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[regression] started to simplify chapter to a 1d problem

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Jan Benda 2020-12-15 23:18:12 +01:00
parent a097b10024
commit be79d450df
7 changed files with 247 additions and 116 deletions
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1
regression/code/meanSquaredErrorLine.m
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@ -1 +0,0 @@
mse = mean((y - y_est).^2);

15
regression/code/meansquarederrorline.m Normal file
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@ -0,0 +1,15 @@
n = 40;
xmin = 2.2;
xmax = 3.9;
c = 6.0;
noise = 50.0;
% generate data:
x = rand(n, 1) * (xmax-xmin) + xmin;
yest = c * x.^3;
y = yest + noise*randn(n, 1);
% compute mean squared error:
mse = mean((y - y_est).^2);
fprintf('the mean squared error is %g kg^2\n', mse))

36
regression/lecture/cubicerrors.py
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@ -15,28 +15,13 @@ def create_data():
return x, y, c
def plot_data(ax, x, y, c):
ax.plot(x, y, zorder=10, **psAm)
xx = np.linspace(2.1, 3.9, 100)
ax.plot(xx, c*xx**3.0, zorder=5, **lsBm)
for cc in [0.25*c, 0.5*c, 2.0*c, 4.0*c]:
ax.plot(xx, cc*xx**3.0, zorder=5, **lsDm)
ax.set_xlabel('Size x', 'm')
ax.set_ylabel('Weight y', 'kg')
ax.set_xlim(2, 4)
ax.set_ylim(0, 400)
ax.set_xticks(np.arange(2.0, 4.1, 0.5))
ax.set_yticks(np.arange(0, 401, 100))
def plot_data_errors(ax, x, y, c):
ax.set_xlabel('Size x', 'm')
#ax.set_ylabel('Weight y', 'kg')
ax.set_ylabel('Weight y', 'kg')
ax.set_xlim(2, 4)
ax.set_ylim(0, 400)
ax.set_xticks(np.arange(2.0, 4.1, 0.5))
ax.set_yticks(np.arange(0, 401, 100))
ax.set_yticklabels([])
ax.annotate('Error',
xy=(x[28]+0.05, y[28]+60), xycoords='data',
xytext=(3.4, 70), textcoords='data', ha='left',
@ -52,31 +37,30 @@ def plot_data_errors(ax, x, y, c):
yy = [c*x[i]**3.0, y[i]]
ax.plot(xx, yy, zorder=5, **lsDm)
def plot_error_hist(ax, x, y, c):
ax.set_xlabel('Squared error')
ax.set_ylabel('Frequency')
bins = np.arange(0.0, 1250.0, 100)
bins = np.arange(0.0, 11000.0, 750)
ax.set_xlim(bins[0], bins[-1])
#ax.set_ylim(0, 35)
ax.set_xticks(np.arange(bins[0], bins[-1], 200))
#ax.set_yticks(np.arange(0, 36, 10))
ax.set_ylim(0, 15)
ax.set_xticks(np.arange(bins[0], bins[-1], 5000))
ax.set_yticks(np.arange(0, 16, 5))
errors = (y-(c*x**3.0))**2.0
mls = np.mean(errors)
ax.annotate('Mean\nsquared\nerror',
xy=(mls, 0.5), xycoords='data',
xytext=(800, 3), textcoords='data', ha='left',
xytext=(4500, 6), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.2),
connectionstyle="angle3,angleA=10,angleB=90") )
ax.hist(errors, bins, **fsC)
if __name__ == "__main__":
x, y, c = create_data()
fig, (ax1, ax2) = plt.subplots(1, 2)
fig.subplots_adjust(wspace=0.2, **adjust_fs(left=6.0, right=1.2))
plot_data(ax1, x, y, c)
plot_data_errors(ax2, x, y, c)
#plot_error_hist(ax2, x, y, c)
fig.subplots_adjust(wspace=0.4, **adjust_fs(left=6.0, right=1.2))
plot_data_errors(ax1, x, y, c)
plot_error_hist(ax2, x, y, c)
fig.savefig("cubicerrors.pdf")
plt.close()

26
regression/lecture/cubicfunc.py
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@ -2,7 +2,7 @@ import matplotlib.pyplot as plt
import numpy as np
from plotstyle import *
if __name__ == "__main__":
def create_data():
# wikipedia:
# Generally, males vary in total length from 250 to 390 cm and
# weigh between 90 and 306 kg
@ -12,10 +12,20 @@ if __name__ == "__main__":
rng = np.random.RandomState(32281)
noise = rng.randn(len(x))*50
y += noise
return x, y, c
fig, ax = plt.subplots(figsize=cm_size(figure_width, 1.4*figure_height))
fig.subplots_adjust(**adjust_fs(left=6.0, right=1.2))
def plot_data(ax, x, y):
ax.plot(x, y, **psA)
ax.set_xlabel('Size x', 'm')
ax.set_ylabel('Weight y', 'kg')
ax.set_xlim(2, 4)
ax.set_ylim(0, 400)
ax.set_xticks(np.arange(2.0, 4.1, 0.5))
ax.set_yticks(np.arange(0, 401, 100))
def plot_data_fac(ax, x, y, c):
ax.plot(x, y, zorder=10, **psA)
xx = np.linspace(2.1, 3.9, 100)
ax.plot(xx, c*xx**3.0, zorder=5, **lsB)
@ -27,6 +37,14 @@ if __name__ == "__main__":
ax.set_ylim(0, 400)
ax.set_xticks(np.arange(2.0, 4.1, 0.5))
ax.set_yticks(np.arange(0, 401, 100))
if __name__ == "__main__":
x, y, c = create_data()
print(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)
plot_data_fac(ax2, x, y, c)
fig.savefig("cubicfunc.pdf")
plt.close()

7
regression/lecture/cubicmse.py
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@ -14,6 +14,7 @@ def create_data():
y += noise
return x, y, c
def gradient_descent(x, y):
n = 20
dc = 0.01
@ -29,6 +30,7 @@ def gradient_descent(x, y):
mses.append(m0)
cc -= eps*dmdc
return cs, mses
def plot_mse(ax, x, y, c, cs):
ms = np.zeros(len(cs))
@ -54,7 +56,8 @@ def plot_mse(ax, x, y, c, cs):
ax.set_ylim(0, 25000)
ax.set_xticks(np.arange(0.0, 10.1, 2.0))
ax.set_yticks(np.arange(0, 30001, 10000))
def plot_descent(ax, cs, mses):
ax.plot(np.arange(len(mses))+1, mses, **lpsBm)
ax.set_xlabel('Iteration')
@ -69,7 +72,7 @@ def plot_descent(ax, cs, mses):
if __name__ == "__main__":
x, y, c = create_data()
cs, mses = gradient_descent(x, y)
fig, (ax1, ax2) = plt.subplots(1, 2)
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=cm_size(figure_width, 1.1*figure_height))
fig.subplots_adjust(wspace=0.2, **adjust_fs(left=8.0, right=0.5))
plot_mse(ax1, x, y, c, cs)
plot_descent(ax2, cs, mses)

111
regression/lecture/regression-chapter.tex
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@ -40,6 +40,113 @@
\item Homework is to do the 2d problem with the straight line!
\end{itemize}
\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
drawing $x$ values between 0 and 120 and \varcode{randn()} for
jittering the $y$ values with a standard deviation of 15. Then
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 documentation to find out how to use \code{surface()}.
\end{exercise}
\begin{exercise}{meanSquaredGradient.m}{}\label{gradientexercise}%
Implement a function \varcode{meanSquaredGradient()}, that takes the
$x$- 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}
\begin{exercise}{errorGradient.m}{}
Extend the script of exercises~\ref{errorsurfaceexercise} to plot
both the error surface and gradients using the
\varcode{meanSquaredGradient()} function from
exercise~\ref{gradientexercise}. Vectors in space can be easily
plotted using the function \code{quiver()}. Use \code{contour()}
instead of \code{surface()} to plot the error surface.
\end{exercise}
\begin{exercise}{gradientDescent.m}{}
Implement the gradient descent for the problem of fitting a straight
line to some measured data. Reuse the data generated in
exercise~\ref{errorsurfaceexercise}.
\begin{enumerate}
\item Store for each iteration the error value.
\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}\vspace{-4.5ex}
\end{exercise}
\begin{figure}[t]
\includegraphics[width=1\textwidth]{lin_regress}\hfill
\titlecaption{Example data suggesting a linear relation.}{A set of
input signals $x$, e.g. stimulus intensities, were used to probe a
system. The system's output $y$ to the inputs are noted
(left). Assuming a linear relation between $x$ and $y$ leaves us
with 2 parameters, the slope (center) and the intercept with the
y-axis (right panel).}\label{linregressiondatafig}
\end{figure}
\begin{figure}[t]
\includegraphics[width=1\textwidth]{linear_least_squares}
\titlecaption{Estimating the \emph{mean square error}.} {The
deviation error (orange) between the prediction (red line) and the
observations (blue dots) is calculated for each data point
(left). Then the deviations are squared and the average is
calculated (right).}
\label{leastsquareerrorfig}
\end{figure}
\begin{figure}[t]
\includegraphics[width=0.75\textwidth]{error_surface}
\titlecaption{Error surface.}{The two model parameters $m$ and $b$
define the base area of the surface plot. For each parameter
combination of slope and intercept the error is calculated. The
resulting surface has a minimum which indicates the parameter
combination that best fits the data.}\label{errorsurfacefig}
\end{figure}
\begin{figure}[t]
\includegraphics[width=0.75\textwidth]{error_gradient}
\titlecaption{Gradient of the error surface.} {Each arrow points
into the direction of the greatest ascend at different positions
of the error surface shown in \figref{errorsurfacefig}. The
contour lines in the background illustrate the error surface. Warm
colors indicate high errors, colder colors low error values. Each
contour line connects points of equal
error.}\label{gradientquiverfig}
\end{figure}
\begin{figure}[t]
\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
sufficiently large.The dots show the positions after each
iteration of the algorithm.} \label{gradientdescentfig}
\end{figure}
\subsection{Linear fits}
\begin{itemize}
\item Polyfit is easy: unique solution! $c x^2$ is also a linear fit.
@ -54,8 +161,8 @@ Fit with matlab functions lsqcurvefit, polyfit
\subsection{Non-linear fits}
\begin{itemize}
\item Example that illustrates the Nebenminima Problem (with error surface)
\item You need got initial values for the parameter!
\item Example that fitting gets harder the more parameter yuo have.
\item You need initial values for the parameter!
\item Example that fitting gets harder the more parameter you have.
\item Try to fix as many parameter before doing the fit.
\item How to test the quality of a fit? Residuals. $\chi^2$ test. Run-test.
\end{itemize}

167
regression/lecture/regression.tex
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@ -2,99 +2,104 @@
\exercisechapter{Optimization and gradient descent}
Optimization problems arise in many different contexts. For example,
to understand the behavior of a given system, the system is probed
with a range of input signals and then the resulting responses are
measured. This input-output relation can be described by a model. Such
a model can be a simple function that maps the input signals to
corresponding responses, it can be a filter, or a system of
differential equations. In any case, the model has some parameters that
specify how input and output signals are related. Which combination
of parameter values are best suited to describe the input-output
relation? The process of finding the best parameter values is an
optimization problem. For a simple parameterized function that maps
input to output values, this is the special case of a \enterm{curve
fitting} problem, where the average distance between the curve and
the response values is minimized. One basic numerical method used for
such optimization problems is the so called gradient descent, which is
introduced in this chapter.
%%% Weiteres einfaches verbales Beispiel? Eventuell aus der Populationsoekologie?
to understand the behavior of a given neuronal system, the system is
probed with a range of input signals and then the resulting responses
are measured. This input-output relation can be described by a
model. Such a model can be a simple function that maps the input
signals to corresponding responses, it can be a filter, or a system of
differential equations. In any case, the model has some parameters
that specify how input and output signals are related. Which
combination of parameter values are best suited to describe the
input-output relation? The process of finding the best parameter
values is an optimization problem. For a simple parameterized function
that maps input to output values, this is the special case of a
\enterm{curve fitting} problem, where the average distance between the
curve and the response values is minimized. One basic numerical method
used for such optimization problems is the so called gradient descent,
which is introduced in this chapter.
\begin{figure}[t]
\includegraphics[width=1\textwidth]{lin_regress}\hfill
\titlecaption{Example data suggesting a linear relation.}{A set of
input signals $x$, e.g. stimulus intensities, were used to probe a
system. The system's output $y$ to the inputs are noted
(left). Assuming a linear relation between $x$ and $y$ leaves us
with 2 parameters, the slope (center) and the intercept with the
y-axis (right panel).}\label{linregressiondatafig}
\includegraphics{cubicfunc}
\titlecaption{Example data suggesting a cubic relation.}{The length
$x$ and weight $y$ of $n=34$ male tigers (blue, left). Assuming a
cubic relation between size and weight leaves us with a single
free parameters, a scaling factor. The cubic relation is shown for
a few values of this scaling factor (orange and red,
right).}\label{cubicdatafig}
\end{figure}
The data plotted in \figref{linregressiondatafig} suggest a linear
relation between input and output of the system. We thus assume that a
straight line
For demonstrating the curve-fitting problem let's take the simple
example of weights and sizes measured for a number of male tigers
(\figref{cubicdatafig}). Weight $y$ is proportional to volume
$V$ via the density $\rho$. The volume $V$ of any object is
proportional to its length $x$ cubed. The factor $\alpha$ relating
volume and size cubed depends on the shape of the object and we do not
know this factor for tigers. For the data set we thus expect a cubic
relation between weight and length
\begin{equation}
\label{straightline}
y = f(x; m, b) = m\cdot x + b
\label{cubicfunc}
y = f(x; c) = c\cdot x^3
\end{equation}
is an appropriate model to describe the system. The line has two free
parameter, the slope $m$ and the $y$-intercept $b$. We need to find
values for the slope and the intercept that best describe the measured
data. In this chapter we use this example to illustrate the gradient
descent and how this methods can be used to find a combination of
slope and intercept that best describes the system.
where $c=\rho\alpha$, the product of a tiger's density and form
factor, is the only free parameter in the relation. We would like to
find out which value of $c$ best describes the measured data. In the
following we use this example to illustrate the gradient descent as a
basic method for finding such an optimal parameter.
\section{The error function --- mean squared error}
Before the optimization can be done we need to specify what exactly is
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}
$y^{est}(x_i)$ 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 $\sum_{i=1}^N y_i -
y^{est}(x_i)$. 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:
$\sum_{i=1}^N |y_i - y^{est}(x_i)|$. 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}(x_i)\}) = \frac{1}{N} \sum_{i=1}^N |y_i - y^{est}(x_i)|
\end{equation}
Instead of the averaged absolute errors, the \enterm[mean squared
error]{mean squared error} (\determ[quadratischer
Fehler!mittlerer]{mittlerer quadratischer Fehler})
Before we go ahead finding the optimal parameter value we need to
specify what exactly we consider as an optimal fit. In our example we
search the parameter that describes the relation of $x$ and $y$
best. What is meant by this? The length $x_i$ of each tiger is
associated with a weight $y_i$ and for each $x_i$ we have a
\emph{prediction} or \emph{estimation} $y^{est}(x_i)$ of the weight by
the model \eqnref{cubicfunc} for a specific value of the parameter
$c$. Prediction and actual data value ideally match (in a perfect
noise-free world), but in general the estimate and measurement are
separated by some distance or error $y_i - y^{est}(x_i)$. In our
example the estimate of the weight for the length $x_i$ is given by
equation \eqref{cubicfunc} $y^{est}(x_i) = f(x_i;c)$. The best fitting
model with parameter $c$ is the one that somehow minimizes the
distances between observations $y_i$ and corresponding estimations
$y^{est}(x_i)$ (\figref{cubicerrorsfig}).
As a first guess we could simply minimize the sum of the distances,
$\sum_{i=1}^N y_i - y^{est}(x_i)$. This, however, does not work
because positive and negative errors would cancel out, no matter how
large they are, and sum up to values close to zero. Better is to sum
over absolute distances: $\sum_{i=1}^N |y_i - y^{est}(x_i)|$. This sum
can only be small if all deviations are indeed small no matter if they
are above or below the prediction. The sum of the squared distances,
$\sum_{i=1}^N (y_i - y^{est}(x_i))^2$, turns out to be an even better
choice. Instead of the sum we could also minimize the average distance
\begin{equation}
\label{meansquarederror}
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, $(y_i - y^{est}(x_i))^2$, is
always positive and thus positive and negative error values do not
This is known as the \enterm[mean squared error]{mean squared error}
(\determ[quadratischer Fehler!mittlerer]{mittlerer quadratischer
Fehler}). Similar to the absolute distance, the square of the errors
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. In
chapter~\ref{maximumlikelihoodchapter} we show that minimizing the
mean square error is equivalent to maximizing the likelihood that the
mean squared 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
corresponding predictions \varcode{y\_est}, compute the \emph{mean
square error} between \varcode{y} and \varcode{y\_est} in a single
line of code.
\begin{exercise}{meansquarederrorline.m}{}\label{mseexercise}
Simulate $n=40$ tigers ranging from 2.2 to 3.9\,m in size and store
these sizes in a vector \varcode{x}. Compute the corresponding
predicted weights \varcode{yest} for each tiger according to
\eqnref{cubicfunc} with $c=6$\,\kilo\gram\per\meter\cubed. From the
predictions generate simulated measurements of the tiger's weights
\varcode{y}, by adding normally distributed random numbers to the
predictions scaled to a standard deviation of 50\,\kilo\gram.
Compute the \emph{mean squared error} between \varcode{y} and
\varcode{yest} in a single line of code.
\end{exercise}
@ -110,13 +115,13 @@ 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}
\titlecaption{Estimating the \emph{mean square error}.} {The
deviation error, orange) between the prediction (red
line) and the observations (blue dots) is calculated for each data
point (left). Then the deviations are squared and the aveage is
\includegraphics{cubicerrors}
\titlecaption{Estimating the \emph{mean squared error}.} {The
deviation error (orange) between the prediction (red line) and the
observations (blue dots) is calculated for each data point
(left). Then the deviations are squared and the average is
calculated (right).}
\label{leastsquareerrorfig}
\label{cubicerrorsfig}
\end{figure}
Replacing $y^{est}$ in the mean squared error \eqref{meansquarederror}
@ -139,7 +144,7 @@ Fehler!kleinster]{Methode der kleinsten Quadrate}).
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.
mean squared error.
\end{exercise}
@ -359,7 +364,7 @@ distance between the red dots in \figref{gradientdescentfig}) is
large.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{gradient_descent}
\includegraphics{cubicmse}
\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