I'm fitting custom functions to my data.
After obtaining the fit I would like to get something like a function handle of my fit function including the parameters set to the ones found by the fit.
I know I can get the model with
formula(fit)
and I can get the parameters with
coeffvalues(fit)
but is there any easy way to combine the two in one step?
This little loop will do the trick:
x = (1:100).'; %'
y = 1*x.^5 + 2*x.^4 + 3*x.^3 + 4*x.^2 + 5*x + 6;
fitobject = fit(x,y,'poly5');
cvalues = coeffvalues(fitobject);
cnames = coeffnames(fitobject);
output = formula(fitobject);
for ii=1:1:numel(cvalues)
cname = cnames{ii};
cvalue = num2str(cvalues(ii));
output = strrep(output, cname , cvalue);
end
output = 1*x^5 + 2*x^4 + 3*x^3 + 4*x^2 + 5*x + 6
The loop needs to be adapted to the number of coefficients of your fit.
Edit: two slight changes in order to fully answer the question.
fhandle = #(x) eval(output)
returns a function handle. Secondly output as given by your procedure doesn't work, as the power operation reads .^ instead of x, which can obviously be replaced by
strrep(output, '^', '.^');
You can use the Matlab curve fitting function, polyfit.
p = polyfit(x,y,n)
So, p contains the coefficients of the polynomial, x and y are the coordinates of the function you're trying to fit. n is the order of the polynomial. For example, n=1 is linear, n=2 is quadratic, etc. For more info, see this documentation centre link. The only issue is that you may not want a polynomial fit, in which case you'll have to use different method.
Oh, and you can use the calculated coefficients p to to re-evaluate the polynomial with:
f = polyval(p,x);
Here, f is the value of the polynomial with coefficients p evaluated at points x.
Related
I am wanting to take a matlab anonymous function of something like #(x) = x + x^2 and split it into a cell array so that
f{1} = #(x) x
f{2} = #(x) x^2
I want to be able to do this with some arbitrary function handle. The best that I have come up with so far is taking in #(x) = x + x^2 as a string and splitting it by the addition signs and appending #(x) to the beginning of this. However, I would like to be able to directly use a function handle as the function argument. It would also be nice as using other variables could lead to some difficulty in the string approach.
I am also considering just taking in a cell array of function handles as an argument, which would be more difficult for the user but easier in my code.
For some background, I'm wanting to do this for some least squares data fitting code that I am writing for a class. This code will be for taking in a model function as an argument and I need to evaluate each term separately for the least squares process. I'm not limiting these models to polynomials and even if a polynomial is the model, I want the option to leave out certain powers in the polynomial. If someone has a better suggestion for taking a model function, that would be great too.
UPDATE: Someone wanted to know what I meant by
I'm not limiting these models to polynomials and even if a polynomial
is the model, I want the option to leave out certain powers in the
polynomial.
For some clarification, I was saying that I didn't want to limit the models to
c0 + c1*x + ... + cn*x^n
in which case I could just take in n as a parameter and create terms from that similar to what happens in polyfit. For example, if I know that my input data fits an even or odd function, I may want one of the following models
c0 + c1*x^2 + c2*x^4 + ... + ck*x^(2k)
c1*x + c2*x^3 + ... + cm*x^(2m-1)
Where k is even and m is odd. Or possibly a model that isn't strictly a polynomial, but keeps the coefficients linear, such as
c0 + exp(x) * ( c1 + c2*x + ... cn*x^(n-1) )
This is an interesting problem. The string should not be split at + signs that are within a parenthesis group. For example, with
f = #(x) x + (x+1)*sqrt(x) + x^2 + exp(x+2);
the string should be split at the first, but not at the second + sign.
This can be accomplished as follows. To detect only + signs that are outside parentheses, add 1 for each opening parenthesis and subtract 1 for each closing parenthesis. Then the desired + signs are those with count 0.
I'm assuming the output should be a cell array of function handles. If it should be a cell array of strings just remove the last line.
F = functions(f);
str = F.function; %// get string from function handle
[pref, body] = regexp(str, '#\(.+?\)', 'match', 'split'); %// pref is the '#(...)' part
body = body{2}; %// body is the main part
ind = cumsum((body=='(')-(body==')'))==0 & body=='+'; %// indices for splitting
body(ind) = '?'; %// '?' will be used as split marker
ff = strcat(pref, strsplit(body, '?')); %// split, and then add prefix
ff = cellfun(#str2func, ff, 'uniformoutput', 0); %// convert strings to functions
Result in this example:
ff{1} =
#(x)x
ff{2} =
#(x)(x+1)*sqrt(x)
ff{3} =
#(x)x^2
ff{4} =
#(x)exp(x+2)
but i want to make a program in which i can generate a function of multiple variables that depend on the number of rows of a matrix.
for k = 1:sizel;
f(k)=(alpha(k,1)+(beta(k,1)*p(k))+(gamma(k,1)*p(k)^2));
end
cost=(sum(f))
this is for the purpose of optimization so i need that at the end the variables are declares as p(1),p(2),p(3)... these will be the input for my function.
Note: i dont want to assign values to the variables because this will be done be the optimization algorithm in the optimization toolbox.
here is the complete code
function cost = cost(p) ;
clc
clear
costfunctionconstantsmatrix;
sizel=size(CostFormulaconstants);
alpha=CostFormulaconstants(:,1);
beta=CostFormulaconstants(:,2);
gamma=CostFormulaconstants(:,3);
for k = 1:sizel;
f(k)=(alpha(k,1)+(beta(k,1)*p(k))+(gamma(k,1)*p(k)^2));
end
cost=(sum(f))
end
i used the symbolic approach and i got the correct answer for the cost indeed, i got something like this: (53*p(1))/10 + (11*p(2))/2 + (29*p(3))/5 + p(1)^2/250 + (3*p(2)^2)/500 + (9*p(3)^2)/1000 + 1100. But when i try to specify my function to be optimized in the optimization toolbox it tells me that the variables p are sym and cannot be converted to double. the trouble is how to convert this expression to double so that the optimization algorithm can input values for the variable p(1), p(2) and p(3)
Can you pass the matrix as an argument to the function?
function cost = fcn(my_mat)
[m,n] = size(my_mat);
f = zeros(m,1);
for k = 1:m % for example
f(k)=(alpha(k,1)+(beta(k,1)*p(k))+(gamma(k,1)*p(k)^2));
end
cost = sum(f);
end
Your problem is not entirely clear to me but I believe you wish to generate a series of functions in which the variables alpha, beta, gamma are constants with different values for each function, and the vector p is an argument.
What confuses me in your question is that you use the index k for both the constants and the arguments, which I think is not what you intended to write. Assuming I understand your goal, a solution may make use of function handles.
The notation f(k) = #(p) p(1)+p(2) for example, generates a function that adds p(1) and p(2). Abbreviating CostFormulaconstants to cf, the following would generate a series of functions, one for each row in cf.
for k = 1 : size(cf, 1)
f{k} = #(p) cf(k,1) + cf(k,2)*p(1) + cf(k,3)*p(2)^2;
end
You can supply individual function handles to callers from the optimization toolbox simply with f{3} for the third function, for example. A call to f{3} would look like
a = f{3}([3,4]);
If your functions are indeed all polynomials, polyval may be worth a look as well.
EDIT: After clarification, the problem seems a bit simpler, no need for function handles. Why not simply
function c = cost(p)
c = 0;
cf = [...]; % your coefficients here.
for k = 1 : size(cf, 1)
c = c + cf(k,1) + cf(k,2)*p(k) + cf(k,3)*p(k)^2;
end
I need to implement Lagrange iterpolation in MATLAB.
I (think I've) understood how it works. I don't get how to implement the x.
lets say I want to calculate for these point: (0,1) (1,1) (2,4)
So I need to do these:
l_0(x) = (x-1)(x-2)/(0-1)(0-2)
l_1(x) = (x-0)(x-2)/(1-0)(1-2)
l_2(x) = (x-0)(x-1)/(2-0)(2-1)
and so on...
So I want to do a MATLAB function that will receive the (x,y) points, and retrieves the coefficients of the resulting Polynomial.
In this case: ( 3/2, 3/2, 1 )
I DON'T WANT A CODE FOR AN ANSWER - just how to implement the above x variant.
Thanks
I'm not sure if this is what you need, but I think that what you are looking for is MATLAB anonymous functions
In your case, you would write
l_0 = #(x) (x-1)(x-2)/(0-1)(0-2)
l_1 = #(x) (x-0)(x-2)/(1-0)(1-2)
l_2 = #(x) (x-0)(x-1)/(2-0)(2-1)
Then you can use your Lagrange polynomials like regular functions:
val = y0 * l_0(x0) + y1 * l_1(x1) + y2 * l_2(x2)
Is that what you were looking for?
Well if you don't want code, then x is simply any value within the range of the input values of your x points. In your case, any value between 0 and 2.
I’m currently a Physics student and for several weeks have been compiling data related to ‘Quantum Entanglement’. I’ve now got to a point where I have to plot my data (which should resemble a cos² graph - and does) to a sort of “best fit” cos² graph. The lab script says the following:
A more precise determination of the visibility V (this is basically how 'clean' the data is) follows from the best fit to the measured data using the function:
f(b) = A/2[1-Vsin(b-b(center)/P)]
Granted this probably doesn’t mean much out of context, but essentially A is the amplitude, b is an angle and P is the periodicity. Hence this is also a “wave” like the experimental data I have found.
From this I understand, as previously mentioned, I am making a “best fit” curve. However, I have been told that this isn’t possible with Excel and that the best approach is Matlab.
I know intermediate JavaScript but do not know Matlab and was hoping for some direction.
Is there a tutorial I can read for this? Is it possible for someone to go through it with me? I really have no idea what it entails, so any feed back would be greatly appreciated.
Thanks a lot!
Initial steps
I guess we should begin by getting a representation in Matlab of the function that you're trying to model. A direct translation of your formula looks like this:
function y = targetfunction(A,V,P,bc,b)
y = (A/2) * (1 - V * sin((b-bc) / P));
end
Getting hold of the data
My next step is going to be to generate some data to work with (you'll use your own data, naturally). So here's a function that generates some noisy data. Notice that I've supplied some values for the parameters.
function [y b] = generateData(npoints,noise)
A = 2;
V = 1;
P = 0.7;
bc = 0;
b = 2 * pi * rand(npoints,1);
y = targetfunction(A,V,P,bc,b) + noise * randn(npoints,1);
end
The function rand generates random points on the interval [0,1], and I multiplied those by 2*pi to get points randomly on the interval [0, 2*pi]. I then applied the target function at those points, and added a bit of noise (the function randn generates normally distributed random variables).
Fitting parameters
The most complicated function is the one that fits a model to your data. For this I use the function fminunc, which does unconstrained minimization. The routine looks like this:
function [A V P bc] = bestfit(y,b)
x0(1) = 1; %# A
x0(2) = 1; %# V
x0(3) = 0.5; %# P
x0(4) = 0; %# bc
f = #(x) norm(y - targetfunction(x(1),x(2),x(3),x(4),b));
x = fminunc(f,x0);
A = x(1);
V = x(2);
P = x(3);
bc = x(4);
end
Let's go through line by line. First, I define the function f that I want to minimize. This isn't too hard. To minimize a function in Matlab, it needs to take a single vector as a parameter. Therefore we have to pack our four parameters into a vector, which I do in the first four lines. I used values that are close, but not the same, as the ones that I used to generate the data.
Then I define the function I want to minimize. It takes a single argument x, which it unpacks and feeds to the targetfunction, along with the points b in our dataset. Hopefully these are close to y. We measure how far they are from y by subtracting from y and applying the function norm, which squares every component, adds them up and takes the square root (i.e. it computes the root mean square error).
Then I call fminunc with our function to be minimized, and the initial guess for the parameters. This uses an internal routine to find the closest match for each of the parameters, and returns them in the vector x.
Finally, I unpack the parameters from the vector x.
Putting it all together
We now have all the components we need, so we just want one final function to tie them together. Here it is:
function master
%# Generate some data (you should read in your own data here)
[f b] = generateData(1000,1);
%# Find the best fitting parameters
[A V P bc] = bestfit(f,b);
%# Print them to the screen
fprintf('A = %f\n',A)
fprintf('V = %f\n',V)
fprintf('P = %f\n',P)
fprintf('bc = %f\n',bc)
%# Make plots of the data and the function we have fitted
plot(b,f,'.');
hold on
plot(sort(b),targetfunction(A,V,P,bc,sort(b)),'r','LineWidth',2)
end
If I run this function, I see this being printed to the screen:
>> master
Local minimum found.
Optimization completed because the size of the gradient is less than
the default value of the function tolerance.
A = 1.991727
V = 0.979819
P = 0.695265
bc = 0.067431
And the following plot appears:
That fit looks good enough to me. Let me know if you have any questions about anything I've done here.
I am a bit surprised as you mention f(a) and your function does not contain an a, but in general, suppose you want to plot f(x) = cos(x)^2
First determine for which values of x you want to make a plot, for example
xmin = 0;
stepsize = 1/100;
xmax = 6.5;
x = xmin:stepsize:xmax;
y = cos(x).^2;
plot(x,y)
However, note that this approach works just as well in excel, you just have to do some work to get your x values and function in the right cells.
I'm really new at matlab, and am trying to fit a line or curve to data points for homework (that part is actually done). Now, I want to take this a little further than the homework asked, I have constructed a function that takes in a text file with coordinates and any number of functions (1,x,x^2... e.g.) and determines the coefficients.
So in the end I'm left with two vectors: one with the coefficients: C = [a,b,c] and another one with functions: F = {'1','x','x^2'}, and I'd like to create a linear combination of them: l = a + b*x + c*x^2, to plot the curve on a graph, and for some reason I can't figure out how to get that to work. Is there something obvious I'm overlooking, or do I have to rethink this in some way?
a=1; b=2; c=3;
C=[a,b,c];
CS = cellfun(#num2str,num2cell(C),'uniformoutput',0)
M={'*','*','*'};
F={'1','x','x^2'};
P={' + ',' + ',''};
S=reshape([CS; M; F; P],1,[]);
cat(2,S{:})
Output:
ans =
1*1 + 2*x + 3*x^2
Are you sure you want to print 'a', 'b' and 'c' as chars?