MATLAB: Does polyfit require data in x and y to be monotonically increasing

MATLABplotpolyfitpolyval

I am using "polyfit" to fit a polynomial to my data and then I visualize the result with "plot" after evaluating the polynomial at my data points with "polyval." The plot does not look right: it looks all scrambled instead of looking like an injective function.

My data is not monotonically increasing and I suspect this is the cause. Therefore, I am wondering if "polyfit" requires the input data to be monotonically increasing (i.e., sorted)?

Best Answer

It does not matter to the "polyfit" function whether the input data is sorted or not. The underlying algorithm is a simple least square problem on a Vandermonde matrix (with QR decomposition). The coefficients of the polynomial found by "polyfit" might differ slightly if the order of the input points changes, but, as long as the input data points form a well-conditioned set of equations, the difference should not be significant.

First, a Vandermonde matrix is constructed from the input data points "x" and the degree of the polynomial "n". Then, the matrix is factored as QR (orthogonal Q and upper triangular R). Finally, the system of equations is solved using "Q", "R" and the observations "y".

The script attached below (test_polyfit.m) shows the difference between a polynomial obtained when sorting the input data and a polynomial obtained without sorting the data. Both polynomials have degree 6.

Note that the data has to be sorted when calling "plot". The "plot" function draws straight lines between adjacent points in the input. If the inputs to "plot" are not rearranged so that "x" is increasing, it will draw something that does not make sense here.

If you are trying to fit a polynomial of higher degree, it might be necessary to center and scale the input data. Also, I noticed that some x-values are redundant. This can lead to a poorly conditioned system.

Related Solutions

MATLAB: Fit a monotonically increasing curve to inexact data

One option is to use the Curve Fitting Tool within MATLAB. You can open this tool using the function "cftool". From this point you have several choices:

1. You can select "Polynomial" and test out some different degrees to see if the fit is good, and if they appear monotonic.

2. You can select "Smoothing Spline" and click "Smoother" until the function appears monotonic.

Note that neither of these methods would strictly enforce monotonicity.

Once you have created a good fit to the data, you can saving the generated fit to your workspace (found under the Fit tab of the CFTool) and then extract the polynomial coefficients of the curve in order to create a symbolic equation.

Another option is to alter the raw data before creating a fit. The following example code is from the comp.soft-sys-matlab newsgroup with thanks to John D'Errico.

This example will alter your raw data ('x' and 'y') to become monotonically increasing via minimal changes:

n = length(x);
C = eye(n);
D = y;
A = diag(ones(n,1),0) - diag(ones(n-1,1),1);
A(end,:) = [];
b = zeros(n-1,1);
opts = optimset('lsqlin');
opts.LargeScale = 'off';
opts.Display = 'none';
yhat = lsqlin(C,D,A,b,[],[],[],[],[],opts);
plot(x,y,'ro',x,yhat,'b-*')

After performing the above calculations, we can consider interpolation methods, as these methods would not produce a monotonic curve from non-monotonic data. From the "cftool", the variable "yhat" is now monotonically increasing. From this point, you can:

1. Run any of the above options

2. Select Interpolation, and then select "Cubic" or "PCHIP".

Note that PCHIP Interpolation on monotonic data will be strictly monotonic. Once again, you will be able to extract the corresponding polynomial coefficients after saving the generated fit to your workspace.

Finally, you could download the following 3rd party toolbox from the MATLAB File Exchange:

http://www.mathworks.com/matlabcentral/fileexchange/24443-slm-shape-language-modeling

Using this toolbox, you can calculate a spline fit to your data, and you can specify that the result is strictly increasing using the following command:

>> pp = slmengine(x,y,'increasing','on','result','pp');

If you would prefer to fit a cubic function to your data, that is equivalent to specifying exactly two knots, one at either end of your data:

>> pp = slmengine(x,y,'knots',[1 10],'increasing','on','result','pp');

From this point, you can then extract the polynomial coefficients with the command "slmpar(pp, 'symabs' )".

MATLAB: How to plot a diagonal line on a log-log scale plot

The most straight forward way to do this is to "loglog" plot a "logspace" array across the bounds that you are working with. For example:

x = logspace(0,15);
scatter(x,x)
grid on
hold on
loglog(x,x)
hold off

Alternatively, you can call "plot" with a "linspace" array as both the "x" and "y" arguments.

x = logspace(0,15);
scatter(x,x)
grid on
hold on
y = linspace(1,1e15);
plot(y,y)
hold off

Best Answer

Related Solutions

MATLAB: Fit a monotonically increasing curve to inexact data

MATLAB: How to plot a diagonal line on a log-log scale plot

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