I am looking for a command in MATLAB which could help me to plot a graph given the adjacency matrix. Can anyone help me? Further I need some graph tools to compute shortest distances between points on a graph, diameter of a set, distance between sets etc. Thanks
Check this Matlab Function by Haruna Matsushita
function [Xout,Yout,Zout]=gplot3(A,xy,lc)
% gplot‚Ì3ŽŸŒ³•\Ž¦
%
% 2005/04/11 Haruna MATSUSHITA
%GPLOT Plot graph, as in "graph theory".
% GPLOT(A,xy) plots the graph specified by A and xy. A graph, G, is
% a set of nodes numbered from 1 to n, and a set of connections, or
% edges, between them.
%
% In order to plot G, two matrices are needed. The adjacency matrix,
% A, has a(i,j) nonzero if and only if node i is connected to node
% j. The coordinates array, xy, is an n-by-2 matrix with the
% position for node i in the i-th row, xy(i,:) = [x(i) y(i)].
%
% GPLOT(A,xy,LineSpec) uses line type and color specified in the
% string LineSpec. See PLOT for possibilities.
%
% [X,Y] = GPLOT(A,xy) returns the NaN-punctuated vectors
% X and Y without actually generating a plot. These vectors
% can be used to generate the plot at a later time if desired.
%
% See also SPY, TREEPLOT.
% John Gilbert, 1991.
% Modified 1-21-91, LS; 2-28-92, 6-16-92 CBM.
% Copyright 1984-2002 The MathWorks, Inc.
% $Revision: 5.12 $ $Date: 2002/04/09 00:26:12 $
[i,j] = find(A);
[ignore, p] = sort(max(i,j));
i = i(p);
j = j(p);
% Create a long, NaN-separated list of line segments,
% rather than individual segments.
X = [ xy(i,1) xy(j,1) repmat(NaN,size(i))]';
Y = [ xy(i,2) xy(j,2) repmat(NaN,size(i))]';
Z = [ xy(i,3) xy(j,3) repmat(NaN,size(i))]';
X = X(:);
Y = Y(:);
Z = Z(:);
if nargout==0,
if nargin<3,
plot3(X, Y, Z)
else
plot3(X, Y, Z,lc,'MarkerFaceColor','none','MarkerEdgeColor','b','MarkerSize',5);
end
else
Xout = X;
Yout = Y;
Zout = Z;
end
Given the adjacency matrix M, plotting the corresponding directed graph in Matlab will be as easy as:
G = digraph(M);
plot(G)
Related
In MATLAB there is a function called cov. If I insert a matrix X into cov like this cov(X), then cov will return a square matrix of covariance.
My question is very simple:
How can I, with MATLAB, plot that matrix cov(X) onto a 2D plot like this.
I can see a lot of covariance matrix plots at Google. But how do they create them?
My best guess is that you're trying to add the principal components to the plot. To do that, you could do something like this.
%% generate data points
S_tru = [2 1; 1 1];
N = 1000;
%% compute mean, covariance, principal components
X = mvnrnd([0,0],S_tru,N);
mu = mean(X);
S = cov(X);
[U,D] = eig(S);
%% specify base points/directions for arrows
base = [mu;mu];
vecs = sqrt(D)*U';
vecs = 2 * vecs;
%% plot
plot(X(:,1),X(:,2), 'r.')
axis equal
hold on
quiver(base(:,1),base(:,2),vecs(:,1),vecs(:,2),'blue','LineWidth',2)
Resulting graph:
I am trying to fit a 3D surface polynomial of n-degrees to some data points in 3D space. My system requires the surface to be monotonically increasing in the area of interest, that is the partial derivatives must be non-negative. This can be achieved using Matlab's built in lsqlin function.
So here's what I've done to try and achieve this:
I have a function that takes in four parameters;
x1 and x2 are my explanatory variables and y is my dependent variable. Finally, I can specify order of polynomial fit. First I build the design matrix A using data from x1 and x2 and the degree of fit I want. Next I build the matrix D that is my container for the partial derivatives of my datapoints. NOTE: the matrix D is double the length of matrix A since all datapoints must be differentiated with respect to both x1 and x2. I specify that Dx >= 0 by setting b to be zeroes.
Finally, I call lsqlin. I use "-D" since Matlab defines the function as Dx <= b.
function w_mono = monotone_surface_fit(x1, x2, y, order_fit)
% Initialize design matrix
A = zeros(length(x1), 2*order_fit+2);
% Adjusting for bias term
A(:,1) = ones(length(x1),1);
% Building design matrix
for i = 2:order_fit+1
A(:,(i-1)*2:(i-1)*2+1) = [x1.^(i-1), x2.^(i-1)];
end
% Initialize matrix containing derivative constraint.
% NOTE: Partial derivatives must be non-negative
D = zeros(2*length(y), 2*order_fit+1);
% Filling matrix that holds constraints for partial derivatives
% NOTE: Matrix D will be double length of A since all data points will have a partial derivative constraint in both x1 and x2 directions.
for i = 2:order_fit+1
D(:,(i-1)*2:(i-1)*2+1) = [(i-1)*x1.^(i-2), zeros(length(x2),1); ...
zeros(length(x1),1), (i-1)*x2.^(i-2)];
end
% Limit of derivatives
b = zeros(2*length(y), 1);
% Constrained LSQ fit
options = optimoptions('lsqlin','Algorithm','interior-point');
% Final weights of polynomial
w_mono = lsqlin(A,y,-D,b,[],[],[],[],[], options);
end
So now I get some weights out, but unfortunately they do not at all capture the structure of the data. I've attached an image so you can just how bad it looks. .
I'll give you my plotting script with some dummy data, so you can try it.
%% Plot different order polynomials to data with constraints
x1 = [-5;12;4;9;18;-1;-8;13;0;7;-5;-8;-6;14;-1;1;9;14;12;1;-5;9;-10;-2;9;7;-1;19;-7;12;-6;3;14;0;-8;6;-2;-7;10;4;-5;-7;-4;-6;-1;18;5;-3;3;10];
x2 = [81.25;61;73;61.75;54.5;72.25;80;56.75;78;64.25;85.25;86;80.5;61.5;79.25;76.75;60.75;54.5;62;75.75;80.25;67.75;86.5;81.5;62.75;66.25;78.25;49.25;82.75;56;84.5;71.25;58.5;77;82;70.5;81.5;80.75;64.5;68;78.25;79.75;81;82.5;79.25;49.5;64.75;77.75;70.25;64.5];
y = [-6.52857142857143;-1.04736842105263;-5.18750000000000;-3.33157894736842;-0.117894736842105;-3.58571428571429;-5.61428571428572;0;-4.47142857142857;-1.75438596491228;-7.30555555555556;-8.82222222222222;-5.50000000000000;-2.95438596491228;-5.78571428571429;-5.15714285714286;-1.22631578947368;-0.340350877192983;-0.142105263157895;-2.98571428571429;-4.35714285714286;-0.963157894736842;-9.06666666666667;-4.27142857142857;-3.43684210526316;-3.97894736842105;-6.61428571428572;0;-4.98571428571429;-0.573684210526316;-8.22500000000000;-3.01428571428571;-0.691228070175439;-6.30000000000000;-6.95714285714286;-2.57232142857143;-5.27142857142857;-7.64285714285714;-2.54035087719298;-3.45438596491228;-5.01428571428571;-7.47142857142857;-5.38571428571429;-4.84285714285714;-6.78571428571429;0;-0.973684210526316;-4.72857142857143;-2.84285714285714;-2.54035087719298];
% Used to plot the surface in all points in the grid
X1 = meshgrid(-10:1:20);
X2 = flipud(meshgrid(30:2:90).');
figure;
for i = 1:4
w_mono = monotone_surface_fit(x1, x2, y, i);
y_nr = w_mono(1)*ones(size(X1)) + w_mono(2)*ones(size(X2));
for j = 1:i
y_nr = w_mono(j*2)*X1.^j + w_mono(j*2+1)*X2.^j;
end
subplot(2,2,i);
scatter3(x1, x2, y); hold on;
axis tight;
mesh(X1, X2, y_nr);
set(gca, 'ZDir','reverse');
xlabel('x1'); ylabel('x2');
zlabel('y');
% zlim([-10 0])
end
I think it may have something to do with the fact that I haven't specified anything about the region of interest, but really I don't know. Thanks in advance for any help.
Alright I figured it out.
The main problem was simply an error in the plotting script. The value of y_nr should be updated and not overwritten in the loop.
Also I figured out that the second derivative should be monotonically decreasing. Here's the updated code if anybody is interested.
%% Plot different order polynomials to data with constraints
x1 = [-5;12;4;9;18;-1;-8;13;0;7;-5;-8;-6;14;-1;1;9;14;12;1;-5;9;-10;-2;9;7;-1;19;-7;12;-6;3;14;0;-8;6;-2;-7;10;4;-5;-7;-4;-6;-1;18;5;-3;3;10];
x2 = [81.25;61;73;61.75;54.5;72.25;80;56.75;78;64.25;85.25;86;80.5;61.5;79.25;76.75;60.75;54.5;62;75.75;80.25;67.75;86.5;81.5;62.75;66.25;78.25;49.25;82.75;56;84.5;71.25;58.5;77;82;70.5;81.5;80.75;64.5;68;78.25;79.75;81;82.5;79.25;49.5;64.75;77.75;70.25;64.5];
y = [-6.52857142857143;-1.04736842105263;-5.18750000000000;-3.33157894736842;-0.117894736842105;-3.58571428571429;-5.61428571428572;0;-4.47142857142857;-1.75438596491228;-7.30555555555556;-8.82222222222222;-5.50000000000000;-2.95438596491228;-5.78571428571429;-5.15714285714286;-1.22631578947368;-0.340350877192983;-0.142105263157895;-2.98571428571429;-4.35714285714286;-0.963157894736842;-9.06666666666667;-4.27142857142857;-3.43684210526316;-3.97894736842105;-6.61428571428572;0;-4.98571428571429;-0.573684210526316;-8.22500000000000;-3.01428571428571;-0.691228070175439;-6.30000000000000;-6.95714285714286;-2.57232142857143;-5.27142857142857;-7.64285714285714;-2.54035087719298;-3.45438596491228;-5.01428571428571;-7.47142857142857;-5.38571428571429;-4.84285714285714;-6.78571428571429;0;-0.973684210526316;-4.72857142857143;-2.84285714285714;-2.54035087719298];
% Used to plot the surface in all points in the grid
X1 = meshgrid(-10:1:20);
X2 = flipud(meshgrid(30:2:90).');
figure;
for i = 1:4
w_mono = monotone_surface_fit(x1, x2, y, i);
% NOTE: Should only have 1 bias term
y_nr = w_mono(1)*ones(size(X1));
for j = 1:i
y_nr = y_nr + w_mono(j*2)*X1.^j + w_mono(j*2+1)*X2.^j;
end
subplot(2,2,i);
scatter3(x1, x2, y); hold on;
axis tight;
mesh(X1, X2, y_nr);
set(gca, 'ZDir','reverse');
xlabel('x1'); ylabel('x2');
zlabel('y');
% zlim([-10 0])
end
And here's the updated function
function [w_mono, w] = monotone_surface_fit(x1, x2, y, order_fit)
% Initialize design matrix
A = zeros(length(x1), 2*order_fit+1);
% Adjusting for bias term
A(:,1) = ones(length(x1),1);
% Building design matrix
for i = 2:order_fit+1
A(:,(i-1)*2:(i-1)*2+1) = [x1.^(i-1), x2.^(i-1)];
end
% Initialize matrix containing derivative constraint.
% NOTE: Partial derivatives must be non-negative
D = zeros(2*length(y), 2*order_fit+1);
for i = 2:order_fit+1
D(:,(i-1)*2:(i-1)*2+1) = [(i-1)*x1.^(i-2), zeros(length(x2),1); ...
zeros(length(x1),1), -(i-1)*x2.^(i-2)];
end
% Limit of derivatives
b = zeros(2*length(y), 1);
% Constrained LSQ fit
options = optimoptions('lsqlin','Algorithm','active-set');
w_mono = lsqlin(A,y,-D,b,[],[],[],[],[], options);
w = lsqlin(A,y);
end
Finally a plot of the fitting (Have used a new simulation, but fit also works on given dummy data).
I want to plot circles in Hadamard matrix pattern of order 8,16, and 32. So far, I have a code for plotting 2D arrays of circles.
%Plotting an N by N arrays of circles
clc; clear;
n_circles = 8; % Define the number of circles to be plotted
R = 40; % Define the radius of the basic circle
Len=1024;
M=zeros(Len); % Create the hole mask
% Get the indices of the points inside the basic circle
M0 = zeros(2*R+1); % Initialize the basic mask
I = 1:(2*R+1); % Define the x and y coordinates of the basic mask
x = (I - R)-1;
y = (R - I)+1;
[X,Y] = meshgrid(x,y); % Create the mask
A = (X.^2 + Y.^2 <= R^2);
[xx,yy]=ind2sub(size(M0),find(A == true));
%plot
for ii=1:n_circles
for jj=1:n_circles
MidX=Len/2+(ii-n_circles/2-0.5)*(2*R);
MidY=Len/2+(jj-n_circles/2-0.5)*(2*R);
% [MidX MidY]
M(sub2ind(size(M),MidX+xx-R-1,MidY+yy-R-1))=1;
end
end
figure(1)
imshow(M)
I searched on how to plot a Hadamard matrix, and from the Mathworks documentation, the hadamard matrix function
H = hadamard(n)
returns the Hadamard matrix of order n. How do I incorporate this in my original code so that the final result will generate an image of circles plotted in a Hadamard pattern, where the value of 1 indicates a circle while -1 is null (absence of circle)?
Thanks,
add in th begining
H = hadamard(n_circles);
and inside the loops change to:
M(sub2ind(size(M),MidX+xx-R-1,MidY+yy-R-1))=H(ii,jj);
I am coding a Gaussian Process regression algorithm. Here is the code:
% Data generating function
fh = #(x)(2*cos(2*pi*x/10).*x);
% range
x = -5:0.01:5;
N = length(x);
% Sampled data points from the generating function
M = 50;
selection = boolean(zeros(N,1));
j = randsample(N, M);
% mark them
selection(j) = 1;
Xa = x(j);
% compute the function and extract mean
f = fh(Xa) - mean(fh(Xa));
sigma2 = 1;
% computing the interpolation using all x's
% It is expected that for points used to build the GP cov. matrix, the
% uncertainty is reduced...
K = squareform(pdist(x'));
K = exp(-(0.5*K.^2)/sigma2);
% upper left corner of K
Kaa = K(selection,selection);
% lower right corner of K
Kbb = K(~selection,~selection);
% upper right corner of K
Kab = K(selection,~selection);
% mean of posterior
m = Kab'*inv(Kaa+0.001*eye(M))*f';
% cov. matrix of posterior
D = Kbb - Kab'*inv(Kaa + 0.001*eye(M))*Kab;
% sampling M functions from from GP
[A,B,C] = svd(Kaa);
F0 = A*sqrt(B)*randn(M,M);
% mean from GP using sampled points
F0m = mean(F0,2);
F0d = std(F0,0,2);
%%
% put together data and estimation
F = zeros(N,1);
S = zeros(N,1);
F(selection) = f' + F0m;
S(selection) = F0d;
% sampling M function from posterior
[A,B,C] = svd(D);
a = A*sqrt(B)*randn(N-M,M);
% mean from posterior GPs
Fm = m + mean(a,2);
Fmd = std(a,0,2);
F(~selection) = Fm;
S(~selection) = Fmd;
%%
figure;
% show what we got...
plot(x, F, ':r', x, F-2*S, ':b', x, F+2*S, ':b'), grid on;
hold on;
% show points we got
plot(Xa, f, 'Ok');
% show the whole curve
plot(x, fh(x)-mean(fh(x)), 'k');
grid on;
I expect to get some nice figure where the uncertainty of unknown data points would be big and around sampled data points small. I got an odd figure and even odder is that the uncertainty around sampled data points is bigger than on the rest. Can someone explain to me what I am doing wrong? Thanks!!
There are a few things wrong with your code. Here are the most important points:
The major mistake that makes everything go wrong is the indexing of f. You are defining Xa = x(j), but you should actually do Xa = x(selection), so that the indexing is consistent with the indexing you use on the kernel matrix K.
Subtracting the sample mean f = fh(Xa) - mean(fh(Xa)) does not serve any purpose, and makes the circles in your plot be off from the actual function. (If you choose to subtract something, it should be a fixed number or function, and not depend on the randomly sampled observations.)
You should compute the posterior mean and variance directly from m and D; no need to sample from the posterior and then obtain sample estimates for those.
Here is a modified version of the script with the above points fixed.
%% Init
% Data generating function
fh = #(x)(2*cos(2*pi*x/10).*x);
% range
x = -5:0.01:5;
N = length(x);
% Sampled data points from the generating function
M = 5;
selection = boolean(zeros(N,1));
j = randsample(N, M);
% mark them
selection(j) = 1;
Xa = x(selection);
%% GP computations
% compute the function and extract mean
f = fh(Xa);
sigma2 = 2;
sigma_noise = 0.01;
var_kernel = 10;
% computing the interpolation using all x's
% It is expected that for points used to build the GP cov. matrix, the
% uncertainty is reduced...
K = squareform(pdist(x'));
K = var_kernel*exp(-(0.5*K.^2)/sigma2);
% upper left corner of K
Kaa = K(selection,selection);
% lower right corner of K
Kbb = K(~selection,~selection);
% upper right corner of K
Kab = K(selection,~selection);
% mean of posterior
m = Kab'/(Kaa + sigma_noise*eye(M))*f';
% cov. matrix of posterior
D = Kbb - Kab'/(Kaa + sigma_noise*eye(M))*Kab;
%% Plot
figure;
grid on;
hold on;
% GP estimates
plot(x(~selection), m);
plot(x(~selection), m + 2*sqrt(diag(D)), 'g-');
plot(x(~selection), m - 2*sqrt(diag(D)), 'g-');
% Observations
plot(Xa, f, 'Ok');
% True function
plot(x, fh(x), 'k');
A resulting plot from this with 5 randomly chosen observations, where the true function is shown in black, the posterior mean in blue, and confidence intervals in green.
My question is pretty standard but can't find a solution of that.
I have points=[x,y,z] and want to plot best fit line.
I am using function given below (and Thanx Smith)
% LS3DLINE.M Least-squares line in 3 dimensions.
%
% Version 1.0
% Last amended I M Smith 27 May 2002.
% Created I M Smith 08 Mar 2002
% ---------------------------------------------------------------------
% Input
% X Array [x y z] where x = vector of x-coordinates,
% y = vector of y-coordinates and z = vector of
% z-coordinates.
% Dimension: m x 3.
%
% Output
% x0 Centroid of the data = point on the best-fit line.
% Dimension: 3 x 1.
%
% a Direction cosines of the best-fit line.
% Dimension: 3 x 1.
%
% <Optional...
% d Residuals.
% Dimension: m x 1.
%
% normd Norm of residual errors.
% Dimension: 1 x 1.
% ...>
%
% [x0, a <, d, normd >] = ls3dline(X)
I have a.
So equation may be
points*a+dist=0
where dist is min. distance from origon.
Now my question is how to plot best filt line in 3D.
It helps to actually read the content of the function, which uses Singular Value Decomposition.
% calculate centroid
x0 = mean(X)';
% form matrix A of translated points
A = [(X(:, 1) - x0(1)) (X(:, 2) - x0(2)) (X(:, 3) - x0(3))];
% calculate the SVD of A
[U, S, V] = svd(A, 0);
% find the largest singular value in S and extract from V the
% corresponding right singular vector
[s, i] = max(diag(S));
a = V(:, i);
The best orthogonal fitting line is
P = x0 + a.*t
as the parameter t varies. This is the direction of maximum variation which means that variation in the orthogonal direction is minimum. The sum of the squares of the points' orthogonal distances to this line is minimized.
This is distinct from linear regression which minimizes the y variation from the line of regression. That regression assumes that all errors are in the y coordinates, whereas orthogonal fitting assumes the errors in both the x and y coordinates are of equal expected magnitudes.
[Credit: Roger Stafford , http://www.mathworks.com/matlabcentral/newsreader/view_thread/294030]
Then you only need to create some t and plot it:
for t=0:100,
P(t,:) = x0 + a.*t;
end
scatter3(P(:,1),P(:,2),P(:,3));
You may want to use plot3() instead, in which case you need only a pair of points. Since a line is infinite by definition, it is up to you to determine where it should begin and end (depends on application).