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[regression] further improved the chapter

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Jan Benda
2019-12-11 09:48:19 +01:00
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7 changed files with 91 additions and 107 deletions
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14
regression/lecture/regression-chapter.tex
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@@ -16,9 +16,11 @@
\include{regression}
\section{Improvements}
Adapt function arguments to matlabs polyfit. That is: first the data
(x,y) and then the parameter vector. p(1) is slope, p(2) is intercept.
\subsection{Linear fits}
\begin{itemize}
\item Polyfit is easy: unique solution!
\item Example for overfitting with polyfit of a high order (=number of data points)
\end{itemize}
\section{Fitting in practice}
@@ -34,11 +36,5 @@ Fit with matlab functions lsqcurvefit, polyfit
\item How to test the quality of a fit? Residuals. $\chi^2$ test. Run-test.
\end{itemize}
\subsection{Linear fits}
\begin{itemize}
\item Polyfit is easy: unique solution!
\item Example for overfitting with polyfit of a high order (=number of data points)
\end{itemize}
\end{document}

143
regression/lecture/regression.tex
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@@ -52,40 +52,43 @@ considered an optimal fit. In our example we search the parameter
combination that describe the relation of $x$ and $y$ best. What is
meant by this? Each input $x_i$ leads to an measured output $y_i$ and
for each $x_i$ there is a \emph{prediction} or \emph{estimation}
$y^{est}_i$ of the output value by the model. At each $x_i$ estimation
and measurement have a distance or error $y_i - y_i^{est}$. In our
example the estimation is given by the equation $y_i^{est} =
f(x;m,b)$. The best fitting model with parameters $m$ and $b$ is the
one that minimizes the distances between observation $y_i$ and
estimation $y_i^{est}$ (\figref{leastsquareerrorfig}).
$y^{est}(x_i)$ of the output value by the model. At each $x_i$
estimation and measurement have a distance or error $y_i -
y^{est}(x_i)$. In our example the estimation is given by the equation
$y^{est}(x_i) = f(x_i;m,b)$. The best fitting model with parameters
$m$ and $b$ is the one that minimizes the distances between
observation $y_i$ and estimation $y^{est}(x_i)$
(\figref{leastsquareerrorfig}).
As a first guess we could simply minimize the sum $\sum_{i=1}^N y_i -
y^{est}_i$. This approach, however, will not work since a minimal sum
y^{est}(x_i)$. This approach, however, will not work since a minimal sum
can also be achieved if half of the measurements is above and the
other half below the predicted line. Positive and negative errors
would cancel out and then sum up to values close to zero. A better
approach is to sum over the absolute values of the distances:
$\sum_{i=1}^N |y_i - y^{est}_i|$. This sum can only be small if all
$\sum_{i=1}^N |y_i - y^{est}(x_i)|$. This sum can only be small if all
deviations are indeed small no matter if they are above or below the
predicted line. Instead of the sum we could also take the average
\begin{equation}
\label{meanabserror}
f_{dist}(\{(x_i, y_i)\}|\{y^{est}_i\}) = \frac{1}{N} \sum_{i=1}^N |y_i - y^{est}_i|
f_{dist}(\{(x_i, y_i)\}|\{y^{est}(x_i)\}) = \frac{1}{N} \sum_{i=1}^N |y_i - y^{est}(x_i)|
\end{equation}
For reasons that are explained in
chapter~\ref{maximumlikelihoodchapter}, instead of the averaged
absolute errors, the \enterm[mean squared error]{mean squared error}
(\determ[quadratischer Fehler!mittlerer]{mittlerer quadratischer
Fehler})
Instead of the averaged absolute errors, the \enterm[mean squared
error]{mean squared error} (\determ[quadratischer
Fehler!mittlerer]{mittlerer quadratischer Fehler})
\begin{equation}
\label{meansquarederror}
f_{mse}(\{(x_i, y_i)\}|\{y^{est}_i\}) = \frac{1}{N} \sum_{i=1}^N (y_i - y^{est}_i)^2
f_{mse}(\{(x_i, y_i)\}|\{y^{est}(x_i)\}) = \frac{1}{N} \sum_{i=1}^N (y_i - y^{est}(x_i))^2
\end{equation}
is commonly used (\figref{leastsquareerrorfig}). Similar to the
absolute distance, the square of the errors, $(y_i - y_i^{est})^2$, is
absolute distance, the square of the errors, $(y_i - y^{est}(x_i))^2$, is
always positive and thus positive and negative error values do not
cancel each other out. In addition, the square punishes large
deviations over small deviations.
deviations over small deviations. In
chapter~\ref{maximumlikelihoodchapter} we show that minimizing the
mean square error is equivalent to maximizing the likelihood that the
observations originate from the model, if the data are normally
distributed around the model prediction.
\begin{exercise}{meanSquaredErrorLine.m}{}\label{mseexercise}%
Given a vector of observations \varcode{y} and a vector with the
@@ -98,20 +101,13 @@ deviations over small deviations.
\section{Objective function}
The mean squared error is a so called \enterm{objective function} or
\enterm{cost function} (\determ{Kostenfunktion}), $f_{cost}(\{(x_i,
y_i)\}|\{y^{est}_i\})$. A cost function assigns to the given data set
$\{(x_i, y_i)\}$ and corresponding model predictions $\{y^{est}_i\}$ 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 chapter~\ref{maximumlikelihoodchapter} we
show that the minimization of the mean square error is equivalent to
maximizing the likelihood that the observations originate from the
model (assuming a normal distribution of the data around the model
prediction). The \enterm{cost function} does not have to be the mean
square error but can be any function that maps the data and the
predictions to a scalar value describing the quality of the fit. In
the optimization process we aim for the paramter combination that
minimizes the costs.
\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.
\begin{figure}[t]
\includegraphics[width=1\textwidth]{linear_least_squares}
@@ -123,41 +119,40 @@ minimizes the costs.
\label{leastsquareerrorfig}
\end{figure}
Replacing $y^{est}$ with our model, the straight line
\eqref{straightline}, yields
Replacing $y^{est}$ in the mean squared error \eqref{meansquarederror}
with our model, the straight line \eqref{straightline}, the cost
function reads
\begin{eqnarray}
f_{cost}(\{(x_i, y_i)\}|m,b) & = & \frac{1}{N} \sum_{i=1}^N (y_i - f(x_i;m,b))^2 \label{msefunc} \\
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}
\end{eqnarray}
That is, the mean square error is given by the pairs $(x_i, y_i)$ of
measurements and the parameters $m$ and $b$ of the straight line. The
optimization process tries to find $m$ and $b$ 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 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
Fehler!kleinster]{Methode der kleinsten Quadrate}).
\begin{exercise}{meanSquaredError.m}{}
Implement the objective function \varcode{meanSquaredError()} that
uses a straight line, \eqnref{straightline}, as a model. The
function takes three arguments. The first is a 2-element vector that
contains the values of parameters \varcode{m} and \varcode{b}. The
second is a vector of x-values, and the third contains the
measurements for each value of $x$, the respective $y$-values. The
function returns the mean square error \eqnref{mseline}.
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}
\section{Error surface}
For each combination of the two parameters $m$ and $b$ of the model we
can use \eqnref{mseline} to calculate the corresponding value of the
cost function. We thus consider the cost function $f_{cost}(\{(x_i,
y_i)\}|m,b)$ as a function $f_{cost}(m,b)$, that maps the parameter
values $m$ and $b$ to an error value. The error values describe a
landscape over the $m$-$b$ plane, the error surface, that can be
illustrated graphically using a 3-d surface-plot. $m$ and $b$ are
plotted on the $x-$ and $y-$ axis while the third dimension indicates
the error value (\figref{errorsurfacefig}).
cost function. The cost function $f_{cost}(m,b|\{(x_i, y_i)\}|)$ is a
function $f_{cost}(m,b)$, that maps the parameter values $m$ and $b$
to a scalar error value. The error values describe a landscape over the
$m$-$b$ plane, the error surface, that can be illustrated graphically
using a 3-d surface-plot. $m$ and $b$ are plotted on the $x$- and $y$-
axis while the third dimension indicates the error value
(\figref{errorsurfacefig}).
\begin{figure}[t]
\includegraphics[width=0.75\textwidth]{error_surface}
@@ -176,8 +171,8 @@ the error value (\figref{errorsurfacefig}).
calculate the mean squared error between the data and straight lines
for a range of slopes and intercepts using the
\varcode{meanSquaredError()} function from the previous exercise.
Illustrates the error surface using the \code{surface()} function
(consult the help to find out how to use \code{surface()}).
Illustrates the error surface using the \code{surface()} function.
Consult the documentation to find out how to use \code{surface()}.
\end{exercise}
By looking at the error surface we can directly see the position of
@@ -185,19 +180,21 @@ the minimum and thus estimate the optimal parameter combination. How
can we use the error surface to guide an automatic optimization
process?
The obvious approach would be to calculate the error surface and then
find the position of the minimum using the \code{min} function. This
approach, however has several disadvantages: (i) it is computationally
very expensive to calculate the error for each parameter
combination. The number of combinations increases exponentially with
the number of free parameters (also known as the ``curse of
dimensionality''). (ii) the accuracy with which the best parameters
can be estimated is limited by the resolution used to sample the
parameter space. The coarser the parameters are sampled the less
precise is the obtained position of the minimum.
The obvious approach would be to calculate the error surface for any
combination of slope and intercept values and then find the position
of the minimum using the \code{min} function. This approach, however
has several disadvantages: (i) it is computationally very expensive to
calculate the error for each parameter combination. The number of
combinations increases exponentially with the number of free
parameters (also known as the ``curse of dimensionality''). (ii) the
accuracy with which the best parameters can be estimated is limited by
the resolution used to sample the parameter space. The coarser the
parameters are sampled the less precise is the obtained position of
the minimum.
We want a procedure that finds the minimum of the cost function with a minimal number
of computations and to arbitrary precision.
So we need a different approach. We want a procedure that finds the
minimum of the cost function with a minimal number of computations and
to arbitrary precision.
\begin{ibox}[t]{\label{differentialquotientbox}Difference quotient and derivative}
\includegraphics[width=0.33\textwidth]{derivative}
@@ -308,9 +305,9 @@ choose the opposite direction.
\begin{exercise}{meanSquaredGradient.m}{}\label{gradientexercise}%
Implement a function \varcode{meanSquaredGradient()}, that takes the
set of parameters $(m, b)$ of a straight line as a two-element
vector and the $x$- and $y$-data as input arguments. The function
should return the gradient at that position as a vector with two
$x$- and $y$-data and the set of parameters $(m, b)$ of a straight
line as a two-element vector as input arguments. The function should
return the gradient at the position $(m, b)$ as a vector with two
elements.
\end{exercise}
@@ -359,7 +356,7 @@ distance between the red dots in \figref{gradientdescentfig}) is
large.
\begin{figure}[t]
\includegraphics[width=0.55\textwidth]{gradient_descent}
\includegraphics[width=0.45\textwidth]{gradient_descent}
\titlecaption{Gradient descent.}{The algorithm starts at an
arbitrary position. At each point the gradient is estimated and
the position is updated as long as the length of the gradient is
@@ -376,7 +373,7 @@ large.
\item Plot the error values as a function of the iterations, the
number of optimization steps.
\item Plot the measured data together with the best fitting straight line.
\end{enumerate}
\end{enumerate}\vspace{-4.5ex}
\end{exercise}