Add R² (coeff of determination) to output

README.md

@@ -48,6 +48,7 @@ Note how a few "unlucky" outliers can bias the least squares estimate (LS), but
 - October 2014 by Z. Danziger: Original version.
 - September 2015 by Z. Danziger: Updated help, speed increase for 2D case
 - March 2022 by J. Keyser: Adjusted formatting, added documentation, improved example and added plot, replaced `nanmedian(X)` with `median(X, 'omitnan')`, removed 2D special case, restructured input and output parameters.
+- August 2022 by J. Keyser: Explicit omission of identical x coordinates, added coefficient of determination (unadjusted R²) as optional output.
 
 ## Contributing and project status
 

TheilSen.m

@@ -1,4 +1,4 @@
-function coef = TheilSen(X, y)
+function [coef, rsqrd] = TheilSen(X, y)
 % THEILSEN performs Theil-Sen robust simple linear regression(s) of X on y.
 % 
 % For convenience, the input X may contain more than one predictor variable;
@@ -18,13 +18,14 @@
 %   y: A column vector containing the observations of the response variable.
 %
 % OUTPUT
-%   coef: Estimated regression coefficients for each predictor column in X, with
-%         respect to the response variable y. Each column in coef corresponds
-%         to one predictor in X, i.e. it will have as many columns as X.
-%         The first row, i.e. coef(1, :), contains the estimated offset(s).
-%         The second row, i.e. coef(2, :), contains the estimated slope(s).
-%         (This output format was chosen to avoid confusion, e.g. with previous
-%          versions of this code.)
+%    coef: Estimated regression coefficients for each predictor column in X,
+%          with respect to the response variable y. Each column in coef
+%          corresponds to one predictor in X, i.e. it has as many columns as X.
+%          The first row, i.e. coef(1, :), contains the estimated offset(s).
+%          The second row, i.e. coef(2, :), contains the estimated slope(s).
+%          (This output format was chosen to avoid confusion, e.g. with previous
+%           versions of this code.)
+%   rsqrd: Ordinary R² (coefficient of determination) per predictor column in X.
 %
 % EXAMPLE
 %   See accompanying file example.m.
@@ -109,4 +110,17 @@
     warning('TheilSen:NaNoutput', ...
             'Output contains NaN; check for degenerate inputs.')
 end
+
+% If requested, output the ordinary/unadjusted R², per predictor column in X.
+if nargout > 1
+    % compute total sum of squares relative to the mean
+    sum_sq_total = sum(power(y - mean(y), 2));
+    sums_sq_total = repmat(sum_sq_total, 1, Num_Pred);
+    % compute the residual sum(s) of squares relative to the regression line(s)
+    ys_model = b0s + b1s .* X;
+    ys_data = repmat(y, 1, Num_Pred);
+    sums_sq_resid = sum(power(ys_data - ys_model, 2));
+    % compute R² as 1 - SSresid / SStotal (i.e., the unadjusted version of R²)
+    rsqrd = 1 - sums_sq_resid ./ sums_sq_total;
+end
 end

example.m

@@ -30,7 +30,7 @@
 est_ls = [ones(N_total, 1), data_x] \ data_y;
 
 % Estimate Theil-Sen parameters.
-est_ts = TheilSen(data_x, data_y);
+[est_ts, ~] = TheilSen(data_x, data_y);
 
 % Plot everything and add comparison of estimates to title.
 figure()
@@ -52,9 +52,9 @@
 data_x2 = +10 - 2 * data_x1 + randn(size(data_x1)) * SDx_usual;
 data_x3 = -50 + 5 * data_x1 + randn(size(data_x1)) * SDx_usual;
 X = [data_x1, data_x2, data_x3];
-est_ts = TheilSen(X, data_y);
+[est_ts, r_sqrd] = TheilSen(X, data_y);
 
-% plot both simple regressions; note mainly the difference between x-axes
+% plot all simple regressions; note mainly the difference between x-axes
 num_pred = size(X, 2);
 figure()
 for pp = 1:num_pred