[regression] objective function section done

regression/code/meanSquaredError.m

@@ -1,12 +0,0 @@
-function mse = meanSquaredError(x, y, parameter)
-% Mean squared error between a straight line and data pairs.
-%
-% Arguments: x, vector of the input values
-%            y, vector of the corresponding measured output values
-%            parameter, vector containing slope and intercept 
-%                       as the 1st and 2nd element, respectively. 
-%
-% Returns:   mse, the mean-squared-error.
-
-  mse = mean((y - x * parameter(1) - parameter(2)).^2);
-end

regression/code/meanSquaredErrorCubic.m

@@ -0,0 +1,11 @@
+function mse = meanSquaredError(x, y, c)
+% Mean squared error between data pairs and a cubic relation.
+%
+% Arguments: x, vector of the input values
+%            y, vector of the corresponding measured output values
+%            c, the factor for the cubic relation. 
+%
+% Returns:   mse, the mean-squared-error.
+
+  mse = mean((y - c*x.^3).^2);
+end

regression/code/meansquarederrorline.m

@@ -10,6 +10,6 @@ yest = c * x.^3;
 y = yest + noise*randn(n, 1);
 
 % compute mean squared error:
-mse = mean((y - y_est).^2);
+mse = mean((y - yest).^2);
 
-fprintf('the mean squared error is %g kg^2\n', mse))
+fprintf('the mean squared error is %.0f kg^2\n', mse)

regression/lecture/cubicfunc.py

@@ -41,7 +41,7 @@ def plot_data_fac(ax, x, y, c):
     
 if __name__ == "__main__":
     x, y, c = create_data()
-    print(len(x))
+    print('n=%d' % len(x))
     fig, (ax1, ax2) = plt.subplots(1, 2)
     fig.subplots_adjust(wspace=0.5, **adjust_fs(fig, left=6.0, right=1.5))
     plot_data(ax1, x, y)

regression/lecture/regression-chapter.tex

@@ -42,17 +42,6 @@
 
 \subsection{2D fit}
 
-\begin{exercise}{meanSquaredError.m}{}
-  Implement the objective function \eqref{mseline} as a function
-  \varcode{meanSquaredError()}.  The function takes three
-  arguments. The first is a vector of $x$-values and the second
-  contains the measurements $y$ for each value of $x$. The third
-  argument is a 2-element vector that contains the values of
-  parameters \varcode{m} and \varcode{b}. The function returns the
-  mean square error.
-\end{exercise}
-
-
 \begin{exercise}{errorSurface.m}{}\label{errorsurfaceexercise}
   Generate 20 data pairs $(x_i|y_i)$ that are linearly related with
   slope $m=0.75$ and intercept $b=-40$, using \varcode{rand()} for

regression/lecture/regression.tex

@@ -109,10 +109,10 @@ The mean squared error is a so called \enterm{objective function} or
 \enterm{cost function} (\determ{Kostenfunktion}). A cost function
 assigns to a model prediction $\{y^{est}(x_i)\}$ for a given data set
 $\{(x_i, y_i)\}$ a single scalar value that we want to minimize.  Here
-we aim to adapt the model parameters to minimize the mean squared
-error \eqref{meansquarederror}. In general, the \enterm{cost function}
-can be any function that describes the quality of the fit by mapping
-the data and the predictions to a single scalar value.
+we aim to adapt the model parameter to minimize the mean squared error
+\eqref{meansquarederror}. In general, the \enterm{cost function} can
+be any function that describes the quality of a fit by mapping the
+data and the predictions to a single scalar value.
 
 \begin{figure}[t]
   \includegraphics{cubicerrors}
@@ -125,25 +125,23 @@ the data and the predictions to a single scalar value.
 \end{figure}
 
 Replacing $y^{est}$ in the mean squared error \eqref{meansquarederror}
-with our model, the straight line \eqref{straightline}, the cost
-function reads
+with our cubic model \eqref{cubicfunc}, the cost function reads
 \begin{eqnarray}
-  f_{cost}(m,b|\{(x_i, y_i)\}) & = & \frac{1}{N} \sum_{i=1}^N (y_i - f(x_i;m,b))^2 \label{msefunc} \\
-  & = & \frac{1}{N} \sum_{i=1}^N (y_i - m x_i - b)^2 \label{mseline}
+  f_{cost}(c|\{(x_i, y_i)\}) & = & \frac{1}{N} \sum_{i=1}^N (y_i - f(x_i;c))^2 \label{msefunc} \\
+  & = & \frac{1}{N} \sum_{i=1}^N (y_i - c x_i^3)^2 \label{msecube}
 \end{eqnarray}
-The optimization process tries to find the slope $m$ and the intercept
-$b$ such that the cost function is minimized. With the mean squared
-error as the cost function this optimization process is also called
-method of the \enterm{least square error} (\determ[quadratischer
+The optimization process tries to find a value for the factor $c$ such
+that the cost function is minimized. With the mean squared error as
+the cost function this optimization process is also called method of
+ \enterm{least squares} (\determ[quadratischer
 Fehler!kleinster]{Methode der kleinsten Quadrate}).
 
-\begin{exercise}{meanSquaredError.m}{}
-  Implement the objective function \eqref{mseline} as a function
-  \varcode{meanSquaredError()}.  The function takes three
+\begin{exercise}{meanSquaredErrorCubic.m}{}
+  Implement the objective function \eqref{msecube} as a function
+  \varcode{meanSquaredErrorCubic()}.  The function takes three
   arguments. The first is a vector of $x$-values and the second
   contains the measurements $y$ for each value of $x$. The third
-  argument is a 2-element vector that contains the values of
-  parameters \varcode{m} and \varcode{b}. The function returns the
+  argument is the value of the factor $c$. The function returns the
   mean squared error.
 \end{exercise}