How to get a linear trendline/line of best fit with a fixed y-intercept?

I don't have access to fit, the rest of the curve fitting toolbox or any additional paid packages.

Currently I am using polyfit to produce a line through my scatter plot however based on my data I know it should go through (0,0). I know that your basic trendline is calculated using something like this LINK or using a method with squares.

I'm not sure how to adapt this to get the same functionality that you would get in Excel when you set the y-intercept equal to a constant ie: 0. Is there a matlab function for trendlines? Does someone have a nice .m that does this? Can someone explain how the linked math can be adapted to suit a fixed y-intercept?

Thanks in advance!

Answer 1

[dpb]

Just build the design matrix and use \ is the simple way; cfit can do it in Curve Fitting TB if you set the specific model as well.

Example:

>> x=1:10;   **** ERRATUM: LINE MISSED INITIAL POSTING -dpb ****
>> y=x+rand(size(x)); plot(x,y,'*')  % arbitrary data, show the data points
>> b=polyfit(x,y,1);                 % fit the OLS line
>> b                                 % coefficients
b =
  0.9991    0.5186
>> 
>> xlim([0 10])                   % expand axis to cover origin
>> yhat=polyval(b,[0 x]);         % evaluate at origin and points
>> hold on; plot([0 x],yhat,'b-') % and add to plot the fitted line

Now fit without intercept term--design matrix is just the x vector without a ones column for intercept term--

>> X=x.';                         % design "matrix" as column
>> b0=X\y.'                       % solve for slope, zero intercept
b0 =
  1.0731
>> yhat0=b0*[0 x];                % evaluate over same range from origin
>> plot([0 x],yhat0,'r-')         % add that line, too...
>> legend('Data points','Poly 1','Zero Intercept')
>>

Result is

You'll get a somewhat different-looking plot as the random noise won't match, of course.

If you do want the other information besides just the coefficient, then fit is probably the way to go -- again, assuming have the C-F ToolBox.

 >> MZ=fit(x,y,fittype({'x'}))
MZ = 
     Linear model:
     MZ(x) = a*x
     Coefficients (with 95% confidence bounds):
       a =       1.073  (1.029, 1.117)
>>

See the doc fit for the optional output arguments, etc., ...

[KMFSousa]

This is 110% what I was looking for. Thank you so much for the amazing quality answer, code and example!

The only thing I noticed was that plot([0 x],yhat0,'r-') was plotting a lot of lines so I changed it to lot([0 max(x)],[0 max(yhat0)],'r-') to get just one line, which may not be a perfect solution.

[dpb]

Thnx, we aim to please... :)

[dpb]

"plot([0 x],yhat0,'r-') was plotting a lot of lines..."

Shouldn't be but one if x is a vector. The line

x=1:10;

got missed in the cut 'n paste I see...if you had something else already defined for x that was an array, then it could happen I guess altho I'd think you'd have size issues elsewhere.

Would have to see actual code to decipher the specific problem I think; obvious answer doesn't strike me at the moment.

[KMFSousa]

Where x_data and y_data are the data.

b = x_data\y_data;
yhat = b*[0 x_data'];
h2 = plot([0 max(x_data)],[0 max(yhat)],'m-');

Produces:

Versus

b = x_data\y_data;
yhat = b*[0 x_data'];
h2 = plot([0 max(x_data')],[0 max(yhat)],'r-');

You can see that the second option creates a much thicker line until the space between the second last and last point which I interpreted as many overlapping lines but I'm not sure that's correct. It's not a important point but I was curious on why it happened. Do you know?

[dpb]

I can't see any difference between the two code snippets other than the color in the linespec position; probably the magenta vs red makes for a visual difference.

size(h2) will have to be the same for both (although you overwrite it between by using same variable name so will have to check in between).

The only thing that could be a glitch is that you're using max(xdata), max(yhat) as the line points but those could conceivably get out of synch; it would be better to use b0*max(xhat) for the y-value corresponding to be sure they're going together.

I don't see any way to have gotten multiple lines here; you could also query the 'linewidth' property to check it's the same default value.

I think it's just a fignewton of a visual perception, not anything actually real.