MATLAB: Using the deviation of a linear regression calculation to trim the calculated data in the linear regression itself.

deviatedeviationlinear regressionMATLABplotting

I have calculated the linear regression of a set of data:

-42.1274227227296  -6.79715991020203
-46.6343683929176  -12.2348878383636
-47.8862977457476  -13.5943198204041
-49.1382270985776  -14.9537518024445
-53.6451727687656  -20.3914797306061
-55.7734526685766  -25.8292076587677
-56.3994173449916  -27.1886396408081
-57.1505749566896  -28.5480716228485
-58.4025043095196  -31.2669355869293

where the 1st column is the X coordinates and the 2nd column is the Y coordinates.

I would like the linear regression to be calculated in a loop (unless there is a better way of doing this) so that it can be calculated using a minimum of the first 5 points, and then keep plotting until it reaches a point which deviates away from the current linear regression by a fixed (user defined) threshold.

The current code used for the linear regression is:

LinearRegression = [ones(length(DATA(:,1)),1) DATA(:,2)]\DATA(:,1);

Plot of linear regression:

 figure
 scatter(DATA(:,1),DATA(:,2),'o','b')
 hold on
 plot(DATA(:,1),1/LinearRegression(2)*(DATA(:,1)-LinearRegression(1)),'r')

Thanks in advance!

Best Answer

Thanks for both of your help.

I have implemented the below and it seems to do what I was looking for (note: I have changed variable names, so if it doesn't run for you, check consistency in variable names):

 MaxDev = 3;   % user defined angle (in degrees)
 if length(DATA)<5
    LinearRegression = [ones(length(DATA(:,1)),1) DATA(:,2)]\DATA(:,1);
    disp('Result inaccurate due to limited amount of data points')
 else
    LinearInit = [ones(length(DATA(1:5,1)),1) DATA(1:5,2)]\DATA(1:5,1);
    InitAngle = abs(atan(1/LinearInit(2)))*180/pi;
    for i = 5:length(DATA)
        LinearRegressionTmp = [ones(length(DATA(1:i,1)),1) DATA(1:i,2)]\DATA(1:i,1);
        TmpAngle = abs(atan(1/LinearRegressionTmp (2)))*180/pi;
        if abs(TmpAngle - InitAngle) > MaxDev 
            LinearRegression = [ones(length(DATA(1:i-1,1)),1) DATA(1:i-1,2)]\DATA(1:i-1,1);
            disp('Calculation stopped due to large deviation')
            break
        else
            LinearRegression = [ones(length(DATA(1:i,1)),1) DATA(1:i,2)]\DATA(1:i,1);
        end
    end
 end

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MATLAB: Polynomial 2nd degree

Higher order polynomials aren't going to help here...the following code generated the figure:

for i=3:9
  subplot(4,2,i-2);
  [~,ix]=sort(SB(:,i));
  plot(SB(ix,i),SB(ix,1))
  xlabel(sprintf('B%d',i-2))
end

There's virtually no correlation with most of the corollary variables; what there is in pieces breaks down in other areas of every variable.

Just taking a stab; ran one that uses all variables; another with the only two that were staistically significant -- those results are

>> fitlm(SB(:,3:end),SB(:,1))
ans = 
Linear regression model:
    y ~ 1 + x1 + x2 + x3 + x4 + x5 + x6 + x7
Estimated Coefficients:
                   Estimate     SE      tStat    pValue
                   ________    _____    _____    ______
    (Intercept)     63.33      13.62     4.65    0.13  
    x1             -29.51      18.92    -1.56    0.36  
    x2              20.66      15.52     1.33    0.41  
    x3               4.74      14.23     0.33    0.80  
    x4              -4.83      15.69    -0.31    0.81  
    x5              20.96      33.11     0.63    0.64  
    x6              -0.02      21.87    -0.00    1.00  
    x7               1.42      16.83     0.08    0.95  
Number of observations: 9, Error degrees of freedom: 1
Root Mean Squared Error: 2.49
R-squared: 0.934,  Adjusted R-Squared 0.474
F-statistic vs. constant model: 2.03, p-value = 0.495
>> figure
>> LMA=ans;
>> plot(LMA)
>> title('Salinity ~ 1 + B1 +B2 + B3 +B4 + B5 +B6 + B7')
>> LM12=fitlm(SB(:,3:4),SB(:,1))
LM12 = 
Linear regression model:
    y ~ 1 + x1 + x2
Estimated Coefficients:
                   Estimate     SE     tStat    pValue
                   ________    ____    _____    ______
    (Intercept)     67.74      2.13    31.82    0.00  
    x1             -20.86      5.90    -3.53    0.01  
    x2              21.88      3.21     6.82    0.00  
Number of observations: 9, Error degrees of freedom: 6
Root Mean Squared Error: 1.33
R-squared: 0.887,  Adjusted R-Squared 0.85
F-statistic vs. constant model: 23.6, p-value = 0.00144
>> figure
>> plot(LM12)
>> title('Salinity ~ 1 + B1 +B2')
>> ylim([40 100])

The last puts the plots on the same scale; notice the intervals are much tighter with only two predictors. Whether this would be worth a hoot for future predictions is pretty much pure luck I'd guess...

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Well, you COULD do this, in a sense. Not using fit however, because tools to do curve fitting do not have anything like that as an option, nor realistically, should they. And weighting will not solve the problem either.

As I said, you COULD do it. Does it make mathematical, statistical sense? No, not a lot. You need to use a maximum likelihood estimation, so it will become an optimization problem, using a tool like fminunc or fmincon, from the optimization toolbox. (No, lsqnonlin, lsqcurvefit, lsqlin all cannot solve the problem you want to solve.) fminsearch could solve the problem though, since it becomes a simple optimization.

Is there any reason you want to do this? Because I don't see any reason to formulate the problem as a maximum likelihood estimation, since it wont give you anything better out the end.

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