
for n = 1:Nt
t = n*dt; % current time
% boundary condition at left side
gl = 0;
% boundary condition at right side
gr = 0;
for j=2:J
if n==1 % first time step
u(j,n) = j;
else % interior nodes
u(j,n)=u(j,n-1) + mu*(u(j+1,n-1) - 2*u(j,n-1) + u(j-1,n-1));
end
end
u(1,n) = gl; % the left-end point
u(J+1,n) = gr; % the right-end point
% calculate the analytic solution
for j=1:J+1
xj = xl + (j-1)*dx;
suma = zeros(100 , 1);
for k= 1:100
suma(k) = 4/(((2*k-1)^2) *pi*pi);
suma(k) = suma(k) * exp(-((2*k-1)^2) *pi*pi*t) * cos(2*k-1)*pi*xj;
end
m = sum(suma);
u_ex(j, n)= 0.5 - m;
end
end
% Plot the results
tt = dt : dt : Nt*dt;
figure(1)
colormap(gray); % draw gray figure
surf(x,tt, u'); % 3-D surface plot
xlabel('x')
ylabel('t')
zlabel('u')
title('Numerical solution of 1-D parabolic equation')
figure(2)
surf(x,tt, u_ex'); % 3-D surface plot
xlabel('x')
ylabel('t')
zlabel('u')
title('Analytic solution of 1-D parabolic equation')
maxerr=max(max(abs(u-u_ex))),
The code is taken from the book "Computational Partial Differential Equations Using MATLAB" by Yi-Tung Chen, Jichun Li, chapter 2, exercise 3.
In short: I'm not asking about the differential equation or the boundary problem, I want to know is: Why am I getting a flat surface when plotting Equation 1? Am I missing a parenthesis?
I do not want to use the symsum function because it never stop the script execution and I want to learn how to plot Equation 1 with no using symsum.
I've tested this code with Matlab R2008b and Octave 4.2.1. I got the same results (even with sums of 1000, 10000 and 50000 terms in the for loop with the k variable).
Edit!
Thanks, Steve!
I was missing a couple of parenthesis near the cosine, the right code is:
clear all; % clear all variables in memory
xl=0; xr=1; % x domain [xl,xr]
J = 10; % J: number of division for x
dx = (xr-xl) / J; % dx: mesh size
tf = 0.1; % final simulation time
Nt = 60; % Nt: number of time steps
dt = tf/Nt/4;
mu = dt/(dx)^2;
if mu > 0.5 % make sure dt satisy stability condition
error('mu should < 0.5!')
end
% Evaluate the initial conditions
x = xl : dx : xr; % generate the grid point
% store the solution at all grid points for all time steps
u = zeros(J+1,Nt);
u_ex = zeros(J+1,Nt);
% Find the approximate solution at each time step
for n = 1:Nt
t = n*dt; % current time
% boundary condition at left side
gl = 0;
% boundary condition at right side
gr = 0;
for j=2:J
if n==1 % first time step
u(j,n) = j;
else % interior nodes
u(j,n)=u(j,n-1) + mu*(u(j+1,n-1) - 2*u(j,n-1) + u(j-1,n-1));
end
end
u(1,n) = gl; % the left-end point
u(J+1,n) = gr; % the right-end point
% calculate the analytic solution
for j=1:J+1
xj = xl + (j-1)*dx;
suma = zeros(1000 , 1);
for k= 1:1000
suma(k) = 4/(((2*k-1)^2) *pi*pi);
suma(k) *= exp(-((2*k-1)^2) *pi*pi*t) * cos((2*k-1)*pi*xj);
end
m = sum(suma);
u_ex(j, n)= 0.5 - m;
end
end
% Plot the results
tt = dt : dt : Nt*dt;
figure(1)
colormap(gray); % draw gray figure
surf(x,tt, u'); % 3-D surface plot
xlabel('x')
ylabel('t')
zlabel('u')
title('Numerical solution of 1-D parabolic equation')
figure(2)
surf(x,tt, u_ex'); % 3-D surface plot
xlabel('x')
ylabel('t')
zlabel('u')
title('Analytic solution of 1-D parabolic equation')
Now my Equation 1 looks much better:
Also Steve was right when pointing out that my numerical solution may be wrong. I didn't notice that the boundary values are for the derivatives of my function, not the actual values of the function. I'll ask my teacher about this.
Edit2!
Ok, I got it. To calculate the derivatives at the boundaries you have to use hint 2.21 in the same book:
% hint 2.21 given by the book
% it is better to calculate the boundary values after calculating the inner points inside the for j = 1:m loop because you will need them:
u(1, n) = u(2, n) - dx * gl; % the left-end point
u(J+1,n) = u(J, n) + dx * gr; % the right-end point
Now my numerical solution looks like my analytic solution :D
Matlab R2008b can't recognize the *= operator that Octave does. I'm not tested this operator in other versions of Matlab because I'm too poor.
Yvon: I think the analytical solution formula comes from the real part of a Fourier expansion, but authors don't tell how they got it.
I'm looking for a simple way for creating a random unit vector constrained by a conical region. The origin is always the [0,0,0].
My solution up to now:
function v = GetRandomVectorInsideCone(coneDir,coneAngleDegree)
coneDir = normc(coneDir);
ang = coneAngleDegree + 1;
while ang > coneAngleDegree
v = [randn(1); randn(1); randn(1)];
v = v + coneDir;
v = normc(v);
ang = atan2(norm(cross(v,coneDir)), dot(v,coneDir))*180/pi;
end
My code loops until the random generated unit vector is inside the defined cone. Is there a better way to do that?
Resultant image from test code bellow
Resultant frequency distribution using Ahmed Fasih code (in comments).
I wonder how to get a rectangular or normal distribution.
c = [1;1;1]; angs = arrayfun(#(i) subspace(c, GetRandomVectorInsideCone(c, 30)), 1:1e5) * 180/pi; figure(); hist(angs, 50);
Testing code:
clearvars; clc; close all;
coneDir = [randn(1); randn(1); randn(1)];
coneDir = [0 0 1]';
coneDir = normc(coneDir);
coneAngle = 45;
N = 1000;
vAngles = zeros(N,1);
vs = zeros(3,N);
for i=1:N
vs(:,i) = GetRandomVectorInsideCone(coneDir,coneAngle);
vAngles(i) = subspace(vs(:,i),coneDir)*180/pi;
end
maxAngle = max(vAngles);
minAngle = min(vAngles);
meanAngle = mean(vAngles);
AngleStd = std(vAngles);
fprintf('v angle\n');
fprintf('Direction: [%.3f %.3f %.3f]^T. Angle: %.2fº\n',coneDir,coneAngle);
fprintf('Min: %.2fº. Max: %.2fº\n',minAngle,maxAngle);
fprintf('Mean: %.2fº\n',meanAngle);
fprintf('Standard Dev: %.2fº\n',AngleStd);
%% Plot
figure;
grid on;
rotate3d on;
axis equal;
axis vis3d;
axis tight;
hold on;
xlabel('X'); ylabel('Y'); zlabel('Z');
% Plot all vectors
p1 = [0 0 0]';
for i=1:N
p2 = vs(:,i);
plot3ex(p1,p2);
end
% Trying to plot the limiting cone, but no success here :(
% k = [0 1];
% [X,Y,Z] = cylinder([0 1 0]');
% testsubject = surf(X,Y,Z);
% set(testsubject,'FaceAlpha',0.5)
% N = 50;
% r = linspace(0, 1, N);
% [X,Y,Z] = cylinder(r, N);
%
% h = surf(X, Y, Z);
%
% rotate(h, [1 1 0], 90);
plot3ex.m:
function p = plot3ex(varargin)
% Plots a line from each p1 to each p2.
% Inputs:
% p1 3xN
% p2 3xN
% args plot3 configuration string
% NOTE: p1 and p2 number of points can range from 1 to N
% but if the number of points are different, one must be 1!
% PVB 2016
p1 = varargin{1};
p2 = varargin{2};
extraArgs = varargin(3:end);
N1 = size(p1,2);
N2 = size(p2,2);
N = N1;
if N1 == 1 && N2 > 1
N = N2;
elseif N1 > 1 && N2 == 1
N = N1
elseif N1 ~= N2
error('if size(p1,2) ~= size(p1,2): size(p1,2) and/or size(p1,2) must be 1 !');
end
for i=1:N
i1 = i;
i2 = i;
if i > N1
i1 = N1;
end
if i > N2
i2 = N2;
end
x = [p1(1,i1) p2(1,i2)];
y = [p1(2,i1) p2(2,i2)];
z = [p1(3,i1) p2(3,i2)];
p = plot3(x,y,z,extraArgs{:});
end
Here’s the solution. It’s based on the wonderful answer at https://math.stackexchange.com/a/205589/81266. I found this answer by googling “random points on spherical cap”, after I learned on Mathworld that a spherical cap is this cut of a 3-sphere with a plane.
Here’s the function:
function r = randSphericalCap(coneAngleDegree, coneDir, N, RNG)
if ~exist('coneDir', 'var') || isempty(coneDir)
coneDir = [0;0;1];
end
if ~exist('N', 'var') || isempty(N)
N = 1;
end
if ~exist('RNG', 'var') || isempty(RNG)
RNG = RandStream.getGlobalStream();
end
coneAngle = coneAngleDegree * pi/180;
% Generate points on the spherical cap around the north pole [1].
% [1] See https://math.stackexchange.com/a/205589/81266
z = RNG.rand(1, N) * (1 - cos(coneAngle)) + cos(coneAngle);
phi = RNG.rand(1, N) * 2 * pi;
x = sqrt(1-z.^2).*cos(phi);
y = sqrt(1-z.^2).*sin(phi);
% If the spherical cap is centered around the north pole, we're done.
if all(coneDir(:) == [0;0;1])
r = [x; y; z];
return;
end
% Find the rotation axis `u` and rotation angle `rot` [1]
u = normc(cross([0;0;1], normc(coneDir)));
rot = acos(dot(normc(coneDir), [0;0;1]));
% Convert rotation axis and angle to 3x3 rotation matrix [2]
% [2] See https://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle
crossMatrix = #(x,y,z) [0 -z y; z 0 -x; -y x 0];
R = cos(rot) * eye(3) + sin(rot) * crossMatrix(u(1), u(2), u(3)) + (1-cos(rot))*(u * u');
% Rotate [x; y; z] from north pole to `coneDir`.
r = R * [x; y; z];
end
function y = normc(x)
y = bsxfun(#rdivide, x, sqrt(sum(x.^2)));
end
This code just implements joriki’s answer on math.stackexchange, filling in all the details that joriki omitted.
Here’s a script that shows how to use it.
clearvars
coneDir = [1;1;1];
coneAngleDegree = 30;
N = 1e4;
sol = randSphericalCap(coneAngleDegree, coneDir, N);
figure;plot3(sol(1,:), sol(2,:), sol(3,:), 'b.', 0,0,0,'rx');
grid
xlabel('x'); ylabel('y'); zlabel('z')
legend('random points','origin','location','best')
title('Final random points on spherical cap')
Here is a 3D plot of 10'000 points from the 30° spherical cap centered around the [1; 1; 1] vector:
Here’s 120° spherical cap:
Now, if you look at the histogram of the angles between these random points at the coneDir = [1;1;1], you will see that the distribution is skewed. Here’s the distribution:
Code to generate this:
normc = #(x) bsxfun(#rdivide, x, sqrt(sum(x.^2)));
mysubspace = #(a,b) real(acos(sum(bsxfun(#times, normc(a), normc(b)))));
angs = arrayfun(#(i) mysubspace(coneDir, sol(:,i)), 1:N) * 180/pi;
nBins = 16;
[n, edges] = histcounts(angs, nBins);
centers = diff(edges(1:2))*[0:(length(n)-1)] + mean(edges(1:2));
figure('color','white');
bar(centers, n);
xlabel('Angle (degrees)')
ylabel('Frequency')
title(sprintf('Histogram of angles between coneDir and random points: %d deg', coneAngleDegree))
Well, this makes sense! If you generate points from the 120° spherical cap around coneDir, of course the 1° cap is going to have very few of those samples, whereas the strip between the 10° and 11° caps will have far more points. So what we want to do is normalize the number of points at a given angle by the surface area of the spherical cap at that angle.
Here’s a function that gives us the surface area of the spherical cap with radius R and angle in radians theta (equation 16 on Mathworld’s spherical cap article):
rThetaToH = #(R, theta) R * (1 - cos(theta));
rThetaToS = #(R, theta) 2 * pi * R * rThetaToH(R, theta);
Then, we can normalize the histogram count for each bin (n above) by the difference in surface area of the spherical caps at the bin’s edges:
figure('color','white');
bar(centers, n ./ diff(rThetaToS(1, edges * pi/180)))
The figure:
This tells us “the number of random vectors divided by the surface area of the spherical segment between histogram bin edges”. This is uniform!
(N.B. If you do this normalized histogram for the vectors generated by your original code, using rejection sampling, the same holds: the normalized histogram is uniform. It’s just that rejection sampling is expensive compared to this.)
(N.B. 2: note that the naive way of picking random points on a sphere—by first generating azimuth/elevation angles and then converting these spherical coordinates to Cartesian coordinates—is no good because it bunches points near the poles: Mathworld, example, example 2. One way to pick points on the entire sphere is sampling from the 3D normal distribution: that way you won’t get bunching near poles. So I believe that your original technique is perfectly appropriate, giving you nice, evenly-distributed points on the sphere without any bunching. This algorithm described above also does the “right thing” and should avoid bunching. Carefully evaluate any proposed algorithms to ensure that the bunching-near-poles problem is avoided.)
it is better to use spherical coordinates and convert it to cartesian coordinates:
coneDirtheta = rand(1) * 2 * pi;
coneDirphi = rand(1) * pi;
coneAngle = 45;
N = 1000;
%perfom transformation preventing concentration of points around the pole
rpolar = acos(cos(coneAngle/2*pi/180) + (1-cos(coneAngle/2*pi/180)) * rand(N, 1));
thetapolar = rand(N,1) * 2 * pi;
x0 = rpolar .* cos(thetapolar);
y0 = rpolar .* sin(thetapolar);
theta = coneDirtheta + x0;
phi = coneDirphi + y0;
r = rand(N, 1);
x = r .* cos(theta) .* sin(phi);
y = r .* sin(theta) .* sin(phi);
z = r .* cos(phi);
scatter3(x,y,z)
if all points should be of length 1 set r = ones(N,1);
Edit:
since intersection of cone with sphere forms a circle first we create random points inside a circle with raduis of (45 / 2) in polar coordinates and as #Ahmed Fasih commented to prevent concentration of points near the pole we should first transform this random points, then convert polar to cartesian 2D coordinates to form x0 and y0
we can use x0 and y0 as phi & theta angle of spherical coordinates and add coneDirtheta & coneDirphi as offsets to these coordinates.
then convert spherical to cartesian 3D coordinates
I am trying to use Matlab's ifftn in 3-dimensions to get solution in physical space. In particular I am trying to use ifftn on 1/k^2. The analytical solution to that in physical space is 1/(4*pi*r). However I am not recovering that. Please note: $r = sqrt(x^2 + y^2 + z^2)$ and $k = sqrt(kx^2 + ky^2 + kz^2)$.
clc; clear;
n = 128; % no. of points for ifft
L = 2*pi; % size of the periodic domain
x = linspace(-L/2,L/2,n); y = x; z = x; % creating vectors for x-y-z direction
[X,Y,Z] = meshgrid(x,y,z); % creating meshgrid for physical space
R = sqrt(X.^2 + Y.^2 + Z.^2); % use for 1/(4*pi*r)
k = (2*pi/L)*[0:n/2 -n/2+1:-1]; % wave vector;
[k1,k2,k3] = meshgrid(k,k,k);
denom = (k1.^2 + k2.^2 + k3.^2); % This is k^2
F = 1./denom; F(1,1,1) = 0; % The first value is set to zero as it is infinite
phi = 1./(4*pi*R); % physical domain solution
phys = fftshift(ifftn(F)); % Using ifftn
ph_abs = abs(phys);
mid = ph_abs(n/2,:,:); % looking at the midplane of the output
mid = permute(mid,[3 2 1]); % permuting for contourplot.
PHI = phi(n/2,:,:); %looking at the midplane of the physical space.
PHI = permute(PHI,[3 2 1]);
figure(1)
surf(x,z,log(mid))
shading flat
colorbar();
figure(2)
surf(x,z,log10(abs(PHI)))
shading flat
colorbar();
I tried to apply the method posted in Find positive solutions to undetermined linear system of equations to the set
A=[0 0.0992 0.315 0.619 1; 0 0.248 0.315 0.248 0]; b=[0.1266 0.4363].
It is supposed that there exists a positive and constrained solution to this problem. The worst thing is that I have an answer code but I can't make it work because my Matlab version doesn`t recognize the anonymous function call and some instructions are obscure for me.
Here is the code:
% example_pt_source_atmos_setup.m
% determine geometry
D2 = 0.5; % diameter of the observation aperture [m]
wvl = 1e-6; % optical wavelength [m]
k = 2*pi / wvl; % optical wavenumber [rad/m]
Dz = 50e3; % propagation distance [m]
% use sinc to model pt source
DROI = 4 * D2; % diam of obs-plane region of interest [m]
D1 = wvl*Dz / DROI; % width of central lobe [m]
R = Dz; % wavefront radius of curvature [m]
% atmospheric properties
Cn2 = 1e-16; % structure parameter [m^-2/3], constant
% SW and PW coherence diameters [m]
r0sw = (0.423 * k^2 * Cn2 * 3/8 * Dz)^(-3/5);
r0pw = (0.423 * k^2 * Cn2 * Dz)^(-3/5);
p = linspace(0, Dz, 1e3);
% log-amplitude variance
rytov = 0.563 * k^(7/6) * sum(Cn2 * (1-p/Dz).^(5/6) ...
.* p.^(5/6) * (p(2)-p(1)));
% screen properties
nscr = 11; % number of screens
A = zeros(2, nscr); % matrix
alpha = (0:nscr-1) / (nscr-1);
A(1,:) = alpha.^(5/3);
A(2,:) = (1 - alpha).^(5/6) .* alpha.^(5/6);
b = [r0sw.^(-5/3); rytov/1.33*(k/Dz)^(5/6)];
% initial guess
x0 = (nscr/3*r0sw * ones(nscr, 1)).^(-5/3);
% objective function
fun = #(X) sum((A*X(:) - b).^2);
% constraints
x1 = zeros(nscr, 1);
rmax = 0.1; % maximum Rytov number per partial prop
x2 = rmax/1.33*(k/Dz)^(5/6) ./ A(2,:);
x2(A(2,:)==0) = 50^(-5/3)
[X,fval,exitflag,output] ...
= fmincon(fun,x0,[],[],[],[],x1,x2)
% check screen r0s
r0scrn = X.^(-3/5)
r0scrn(isinf(r0scrn)) = 1e6;
% check resulting r0sw & rytov
bp = A*X(:); [bp(1)^(-3/5) bp(2)*1.33*(Dz/k)^(5/6)]
[r0sw rytov]
Thanks for your attention.
Carolina Rickenstorff