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[regression] new figures for fit of cubic functions

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This commit is contained in:
Jan Benda 2019-12-19 19:25:01 +01:00
parent 60a8250590
commit ebff6cf5ad
4 changed files with 241 additions and 3 deletions
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102
regression/lecture/cubicerrors.py Normal file
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import matplotlib.pyplot as plt
import numpy as np
def create_data():
# wikipedia:
# Generally, males vary in total length from 250 to 390 cm and
# weigh between 90 and 306 kg
c = 6
x = np.arange(2.2, 3.9, 0.05)
y = c * x**3.0
rng = np.random.RandomState(32281)
noise = rng.randn(len(x))*50
y += noise
return x, y, c
def plot_data(ax, x, y, c):
ax.scatter(x, y, marker='o', color='b', s=40, zorder=10)
xx = np.linspace(2.1, 3.9, 100)
ax.plot(xx, c*xx**3.0, color='#CC0000', lw=2, zorder=5)
for cc in [0.25*c, 0.5*c, 2.0*c, 4.0*c]:
ax.plot(xx, cc*xx**3.0, color='#FF9900', lw=1.5, zorder=5)
ax.spines["right"].set_visible(False)
ax.spines["top"].set_visible(False)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
ax.tick_params(direction="out", width=1.25)
ax.tick_params(direction="out", width=1.25)
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.spines["right"].set_visible(False)
ax.spines["top"].set_visible(False)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
ax.tick_params(direction="out", width=1.25)
ax.tick_params(direction="out", width=1.25)
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))
ax.set_yticklabels([])
ax.annotate('Error',
xy=(x[28]+0.05, y[28]+60), xycoords='data',
xytext=(3.4, 70), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.9,1.0),
connectionstyle="angle3,angleA=50,angleB=-30") )
ax.scatter(x[:40], y[:40], color='b', s=10, zorder=0)
inxs = [3, 10, 11, 17, 18, 21, 28, 30, 33]
ax.scatter(x[inxs], y[inxs], color='b', s=40, zorder=10)
xx = np.linspace(2.1, 3.9, 100)
ax.plot(xx, c*xx**3.0, color='#CC0000', lw=2)
for i in inxs :
xx = [x[i], x[i]]
yy = [c*x[i]**3.0, y[i]]
ax.plot(xx, yy, color='#FF9900', lw=2, zorder=5)
def plot_error_hist(ax, x, y, c):
ax.spines["right"].set_visible(False)
ax.spines["top"].set_visible(False)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
ax.tick_params(direction="out", width=1.25)
ax.tick_params(direction="out", width=1.25)
ax.set_xlabel('Squared error')
ax.set_ylabel('Frequency')
bins = np.arange(0.0, 1250.0, 100)
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))
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',
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.2),
connectionstyle="angle3,angleA=10,angleB=90") )
ax.hist(errors, bins, color='#FF9900')
if __name__ == "__main__":
x, y, c = create_data()
plt.xkcd()
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(7., 2.6))
plot_data(ax1, x, y, c)
plot_data_errors(ax2, x, y, c)
#plot_error_hist(ax2, x, y, c)
fig.set_facecolor("white")
fig.tight_layout()
fig.savefig("cubicerrors.pdf")
plt.close()

40
regression/lecture/cubicfunc.py Normal file
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import matplotlib.pyplot as plt
import numpy as np
if __name__ == "__main__":
# wikipedia:
# Generally, males vary in total length from 250 to 390 cm and
# weigh between 90 and 306 kg
c = 6
x = np.arange(2.2, 3.9, 0.05)
y = c * x**3.0
rng = np.random.RandomState(32281)
noise = rng.randn(len(x))*50
y += noise
plt.xkcd()
fig, ax = plt.subplots(figsize=(7., 3.6))
ax.scatter(x, y, marker='o', color='b', s=40, zorder=10)
xx = np.linspace(2.1, 3.9, 100)
ax.plot(xx, c*xx**3.0, color='#CC0000', lw=3, zorder=5)
for cc in [0.25*c, 0.5*c, 2.0*c, 4.0*c]:
ax.plot(xx, cc*xx**3.0, color='#FF9900', lw=2, zorder=5)
ax.spines["right"].set_visible(False)
ax.spines["top"].set_visible(False)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
ax.tick_params(direction="out", width=1.25)
ax.tick_params(direction="out", width=1.25)
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))
fig.set_facecolor("white")
fig.subplots_adjust(0.11, 0.16, 0.98, 0.97)
fig.savefig("cubicfunc.pdf")
plt.close()

93
regression/lecture/cubicmse.py Normal file
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import matplotlib.pyplot as plt
import numpy as np
def create_data():
# wikipedia:
# Generally, males vary in total length from 250 to 390 cm and
# weigh between 90 and 306 kg
c = 6
x = np.arange(2.2, 3.9, 0.05)
y = c * x**3.0
rng = np.random.RandomState(32281)
noise = rng.randn(len(x))*50
y += noise
return x, y, c
def gradient_descent(x, y):
n = 20
dc = 0.01
eps = 0.0001
cc = 1.1
cs = []
mses = []
for k in range(n):
m0 = np.mean((y-(cc*x**3.0))**2.0)
m1 = np.mean((y-((cc+dc)*x**3.0))**2.0)
dmdc = (m1 - m0)/dc
cs.append(cc)
mses.append(m0)
cc -= eps*dmdc
return cs, mses
def plot_mse(ax, x, y, c, cs):
ms = np.zeros(len(cs))
for i, cc in enumerate(cs):
ms[i] = np.mean((y-(cc*x**3.0))**2.0)
ccs = np.linspace(0.5, 10.0, 200)
mses = np.zeros(len(ccs))
for i, cc in enumerate(ccs):
mses[i] = np.mean((y-(cc*x**3.0))**2.0)
ax.plot(ccs, mses, 'b', lw=2, zorder=10)
ax.scatter(cs, ms, color='#cc0000', s=40, zorder=20)
ax.scatter(cs[-1], ms[-1], color='#FF9900', s=60, zorder=30)
for i in range(4):
ax.annotate('',
xy=(cs[i+1]+0.2, ms[i+1]), xycoords='data',
xytext=(cs[i]+0.3, ms[i]+200), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.0),
connectionstyle="angle3,angleA=10,angleB=70") )
ax.spines["right"].set_visible(False)
ax.spines["top"].set_visible(False)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
ax.tick_params(direction="out", width=1.25)
ax.tick_params(direction="out", width=1.25)
ax.set_xlabel('c')
ax.set_ylabel('mean squared error')
ax.set_xlim(0, 10)
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, '-o', c='#cc0000', mew=0, ms=8)
ax.spines["right"].set_visible(False)
ax.spines["top"].set_visible(False)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
ax.tick_params(direction="out", width=1.25)
ax.tick_params(direction="out", width=1.25)
ax.set_xlabel('iteration')
#ax.set_ylabel('mean squared error')
ax.set_xlim(0, 10.5)
ax.set_ylim(0, 25000)
ax.set_xticks(np.arange(0.0, 10.1, 2.0))
ax.set_yticks(np.arange(0, 30001, 10000))
ax.set_yticklabels([])
if __name__ == "__main__":
x, y, c = create_data()
cs, mses = gradient_descent(x, y)
plt.xkcd()
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(7., 2.6))
plot_mse(ax1, x, y, c, cs)
plot_descent(ax2, cs, mses)
fig.set_facecolor("white")
fig.tight_layout()
fig.savefig("cubicmse.pdf")
plt.close()

9
regression/lecture/regression-chapter.tex
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@ -25,13 +25,16 @@
\subsection{Start with one-dimensional problem!}
\begin{itemize}
\item Let's fit a cubic function $y=cx^3$ (weight versus length of a tiger)
\item Let's fit a cubic function $y=cx^3$ (weight versus length of a tiger)\\
\includegraphics[width=0.8\textwidth]{cubicfunc}
\item Introduce the problem, $c$ is density and form factor
\item How to generate an artificial data set (refer to simulation chapter)
\item How to plot a function (do not use the data x values!)
\item Just the mean square error as a function of the factor c
\item Just the mean square error as a function of the factor c\\
\includegraphics[width=0.8\textwidth]{cubicerrors}
\item Also mention the cost function for a straight line
\item 1-d gradient, NO quiver plot (it is a nightmare to get this right)
\item 1-d gradient, NO quiver plot (it is a nightmare to get this right)\\
\includegraphics[width=0.8\textwidth]{cubicmse}
\item 1-d gradient descend
\item Describe in words the n-d problem.
\item Homework is to do the 2d problem with the straight line!