I have a program that does some work to get a matrix w, which is 3(n+1) by 3(n+1). I have a vector fbar that is 3(n+1) by 1. I want to get the matrix that, when w is multiplied by it, gives fbar.
In mathematical notation, w * A = fbar. I have w and fbar, and I want A.
I tried to solve it with this command:
fsolve({seq(multiply(w, A)[i, 1] = fbar[i, 1], i = 1 .. 3*(n+1))})
but I don't understand the response Maple gave:
fsolve({2.025881905 A1[2,1]+7.814009150 A1[3,1]+...
-7.071067816 10^(-13) A1[3,1]-0.0004999999990
A1[4,1]-0.0007071067294 A1[5,1]-0.0004999999990 A1[6,1])
A3[6,1]=0},{A1[1,1],A1[2,1],A1[3,1],A1[4,1],A1[5,1],A1[6,1],A\
2[1,1],A2[2,1],A2[3,1],A2[4,1],A2[5,1],A2[6,1],A3[1,1],A3[2,1]\
,A3[3,1],A3[4,1],A3[5,1],A3[6,1]})
What does this mean, and how can I get a more meaningful answer?
You can do this directly with the LinearSolve functional from the LinearAlgebra package if w and fbar are defined as a matrix and vector respectively. The below code makes a reproducible example. Note that the solution of LinearSolve should be equal to x.
w := Matrix(<<1,2,3>|<4,5,6>|<7,8,10>>);
LinearAlgebra[ReducedRowEchelonForm](%); ## Full rank => 1 solution)
x := <1,2,3>;
fbar := w.x;
## Solve the equation w.x = fbar
LinearAlgebra[LinearSolve](w,fbar);
Related
Finding m and c for an equation y = mx + c, with the help of math and plots.
y is data_model_1, x is time.
Avoid other MATLAB functions like fitlm as it defeats the purpose.
I am having trouble finding the constants m and c. I am trying to find both m and c by limiting them to a range (based on smart guess) and I need to deduce the m and c values based on the mean error range. The point where mean error range is closest to 0 should be my m and c values.
load(file)
figure
plot(time,data_model_1,'bo')
hold on
for a = 0.11:0.01:0.13
c = -13:0.1:-10
data_a = a * time + c ;
plot(time,data_a,'r');
end
figure
hold on
for a = 0.11:0.01:0.13
c = -13:0.1:-10
data_a = a * time + c ;
mean_range = mean(abs(data_a - data_model_1));
plot(a,mean_range,'b.')
end
A quick & dirty approach
You can quickly get m and c using fminsearch(). In the first example below, the error function is the sum of squared error (SSE). The second example uses the sum of absolute error. The key here is ensuring the error function is convex.
Note that c = Beta(1) and m = Beta(2).
Reproducible example (MATLAB code[1]):
% Generate some example data
N = 50;
X = 2 + 13*random(makedist('Beta',.7,.8),N,1);
Y = 5 + 1.5.*X + randn(N,1);
% Example 1
SSEh =#(Beta) sum((Y - (Beta(1) + (Beta(2).*X))).^2);
Beta0 = [0.5 0.5]; % Initial Guess
[Beta SSE] = fminsearch(SSEh,Beta0)
% Example 2
SAEh =#(Beta) sum(abs(Y-(Beta(1) + Beta(2).*X)));
[Beta SumAbsErr] = fminsearch(SAEh,Beta0)
This is a quick & dirty approach that can work for many applications.
#Wolfie's comment directs you to the analytical approach to solve a system of linear equations with the \ operator or mldivide(). This is the more correct approach (though it will get a similar answer). One caveat is this approach gets the SSE answer.
[1] Tested with MATLAB R2018a
I have a 30x30 matrix as a base matrix (OD_b1), I also have two base vectors (bg and Ag). My aim is to optimize a matrix (X) who's dimensions are 30X30 such that:
1) the squared difference between vector (bg) and vector of sum of all the columns is minimized.
2)the squared difference between vector (Ag) and vector of sum of all rows is minimized.
3)the squared difference between the elements of matrix (X) and matrix (OD_b1) is minimized.
The mathematical form of the equation is as follows:
I have tried this:
fun=#(X)transpose(bg-sum(X,2))*(bg-sum(X,2))+ (Ag-sum(X,1))*transpose(Ag-sum(X,1))+sumsqr(X_b-X);
[val,X]=fmincon(fun,OD_b1,AA,BB,Aeq,beq,LB,UB)
I don't get errors but it seems like it's stuck.
Is it because I have too many variables or is there another reason?
Thanks in advance
This is a simple, unconstrained least squares problem and hence has a simple solution that can be expressed as the solution to a linear system.
I will show you (1) the precise and efficient way to solve this and (2) how to solve with fmincon.
The precise, efficient solution:
Problem setup
Just so we're on the same page, I initialize the variables as follows:
n = 30;
Ag = randn(n, 1); % observe the dimensions
X_b = randn(n, n);
bg = randn(n, 1);
The code:
A1 = kron(ones(1,n), eye(n));
A2 = kron(eye(n), ones(1,n));
A = (A1'*A1 + A2'*A2 + eye(n^2));
b = A1'*bg + A2'*Ag + X_b(:);
x = A \ b; % solves A*x = b
Xstar = reshape(x, n, n);
Why it works:
I first reformulated your problem so the objective is a vector x, not a matrix X. Observe that z = bg - sum(X,2) is equivalent to:
x = X(:) % vectorize X
A1 = kron(ones(1,n), eye(n)); % creates a special matrix that sums up
% stuff appropriately
z = A1*x;
Similarly, A2 is setup so that A2*x is equivalent to Ag'-sum(X,1). Your problem is then equivalent to:
minimize (over x) (bg - A1*x)'*(bg - A1*x) + (Ag - A2*x)'*(Ag - A2*x) + (y - x)'*(y-x) where y = Xb(:). That is, y is a vectorized version of Xb.
This problem is convex and the first order condition is a necessary and sufficient condition for the optimum. Take the derivative with respect to x and that equation will define your solution! Sample example math for almost equivalent (but slightly simpler problem is below):
minimize(over x) (b - A*x)'*(b - A*x) + (y - x)' * (y - x)
rewriting the objective:
b'b- b'Ax - x'A'b + x'A'Ax +y'y - 2y'x+x'x
Is equivalent to:
minimize(over x) (-2 b'A - 2y'*I) x + x' ( A'A + I) * x
the first order condition is:
(A'A+I+(A'A+I)')x -2A'b-2I'y = 0
(A'A+I) x = A'b+I'y
Your problem is essentially the same. It has the first order condition:
(A1'*A1 + A2'*A2 + I)*x = A1'*bg + A2'*Ag + y
How to solve with fmincon
You can do the following:
f = #(X) transpose(bg-sum(X,2))*(bg-sum(X,2)) + (Ag'-sum(X,1))*transpose(Ag'-sum(X,1))+sum(sum((X_b-X).^2));
o = optimoptions('fmincon');%MaxFunEvals',30000);
o.MaxFunEvals = 30000;
Xstar2 = fmincon(f,zeros(n,n),[],[],[],[],[],[],[],o);
You can then check the answers are about the same with:
normdif = norm(Xstar - Xstar2)
And you can see that gap is small, but that the linear algebra based solution is somewhat more precise:
gap = f(Xstar2) - f(Xstar)
If the fmincon approach hangs, try it with a smaller n just to gain confidence that my linear algebra based solution is more precise, way way faster etc... n = 30 is solving a 30^2 = 900 variable optimization problem: not easy. With the linear algebra approach, you can go up to n = 100 (i.e. 10000 variable problem) or even larger.
I would probably solve this as a QP using quadprog using the following reformulation (keeping the objective as simple as possible to make the problem "less nonlinear"):
min sum(i,v(i)^2)+sum(i,w(i)^2)+sum((i,j),z(i,j)^2)
v = bg - sum(c,x)
w = ag - sum(r,x)
Z = xbase-x
The QP solver is more precise (no gradients using finite differences). This approach also allows you to add additional bounds and linear equality and inequality constraints.
The other suggestion to form the first order conditions explicitly is also a good one: it also has no issue with imprecise gradients (the first order conditions are linear). I usually prefer a quadratic model because of its flexibility.
I have data like this:
y = [0.001
0.0042222222
0.0074444444
0.0106666667
0.0138888889
0.0171111111
0.0203333333
0.0235555556
0.0267777778
0.03]
and
x = [3.52E-06
9.72E-05
0.0002822918
0.0004929136
0.0006759156
0.0008199029
0.0009092797
0.0009458332
0.0009749509
0.0009892005]
and I want y to be a function of x with y = a(0.01 − b*n^−cx).
What is the best and easiest computational approach to find the best combination of the coefficients a, b and c that fit to the data?
Can I use Octave?
Your function
y = a(0.01 − b*n−cx)
is in quite a specific form with 4 unknowns. In order to estimate your parameters from your list of observations I would recommend that you simplify it
y = β1 + β2β3x
This becomes our objective function and we can use ordinary least squares to solve for a good set of betas.
In default Matlab you could use fminsearch to find these β parameters (lets call it our parameter vector, β), and then you can use simple algebra to get back to your a, b, c and n (assuming you know either b or n upfront). In Octave I'm sure you can find an equivalent function, I would start by looking in here: http://octave.sourceforge.net/optim/index.html.
We're going to call fminsearch, but we need to somehow pass in your observations (i.e. x and y) and we will do that using anonymous functions, so like example 2 from the docs:
beta = fminsearch(#(x,y) objfun(x,y,beta), beta0) %// beta0 are your initial guesses for beta, e.g. [0,0,0] or [1,1,1]. You need to pick these to be somewhat close to the correct values.
And we define our objective function like this:
function sse = objfun(x, y, beta)
f = beta(1) + beta(2).^(beta(3).*x);
err = sum((y-f).^2); %// this is the sum of square errors, often called SSE and it is what we are trying to minimise!
end
So putting it all together:
y= [0.001; 0.0042222222; 0.0074444444; 0.0106666667; 0.0138888889; 0.0171111111; 0.0203333333; 0.0235555556; 0.0267777778; 0.03];
x= [3.52E-06; 9.72E-05; 0.0002822918; 0.0004929136; 0.0006759156; 0.0008199029; 0.0009092797; 0.0009458332; 0.0009749509; 0.0009892005];
beta0 = [0,0,0];
beta = fminsearch(#(x,y) objfun(x,y,beta), beta0)
Now it's your job to solve for a, b and c in terms of beta(1), beta(2) and beta(3) which you can do on paper.
I am trying to create a custom function for pearson's correlation coefficient with this code in matlab 2010
function [p] = customcorr(o)
x := a
y := b
x_mean := mean(a)
y_mean := mean(b)
x_std := std(a)
y_std := std(b)
n := length(o)
r := (1/(n-1))*((x-x_mean)*(y-y_mean))/(x_std*y_std)
end
But i get an error when trying to execute it
Error in ==> customcorr at 2
x := a
Anybody might know what the problem is? Thank you
First, check the correct MATLAB syntax: a "normal" assignment is done by =, not by :=.
Second, you use a and b but these are not defined as parameters of the function. Replace the function head by function p = customcorr(a,b).
Third, I am not really sure what o should be, I assume it can be replaced by length(a) or length(b).
The estimator for an unbiased correlation coefficient is given by
(from wikipedia)
Thus you need to sum all the (a-a_mean).*(b-b_mean) up with sum. Note that it is required to write .* to get the element-wise multiplication. That way you subtract the mean from each element of the vectors, then multiply the corresponding a's and b's and sum up the results of these multiplications.
Together this is
function p = customcorr(a,b)
a_mean = mean(a);
b_mean = mean(b);
a_std = std(a);
b_std = std(b);
n = length(a);
p = (1/(n-1)) * sum((a-a_mean).*(b-b_mean)) / (a_std*b_std);
end
What MATLAB does in their corr function (besides many other interesting things) is, they check the number of arguments (nargin variable) to see if a and b were supplied or not. You can do that by adding the following code to the function (at the beginning)
if nargin < 2
b = a;
end
I have equation F(f)=a*f^3+b*f+c. I have known vectors of the data, p, independent variable, 'f'. I need to find values of a, b, c.
What I tried:
function [ val ] = myfunc(par_fit,f,p)
% This gives me a,b,c
% p= af^3 +bf +c
val = norm(p - (par_fit(1)*(f.^3))+ (par_fit(2)*f) + (par_fit(3)));
end
my_par = fminsearch(#(par_fit) myfunc(par_fit,f,p),rand(1,3));
This gives me my_par = [1.9808 -2.2170 -24.8039], or a=1.9808, b=-2.2170, and c=-24.8039, but I require that b should be larger than 5, and c should be larger than zero.
I think your problem might be because your objective function is incorrect:
val = norm(p - (par_fit(1)*(f.^3))+ (par_fit(2)*f) + (par_fit(3)));
should probably be:
val = norm(p-(par_fit(1)*f.^3+par_fit(2)*f+par_fit(3)));
But you can constrain the values of variables when you do minimisation by using fmincon rather than fminsearch. By setting the lb input to [-Inf -Inf 0], the first two coefficients are allowed to be any real number, but the third coefficient must be greater than or equal to zero. For example: (I've also shown how to solve the problem (without the non-negativity constraint) using a matrix method)
% Sample data
f=(0:.1:1).';
p=2*f.^3+3*f+1+randn(size(f))
% Create Van der Monde matrix
M=[f.^3 f f.^0];
C=M\p; % Solve the matrix problem in a least squares sense if size(f)>size(F)
my_par=fmincon(#(c) norm(p-(c(1)*f.^3+c(2)*f+c(3))),rand(1,3),[],[],[],[],[-Inf 5 0],[])
C.'
plot(f,p,'o',f,M*C,f,my_par(1)*f.^3+my_par(2)*f+my_par(3))