MATLAB Best fit line for data points

Wrichik Basu

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I have some experimental data, and I would like to plot a graph in matlab and also find the best fit line. I also want to find the distance between each point and the best fit line, and if the distance is greater than 1 unit, an error will be shown.

For this, I have written a function. Here it is:

Matlab:
function [] = plot2d(X,Y);
 
    %The aim of this function is to read
    % X and Y from the user, and plot the points.
    % The best fit line shall be shown.
 
    hold off;
 
    %this plots the given data points
    plot(X,Y,'.k', 'MarkerSize', 20);
 
    hold on;
 
    %here, we are calculating the best fit line
    p = polyfit(X,Y,1);
    f = polyval(p, X);
    plot(X, f, '-r');
 
    %necessary plot settings
    zoom on;
    set(gca, 'xminorgrid', 'on', 'yminorgrid', 'on');
 
    syms x1 y1 x2 y2 x y;
 
    %finding the equation of the best fit line
    x1 = X(1);
    x2 = X(end);
    y1 = f(1);
    y2 = f(end);
    m = (y2 - y1) / (x2 - x1);
 
    eqn = y - y1 == m * (x - x1);
 
    %simplifying the equation
    eqn = simplify(eqn);
 
    %writing the equation in Ax + By + C = 0 form and removing the rhs to make it an algebraic expression
    eqn2 = lhs(eqn) - rhs(eqn);
 
    coefx = coeffs(eqn2,x);
 
    %coefficient of x
    cx = coefx(end);
 
    coefy = coeffs(eqn2,y);
 
    %coefficient of y
    cy = coefy(end);
 
    %finding number of data points given
    sz = numel(X);
 
    for i = 1:1:sz
     
        x = X(i);
        y = f(i);
     
        %evaluating the expression by putting in values
        p = subs(eqn2);
     
        %calculating final distance
        dist = p / sqrt(cx^2 + cy^2);
     
        if dist > 1
            disp("Error for point (" + x + "," + y + ")");
        end
         
             
    end
 
    hold off;
 
end
Matlab gives me a good graph:

Figure 2018-08-01 20_19_01.png


The data is:

Screenshot_20180801-202541.png


But all distances of points are coming to be 0. Because the subs(eqn2) is yielding a 0. From the graph, it is evident that all points do not lie on the line. But I'm still getting a 0. I know this because I printed p from the function. This is what I got:

Screenshot_20180801-202755.png


Screenshot_20180801-202801.png


Any help is appreciated. Where am I going wrong?

Any ideas?

I use matlab mobile, version R2018a.
 

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My suggestion is to make some test data and a small program that illustrates the problem. Focus on the subs and at the Matlab api description of subs to see if you’re using it correctly.

Usually there are examples of the function that you can try too.
 

Wrichik Basu

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My suggestion is to make some test data and a small program that illustrates the problem. Focus on the subs and at the Matlab api description of subs to see if you’re using it correctly.

Usually there are examples of the function that you can try too.
The problem is that, I tried by putting values in eqn2 myself. I did this by printing eqn2, and then manually putting in values. The answers are truly coming to 0. But from the graph, the distance cannot be 0. That is the bottleneck. The points do not lie on the best fit line, but the equation of the best fit line shows that all the points lie on it. I think the problem is coming from the polyfit or polyval functions.
 
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What is this line used for?

Matlab:
    eqn = y - y1 == m * (x - x1);
 

Wrichik Basu

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883
What is this line used for?

Matlab:
    eqn = y - y1 == m * (x - x1);
That line finds the equation of the best fit line using the formula $$y - y_1 = m(x-x_1)$$
 

FactChecker

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Is your "distance" the Y error or is it the perpondicular distance to the line? If it is the former, there are easier ways to get the Y error. If it is the latter, consider principle components.
 

Wrichik Basu

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Is your "distance" the Y error or is it the perpondicular distance to the line? If it is the former, there are easier ways to get the Y error. If it is the latter, consider principle components.
It is the perpendicular distance of each point from the best fit line.

Can you elaborate a bit on the principle components? For computation of distance, I have used the formula $$\dfrac{Ax+By+C}{\sqrt{A^2+B^2}}$$ where ##x## and ##y## are from the data.
 

FactChecker

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The MATLAB function polyfit is not doing what you want to do. It is minimizing the sum-squares of the y errors -- the vertical distance to the line.
To minimize the sum-squared perpendicular distances, the first principle component is what you want. From https://en.wikipedia.org/wiki/Principal_component_analysis#Further_considerations :

"Given a set of points in Euclidean space, the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line."
 

Wrichik Basu

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2018 Award
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The MATLAB function polyfit is not doing what you want to do. It is minimizing the sum-squares of the y errors -- the vertical distance to the line.
To minimize the sum-squared perpendicular distances, the first principle component is what you want. From https://en.wikipedia.org/wiki/Principal_component_analysis#Further_considerations :

"Given a set of points in Euclidean space, the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line."
Two questions:

1. What is the difference between vertical and perpendicular distances?
2. Is there any known function in matlab to make a graph from first PCA?
 

Wrichik Basu

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1) See, for instance, https://benediktehinger.de/blog/science/scatterplots-regression-lines-and-the-first-principal-component/
2) In MATLAB, use [COEFF,SCORE] = princomp(X). The first row of COEFF will give you the first principle component. I think that you will have to plot it yourself.
Thanks for the function. But princomp has been removed; it has now become pca in R2018a.

However, I will not be able to use it, because it requires me to buy the Statistics toolbox. In fact, my matlab is not licensed, and that's why I can use it only on mobile. :frown:
 

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Wrichik Basu

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I am indeed heartbroken with matlab. Just now, I found that they have stopped supporting 32-bit pc. The last supporting version was R2015b, which is no longer sold now. Perhaps I am done with matlab for the moment, and I should shift to some other language.
 
Or perhaps you should upgrade your computer.
 

Wrichik Basu

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Or perhaps you should upgrade your computer.
That's costly. What I have decided is, I will buy the software and additional toolboxes, but keep on using it in the mobile app. I don't need simulink at this moment, so I think I'm fine with whatever I can do on the app.
 

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