Polynomial equation not giving expected result

I have a set of X axis values that are known moisture content numbers. These values give me the Y axis values in a microcontroller as shown below.

MContent = [1.278096078 22.26039162 31.88752617 40.4470463 45.821662 51.19483315 54.14512732 58.15256688 62.08950419 65.25710918 67.5910596 69.40480782 71.08289411 73.09132161];
BitValue = [1020 960 862 657 440 380 341 193 193 186 179 112 106 85];

When plotting this I will get the following output.

I wanted to obtain a polynomial equation that will fit this line. Therefore I created a 5th order polynomial as shown below.

p5 = polyfit(MContent, BitValue, 5);
format long g
disp(p5)
pp5 = polyval(p5, MContent);
hold on
plot(MContent, pp5, '-p')

Which gives this output.

This line is a satisfactory fit and I intended to use the equation obtained from this to get bit values when I add the moisture content into the 'x' value. To get this polynomial equation I used the following command.

p5 = polyfit(MContent, BitValue, 5);
format long g
disp(p5)

Which gives me

Columns 1 through 4
      -1.3759358768261e-05       0.00276912138617569        -0.193482771635317          5.27065703595948
  Columns 5 through 6
         -53.9134842525771           1080.5582927381

Ignoring the x^5 value, as it is too small, I will use the rest and get the following equation.

yAxis = (0.00276912138617569*(MContent.^4))-(0.193482771635317*(MContent.^3))+(5.27065703595948*(MContent.^2))-(53.9134842525771*(MContent.^1))+(1080.5582927381*(MContent.^0));

But when I try to plot

figure;plot(MContent, yAxis, '-x');

I get the folllowing plot

Which is no where near to the original MContent to BitValue plot. I would like to know why, and how I can get an equation that will fit my original 'MContent' to 'BitValue' plot.

The full code is shown below for reference.

close all

MContent = [1.278096078 22.26039162 31.88752617 40.4470463 45.821662 51.19483315 54.14512732 58.15256688 62.08950419 65.25710918 67.5910596 69.40480782 71.08289411 73.09132161];
BitValue = [1020 960 862 657 440 380 341 193 193 186 179 112 106 85];


plot(MContent, BitValue, ':s', 'MarkerSize', 6, 'MarkerFaceColor', 'b')
title('MContent to bit value from 12th March 2019')
xlabel('Moisture Content (%)')
ylabel('Bit value (bits)')


p5 = polyfit(MContent, BitValue, 5);


format long g
disp(p5)

pp5 = polyval(p5, MContent);

hold on

plot(MContent, pp5, '-p')

legend('M.C. (original)', '5th degree polynomial', 'Location', 'NorthEast')

hold off

yAxis = (0.00276912138617569*(MContent.^4))-(0.193482771635317*(MContent.^3))+(5.27065703595948*(MContent.^2))-(53.9134842525771*(MContent.^1))+(1080.5582927381*(MContent.^0));
figure;plot(MContent, yAxis, '-x');

[Pawel Tokarczuk]

What you're seeing is the result of numerical instability.

You should start by centring and scaling your x-variable:

mc = (MContent - 50.0)/50.0;

[dpb]

While standardizing would be agoodthing™, numerical issues aren't the problem here; the problem is the OP just ignored one coefficient that once x gets sizable is by far the dominant factor.

The same effect would be true if the coefficients were computed for normalized/standardized variables.

Answer 1

[dpb]

Just because the coefficient is small doesn't mean it isn't important...it is, after all the coefficient of x^5 and 75^5 = 2.3730e+09 so a factor of 10E-5 on the coefficent is still a 10E4 contribution so you can't just ignore it.

b=polyfit(MContent, BitValue,5);
plot(MContent,BitValue,'bs','MarkerFaceColor','b')
hold on
xh=0:75;
yh4=polyval(b(2:end),0:75);                  % evaluate w/o x^5 term
h4=plot(xh,yh4,'b-');
yh5Only=polyval([b(1) zeros(1,5)],xh);       % and now just that term...
h5O=plot(xh,yh5Only,'r-');
yHat=polyval(b,xh);                          % and the real polynomial...
max(abs(yHat-(yh4+yh5Only)))                 % and that leads to same as adding the x^5 term
ans =
   1.5916e-11
plot(xh,(yhat),'g-')                         % so plot what the real answer is, too...

Above leads to the above plot which clearly illustrates why "you can't DO that!" and expect reasonable results.

[Chamath Vithanawasam]

Yep, all I had to do was add the 5th order polynomial and it worked.

[dpb]

NB: This also illustrates why it's a very bad idea to extrapolate with any arbitrary fitted function but most particularly with polynomials and why higher-order polynomials are especially bad.

I also intended to note that for numerical stability it would be better to standardize the data before doing the fitting--there's doc/example in the polyfit/polyval do on how to do that.

Answer 2

[Chamath Vithanawasam]

So I found out that I am not supposed to ignore any polynomial regardless of how small it may be. Here is the reading with the 5th order.