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[regression] objective function section done

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This commit is contained in:
Jan Benda
2020-12-16 09:25:01 +01:00
parent be79d450df
commit 1b14bca164
7 changed files with 31 additions and 45 deletions
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2
regression/lecture/cubicfunc.py
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@@ -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)

11
regression/lecture/regression-chapter.tex
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@@ -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

32
regression/lecture/regression.tex
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@@ -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}