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sonnenberg 2020-01-16 14:01:29 +01:00
commit 4e7373da38
30 changed files with 545 additions and 732 deletions
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16
bootstrap/lecture/bootstrapsem.py
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@ -1,8 +1,7 @@
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
from plotstyle import *
plt.xkcd()
fig = plt.figure( figsize=(6,3.5) )
rng = np.random.RandomState(637281) rng = np.random.RandomState(637281)
nsamples = 100 nsamples = 100
@ -25,11 +24,8 @@ for i in range(nresamples) :
musrs.append(np.mean(rng.randn(nsamples))) musrs.append(np.mean(rng.randn(nsamples)))
hmusrs, _ = np.histogram(musrs, bins, density=True) hmusrs, _ = np.histogram(musrs, bins, density=True)
ax = fig.add_subplot(1, 1, 1) fig, ax = plt.subplots(figsize=cm_size(figure_width, 1.2*figure_height))
ax.spines['right'].set_visible(False) fig.subplots_adjust(**adjust_fs(left=4.0, bottom=2.7, right=1.5))
ax.spines['top'].set_visible(False)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
ax.set_xlabel('Mean') ax.set_xlabel('Mean')
ax.set_xlim(-0.4, 0.4) ax.set_xlim(-0.4, 0.4)
ax.set_ylabel('Probability density') ax.set_ylabel('Probability density')
@ -45,9 +41,7 @@ ax.annotate('bootstrap\ndistribution',
xytext=(0.25, 4), textcoords='data', xytext=(0.25, 4), textcoords='data',
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5), arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
connectionstyle="angle3,angleA=20,angleB=60") ) connectionstyle="angle3,angleA=20,angleB=60") )
ax.bar(bins[:-1]-0.25*db, hmusrs, 0.5*db, color='r') ax.bar(bins[:-1]-0.25*db, hmusrs, 0.5*db, **fsB)
ax.bar(bins[:-1]+0.25*db, hmus, 0.5*db, color='b') ax.bar(bins[:-1]+0.25*db, hmus, 0.5*db, **fsA)
plt.tight_layout()
plt.savefig('bootstrapsem.pdf') plt.savefig('bootstrapsem.pdf')
#plt.show();

18
bootstrap/lecture/permutecorrelation.py
View File

@ -1,9 +1,8 @@
import numpy as np import numpy as np
import scipy.stats as st import scipy.stats as st
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
from plotstyle import *
plt.xkcd()
fig = plt.figure( figsize=(6,3.5) )
rng = np.random.RandomState(637281) rng = np.random.RandomState(637281)
# generate correlated data: # generate correlated data:
@ -36,11 +35,8 @@ print('Measured correlation coefficient %.2f is at %.4f percentile of bootstrap'
rp, ra = st.pearsonr(x, y) rp, ra = st.pearsonr(x, y)
print('Measured correlation coefficient %.2f is at %.4f percentile of test' % (rp, ra)) print('Measured correlation coefficient %.2f is at %.4f percentile of test' % (rp, ra))
ax = fig.add_subplot(1, 1, 1) fig, ax = plt.subplots(figsize=cm_size(figure_width, 1.2*figure_height))
ax.spines['right'].set_visible(False) fig.subplots_adjust(**adjust_fs(left=4.0, bottom=2.7, right=0.5, top=1.0))
ax.spines['top'].set_visible(False)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
ax.annotate('Measured\ncorrelation\nis significant!', ax.annotate('Measured\ncorrelation\nis significant!',
xy=(rd, 1.1), xycoords='data', xy=(rd, 1.1), xycoords='data',
xytext=(rd, 2.2), textcoords='data', ha='left', xytext=(rd, 2.2), textcoords='data', ha='left',
@ -56,13 +52,11 @@ ax.annotate('Distribution of\nuncorrelated\nsamples',
xytext=(-0.22, 5.0), textcoords='data', ha='left', xytext=(-0.22, 5.0), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.5,0.0), arrowprops=dict(arrowstyle="->", relpos=(0.5,0.0),
connectionstyle="angle3,angleA=150,angleB=100") ) connectionstyle="angle3,angleA=150,angleB=100") )
ax.bar(b[:-1], h, width=b[1]-b[0], color='#ffff66') ax.bar(b[:-1], h, width=b[1]-b[0], **fsC)
ax.bar(b[:-1][b[:-1]>=rq], h[b[:-1]>=rq], width=b[1]-b[0], color='#ff9900') ax.bar(b[:-1][b[:-1]>=rq], h[b[:-1]>=rq], width=b[1]-b[0], **fsB)
ax.plot( [rd, rd], [0, 1], 'b', linewidth=4 ) ax.plot( [rd, rd], [0, 1], **lsA)
ax.set_xlim(-0.25, 0.35) ax.set_xlim(-0.25, 0.35)
ax.set_xlabel('Correlation coefficient') ax.set_xlabel('Correlation coefficient')
ax.set_ylabel('Probability density of H0') ax.set_ylabel('Probability density of H0')
plt.tight_layout()
plt.savefig('permutecorrelation.pdf') plt.savefig('permutecorrelation.pdf')
#plt.show();

65
likelihood/lecture/mlecoding.py
View File

@ -1,24 +1,22 @@
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
import matplotlib.cm as cm import matplotlib.cm as cm
import matplotlib.gridspec as gridspec
from plotstyle import *
plt.xkcd()
fig = plt.figure( figsize=(6,6.8) )
rng = np.random.RandomState(4637281) rng = np.random.RandomState(4637281)
lmarg=0.1 lmarg=0.1
rmarg=0.1 rmarg=0.1
ax = fig.add_axes([lmarg, 0.75, 1.0-rmarg, 0.25]) fig = plt.figure(figsize=cm_size(figure_width, 2.8*figure_height))
ax.spines['bottom'].set_position('zero') spec = gridspec.GridSpec(nrows=4, ncols=1, height_ratios=[4, 4, 1, 3], hspace=0.2,
ax.spines['left'].set_visible(False) **adjust_fs(fig, left=4.0))
ax.spines['right'].set_visible(False) ax = fig.add_subplot(spec[0, 0])
ax.spines['top'].set_visible(False)
ax.xaxis.set_ticks_position('bottom')
ax.get_yaxis().set_visible(False)
ax.set_xlim(0.0, np.pi) ax.set_xlim(0.0, np.pi)
ax.set_xticks(np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi)) ax.set_xticks(np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi))
ax.set_xticklabels([]) ax.set_xticklabels([])
ax.set_ylim(0.0, 3.5) ax.set_ylim(0.0, 3.5)
ax.yaxis.set_major_locator(plt.NullLocator())
ax.text(-0.2, 0.5*3.5, 'Activity', rotation='vertical', va='center') ax.text(-0.2, 0.5*3.5, 'Activity', rotation='vertical', va='center')
ax.annotate('Tuning curve', ax.annotate('Tuning curve',
xy=(0.42*np.pi, 2.5), xycoords='data', xy=(0.42*np.pi, 2.5), xycoords='data',
@ -31,55 +29,49 @@ ax.annotate('',
arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5), arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5),
connectionstyle="angle3,angleA=80,angleB=90") ) connectionstyle="angle3,angleA=80,angleB=90") )
ax.text(0.52*np.pi, 0.7, 'preferred\norientation') ax.text(0.52*np.pi, 0.7, 'preferred\norientation')
ax.plot([0, 0], [0.0, 3.5], 'k', zorder=10, clip_on=False)
xx = np.arange(0.0, 2.0*np.pi, 0.01) xx = np.arange(0.0, 2.0*np.pi, 0.01)
pp = 0.5*np.pi pp = 0.5*np.pi
yy = np.exp(np.cos(2.0*(xx+pp))) yy = np.exp(np.cos(2.0*(xx+pp)))
ax.fill_between(xx, yy+0.25*yy, yy-0.25*yy, color=cm.autumn(0.3, 1), alpha=0.5) ax.fill_between(xx, yy+0.25*yy, yy-0.25*yy, **fsBa)
ax.plot(xx, yy, color=cm.autumn(0.0, 1)) ax.plot(xx, yy, **lsB)
ax = fig.add_axes([lmarg, 0.34, 1.0-rmarg, 0.38]) ax = fig.add_subplot(spec[1, 0])
ax.spines['bottom'].set_position('zero')
ax.spines['left'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.spines['top'].set_visible(False)
ax.xaxis.set_ticks_position('bottom')
ax.get_yaxis().set_visible(False)
ax.set_xlim(0.0, np.pi) ax.set_xlim(0.0, np.pi)
ax.set_xticks(np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi)) ax.set_xticks(np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi))
ax.set_xticklabels([]) ax.set_xticklabels([])
ax.set_ylim(-1.5, 3.0) ax.set_ylim(0.0, 3.0)
ax.text(0.5*np.pi, -1.8, 'Orientation', ha='center') ax.yaxis.set_major_locator(plt.NullLocator())
ax.text(-0.2, 0.5*3.5, 'Activity', rotation='vertical', va='center') ax.text(-0.2, 0.5*3.5, 'Activity', rotation='vertical', va='center')
ax.plot([0, 0], [0.0, 3.0], 'k', zorder=10, clip_on=False)
xx = np.arange(0.0, 1.0*np.pi, 0.01) xx = np.arange(0.0, 1.0*np.pi, 0.01)
prefphases = np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi) prefphases = np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi)
responses = [] responses = []
xresponse = 0.475*np.pi xresponse = 0.475*np.pi
for pp in prefphases : for pp, ls, ps in zip(prefphases, [lsE, lsC, lsD, lsB, lsD, lsC, lsE],
[psE, psC, psD, psB, psD, psC, psE]) :
yy = np.exp(np.cos(2.0*(xx+pp))) yy = np.exp(np.cos(2.0*(xx+pp)))
ax.plot(xx, yy, color=cm.autumn(2.0*np.abs(pp/np.pi-0.5), 1)) #ax.plot(xx, yy, color=cm.autumn(2.0*np.abs(pp/np.pi-0.5), 1))
ax.plot(xx, yy, **ls)
y = np.exp(np.cos(2.0*(xresponse+pp))) y = np.exp(np.cos(2.0*(xresponse+pp)))
responses.append(y + rng.randn()*0.25*y) responses.append(y + rng.randn()*0.25*y)
ax.plot(xresponse, y, '.', markersize=20, color=cm.autumn(2.0*np.abs(pp/np.pi-0.5), 1)) ax.plot(xresponse, y, **ps)
r=0.3
y=-0.8
ax.plot([pp-0.5*r*np.cos(pp), pp+0.5*r*np.cos(pp)], [y-r*np.sin(pp), y+r*np.sin(pp)], 'k', lw=6)
responses = np.array(responses) responses = np.array(responses)
ax = fig.add_axes([lmarg, 0.05, 1.0-rmarg, 0.22]) ax = fig.add_subplot(spec[2, 0])
ax.spines['left'].set_visible(False) ax.show_spines('')
ax.spines['right'].set_visible(False) r = 0.3
ax.spines['top'].set_visible(False) ax.set_ylim(-1.1*r, 1.1*r)
ax.xaxis.set_ticks_position('bottom') for pp in prefphases:
ax.get_yaxis().set_visible(False) ax.plot([pp-0.5*r*np.cos(pp), pp+0.5*r*np.cos(pp)], [-r*np.sin(pp), r*np.sin(pp)],
colors['black'], lw=6, clip_on=False)
ax = fig.add_subplot(spec[3, 0])
ax.set_xlim(0.0, np.pi) ax.set_xlim(0.0, np.pi)
ax.set_xticks(np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi)) ax.set_xticks(np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi))
ax.set_xticklabels([]) ax.set_xticklabels([])
ax.set_ylim(-1600, 0) ax.set_ylim(-1600, 0)
ax.yaxis.set_major_locator(plt.NullLocator())
ax.set_xlabel('Orientation') ax.set_xlabel('Orientation')
ax.text(-0.2, -800, 'Log-Likelihood', rotation='vertical', va='center') ax.text(-0.2, -800, 'Log-Likelihood', rotation='vertical', va='center')
ax.plot([0, 0], [-1600, 0], 'k', zorder=10, clip_on=False)
phases = np.linspace(0.0, 1.1*np.pi, 100) phases = np.linspace(0.0, 1.1*np.pi, 100)
probs = np.zeros((len(responses), len(phases))) probs = np.zeros((len(responses), len(phases)))
for k, (pp, r) in enumerate(zip(prefphases, responses)) : for k, (pp, r) in enumerate(zip(prefphases, responses)) :
@ -95,7 +87,6 @@ ax.annotate('',
arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5), arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5),
connectionstyle="angle3,angleA=80,angleB=90") ) connectionstyle="angle3,angleA=80,angleB=90") )
ax.text(maxp+0.05, -1100, 'most likely\norientation\ngiven the responses') ax.text(maxp+0.05, -1100, 'most likely\norientation\ngiven the responses')
ax.plot(phases, loglikelihood, '-b') ax.plot(phases, loglikelihood, **lsA)
plt.savefig('mlecoding.pdf') plt.savefig('mlecoding.pdf')
#plt.show();

37
likelihood/lecture/mlemean.py
View File

@ -1,8 +1,11 @@
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec
from plotstyle import *
plt.xkcd() fig = plt.figure(figsize=cm_size(figure_width, 1.8*figure_height))
fig = plt.figure( figsize=(6,5) ) spec = gridspec.GridSpec(nrows=2, ncols=2, hspace=0.6,
**adjust_fs(fig, left=5.5))
# the data: # the data:
n = 40 n = 40
@ -17,11 +20,7 @@ g=np.zeros((len(x), len(mus)))
for k, mu in enumerate(mus) : for k, mu in enumerate(mus) :
g[:,k] = np.exp(-0.5*((x-mu)/sigma)**2.0)/np.sqrt(2.0*np.pi)/sigma g[:,k] = np.exp(-0.5*((x-mu)/sigma)**2.0)/np.sqrt(2.0*np.pi)/sigma
# plot it: # plot it:
ax = fig.add_subplot( 2, 1, 1 ) ax = fig.add_subplot(spec[0, :])
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.set_xlim(0.5, 3.5) ax.set_xlim(0.5, 3.5)
ax.set_ylim(-0.02, 0.85) ax.set_ylim(-0.02, 0.85)
ax.set_xticks(np.arange(0, 5)) ax.set_xticks(np.arange(0, 5))
@ -41,9 +40,9 @@ for mu in mus :
ax.text(mu-0.1, 0.04, '?', zorder=1, ha='right') ax.text(mu-0.1, 0.04, '?', zorder=1, ha='right')
else : else :
ax.text(mu+0.1, 0.04, '?', zorder=1) ax.text(mu+0.1, 0.04, '?', zorder=1)
for k in range(len(mus)) : for k, ls in enumerate([lsCm, lsBm, lsDm]) :
ax.plot(x, g[:,k], zorder=5) ax.plot(x, g[:,k], zorder=5, **ls)
ax.scatter(xd, 0.05*rng.rand(len(xd))+0.2, s=30, zorder=10) ax.plot(xd, 0.05*rng.rand(len(xd))+0.2, zorder=10, **psAm)
# likelihood: # likelihood:
thetas=np.arange(1.5, 2.6, 0.01) thetas=np.arange(1.5, 2.6, 0.01)
@ -52,11 +51,7 @@ for i, theta in enumerate(thetas) :
ps[:,i]=np.exp(-0.5*((xd-theta)/sigma)**2.0)/np.sqrt(2.0*np.pi)/sigma ps[:,i]=np.exp(-0.5*((xd-theta)/sigma)**2.0)/np.sqrt(2.0*np.pi)/sigma
p=np.prod(ps,axis=0) p=np.prod(ps,axis=0)
# plot it: # plot it:
ax = fig.add_subplot( 2, 2, 3 ) ax = fig.add_subplot(spec[1, 0])
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.set_xlabel(r'Parameter $\theta$') ax.set_xlabel(r'Parameter $\theta$')
ax.set_ylabel('Likelihood') ax.set_ylabel('Likelihood')
ax.set_xticks(np.arange(1.6, 2.5, 0.4)) ax.set_xticks(np.arange(1.6, 2.5, 0.4))
@ -70,13 +65,9 @@ ax.annotate('',
xytext=(2.0, 5e-11), textcoords='data', xytext=(2.0, 5e-11), textcoords='data',
arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5), arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5),
connectionstyle="angle3,angleA=90,angleB=80")) connectionstyle="angle3,angleA=90,angleB=80"))
ax.plot(thetas,p) ax.plot(thetas, p, **lsAm)
ax = fig.add_subplot( 2, 2, 4 ) ax = fig.add_subplot(spec[1, 1])
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.set_xlabel(r'Parameter $\theta$') ax.set_xlabel(r'Parameter $\theta$')
ax.set_ylabel('Log-Likelihood') ax.set_ylabel('Log-Likelihood')
ax.set_ylim(-50,-20) ax.set_ylim(-50,-20)
@ -92,8 +83,6 @@ ax.annotate('',
xytext=(2.0, -26), textcoords='data', xytext=(2.0, -26), textcoords='data',
arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5), arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5),
connectionstyle="angle3,angleA=80,angleB=100")) connectionstyle="angle3,angleA=80,angleB=100"))
ax.plot(thetas,np.log(p)) ax.plot(thetas,np.log(p), **lsAm)
plt.tight_layout();
plt.savefig('mlemean.pdf') plt.savefig('mlemean.pdf')
#plt.show();

60
likelihood/lecture/mlepdf.py
View File

@ -2,9 +2,10 @@ import numpy as np
import scipy.stats as st import scipy.stats as st
import scipy.optimize as opt import scipy.optimize as opt
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
from plotstyle import *
plt.xkcd() fig, (ax1, ax2) = plt.subplots(1, 2)
fig = plt.figure( figsize=(6,3) ) fig.subplots_adjust(**adjust_fs(fig, right=1.0))
# the data: # the data:
n = 100 n = 100
@ -23,27 +24,23 @@ a = st.gamma.fit(xd, 5.0)
yf = st.gamma.pdf(xx, *a) yf = st.gamma.pdf(xx, *a)
# plot it: # plot it:
ax = fig.add_subplot( 1, 2, 1 ) ax1.set_xlim(0, 10.0)
ax.spines['right'].set_visible(False) ax1.set_ylim(0.0, 0.42)
ax.spines['top'].set_visible(False) ax1.set_xticks(np.arange(0, 11, 2))
ax.yaxis.set_ticks_position('left') ax1.set_yticks(np.arange(0, 0.42, 0.1))
ax.xaxis.set_ticks_position('bottom') ax1.set_xlabel('x')
ax.set_xlim(0, 10.0) ax1.set_ylabel('Probability density')
ax.set_ylim(0.0, 0.42) ax1.plot(xx, yy, label='pdf', **lsB)
ax.set_xticks( np.arange(0, 11, 2)) ax1.plot(xx, yf, label='mle', **lsCm)
ax.set_yticks( np.arange(0, 0.42, 0.1))
ax.set_xlabel('x')
ax.set_ylabel('Probability density')
ax.plot(xx, yy, '-', lw=5, color='#ff0000', label='pdf')
ax.plot(xx, yf, '-', lw=2, color='#ffcc00', label='mle')
kernel = st.gaussian_kde(xd) kernel = st.gaussian_kde(xd)
x = kernel(xd) x = kernel(xd)
x /= np.max(x) x /= np.max(x)
ax.scatter(xd, 0.05*x*(rng.rand(len(xd))-0.5)+0.05, s=30, zorder=10) sigma = 0.07
ax.legend(loc='upper right', frameon=False) ax1.plot(xd, sigma*x*(rng.rand(len(xd))-0.5)+sigma, zorder=10, **psAm)
ax1.legend(loc='upper right')
# histogram: # histogram:
h,b = np.histogram(xd, np.arange(0, 8.5, 1), density=True) h,b = np.histogram(xd, np.arange(0, 8.4, 0.5), density=True)
# fit histogram: # fit histogram:
def gammapdf(x, n, l, s) : def gammapdf(x, n, l, s) :
@ -52,22 +49,15 @@ popt, pcov = opt.curve_fit(gammapdf, b[:-1]+0.5*(b[1]-b[0]), h)
yc = st.gamma.pdf(xx, *popt) yc = st.gamma.pdf(xx, *popt)
# plot it: # plot it:
ax = fig.add_subplot( 1, 2, 2 ) ax2.set_xlim(0, 10.0)
ax.spines['right'].set_visible(False) ax2.set_xticks(np.arange(0, 11, 2))
ax.spines['top'].set_visible(False) ax2.set_xlabel('x')
ax.yaxis.set_ticks_position('left') ax2.set_ylim(0.0, 0.42)
ax.xaxis.set_ticks_position('bottom') ax2.set_yticks(np.arange(0, 0.42, 0.1))
ax.set_xlim(0, 10.0) ax2.set_ylabel('Probability density')
ax.set_xticks( np.arange(0, 11, 2)) ax2.plot(xx, yy, label='pdf', **lsB)
ax.set_xlabel('x') ax2.plot(xx, yc, label='fit', **lsCm)
ax.set_ylim(0.0, 0.42) ax2.bar(b[:-1], h, np.diff(b), **fsA)
ax.set_yticks( np.arange(0, 0.42, 0.1)) ax2.legend(loc='upper right')
ax.set_ylabel('Probability density')
ax.plot(xx, yy, '-', lw=5, color='#ff0000', label='pdf')
ax.plot(xx, yc, '-', lw=2, color='#ffcc00', label='fit')
ax.bar(b[:-1], h, np.diff(b))
ax.legend(loc='upper right', frameon=False)
plt.tight_layout();
plt.savefig('mlepdf.pdf') plt.savefig('mlepdf.pdf')
#plt.show();

66
likelihood/lecture/mlepropline.py
View File

@ -1,9 +1,14 @@
import numpy as np import numpy as np
import scipy.stats as st import scipy.stats as st
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec
from plotstyle import *
plt.xkcd() fig = plt.figure()
fig = plt.figure(figsize=(6, 3)) spec = gridspec.GridSpec(nrows=1, ncols=2, wspace=0.3,
**adjust_fs(fig, left=5.5))
spec1 = gridspec.GridSpecFromSubplotSpec(1, 2, spec[0, 0], width_ratios=[3, 1], wspace=0.0)
spec2 = gridspec.GridSpecFromSubplotSpec(1, 2, spec[0, 1], width_ratios=[3, 1], wspace=0.0)
# the line: # the line:
slope = 2.0 slope = 2.0
@ -20,71 +25,44 @@ slopef = np.sum(x*y)/np.sum(x*x)
yf = slopef*xx yf = slopef*xx
# plot it: # plot it:
ax = fig.add_axes([0.09, 0.02, 0.33, 0.9]) ax = fig.add_subplot(spec1[0, 0])
ax.spines['left'].set_position('zero')
ax.spines['bottom'].set_position('zero')
ax.spines['right'].set_visible(False)
ax.spines['top'].set_visible(False)
ax.get_xaxis().set_tick_params(direction='inout', length=10, width=2)
ax.get_yaxis().set_tick_params(direction='inout', length=10, width=2)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
ax.set_xticks(np.arange(0.0, 4.1)) ax.set_xticks(np.arange(0.0, 4.1))
ax.set_xlim(0.0, 4.2) ax.set_xlim(0.0, 4.2)
ax.set_ylim(-4.0, 12.0) ax.set_ylim(-4.0, 12.0)
ax.set_xlabel('x') ax.set_xlabel('x')
ax.set_ylabel('y') ax.set_ylabel('y')
ax.scatter(x, y, label='data', s=40, zorder=10) ax.plot(x, y, label='data', zorder=10, **psAm)
ax.plot(xx, yy, 'r', lw=5.0, color='#ff0000', label='original', zorder=5) ax.plot(xx, yy, label='original', zorder=5, **lsB)
ax.plot(xx, yf, '--', lw=1.0, color='#ffcc00', label='fit', zorder=7) ax.plot(xx, yf, label='fit', zorder=7, **lsCm)
ax.legend(loc='upper left', bbox_to_anchor=(0.0, 1.15), frameon=False) ax.legend(loc='upper left', bbox_to_anchor=(0.0, 1.15))
ax = fig.add_axes([0.42, 0.02, 0.07, 0.9]) ax = fig.add_subplot(spec1[0, 1])
ax.spines['left'].set_position('zero') ax.show_spines('l')
ax.spines['right'].set_visible(False)
ax.spines['top'].set_visible(False)
ax.spines['bottom'].set_visible(False)
ax.get_yaxis().set_tick_params(direction='inout', length=10, width=2)
ax.yaxis.set_ticks_position('left')
ax.set_xticks([]) ax.set_xticks([])
ax.set_ylim(-4.0, 12.0) ax.set_ylim(-4.0, 12.0)
ax.set_yticks([]) ax.set_yticks([])
bins = np.arange(-4.0, 12.1, 0.75) bins = np.arange(-4.0, 12.1, 0.75)
ax.hist(y, bins, orientation='horizontal', zorder=10) ax.hist(y, bins, orientation='horizontal', zorder=10, **fsA)
ax = fig.add_axes([0.6, 0.02, 0.33, 0.9]) ax = fig.add_subplot(spec2[0, 0])
ax.spines['left'].set_position('zero')
ax.spines['bottom'].set_position('zero')
ax.spines['right'].set_visible(False)
ax.spines['top'].set_visible(False)
ax.get_xaxis().set_tick_params(direction='inout', length=10, width=2)
ax.get_yaxis().set_tick_params(direction='inout', length=10, width=2)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
ax.set_xticks(np.arange(0.0, 4.1)) ax.set_xticks(np.arange(0.0, 4.1))
ax.set_xlim(0.0, 4.2) ax.set_xlim(0.0, 4.2)
ax.set_ylim(-4.0, 12.0) ax.set_ylim(-4.0, 12.0)
ax.set_xlabel('x') ax.set_xlabel('x')
ax.set_ylabel('y - mx') ax.set_ylabel('y - mx')
ax.scatter(x, y - slopef*x, label='residuals', s=40, zorder=10) ax.plot(x, y - slopef*x, label='residuals', zorder=10, **psAm)
#ax.legend(loc='upper left', bbox_to_anchor=(0.0, 1.0), frameon=False) #ax.legend(loc='upper left', bbox_to_anchor=(0.0, 1.0))
ax = fig.add_axes([0.93, 0.02, 0.07, 0.9]) ax = fig.add_subplot(spec2[0, 1])
ax.spines['left'].set_position('zero') ax.show_spines('l')
ax.spines['right'].set_visible(False)
ax.spines['top'].set_visible(False)
ax.spines['bottom'].set_visible(False)
ax.get_yaxis().set_tick_params(direction='inout', length=10, width=2)
ax.yaxis.set_ticks_position('left')
ax.set_xlim(0.0, 11.0) ax.set_xlim(0.0, 11.0)
ax.set_xticks([]) ax.set_xticks([])
ax.set_ylim(-4.0, 12.0) ax.set_ylim(-4.0, 12.0)
ax.set_yticks([]) ax.set_yticks([])
r = y - slopef*x r = y - slopef*x
ax.hist(r, bins, orientation='horizontal', zorder=10) ax.hist(r, bins, orientation='horizontal', zorder=10, **fsA)
gx = np.arange(-4.0, 12.1, 0.1) gx = np.arange(-4.0, 12.1, 0.1)
gy = st.norm.pdf(gx, np.mean(r), np.std(r)) gy = st.norm.pdf(gx, np.mean(r), np.std(r))
ax.plot(1.0+gy*29.0, gx, 'r', lw=2, zorder=5) ax.plot(1.0+gy*29.0, gx, zorder=5, **lsBm)
plt.savefig('mlepropline.pdf') plt.savefig('mlepropline.pdf')
#plt.show();

225
plotstyle.py
View File

@ -1,6 +1,5 @@
import matplotlib as mpl import matplotlib as mpl
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
from cycler import cycler
from mpl_toolkits.mplot3d import Axes3D from mpl_toolkits.mplot3d import Axes3D
xkcd_style = False xkcd_style = False
@ -13,125 +12,103 @@ figure_height = 6.0 # cm, for a 1 x 2 figure
ppi = 72.0 ppi = 72.0
# colors: # colors:
colors = {}
def lighter(color, lightness): colors['red'] = '#DD1000'
""" Make a color lighter. colors['orange'] = '#FF9900'
colors['lightorange'] = '#FFCC00'
Parameters colors['yellow'] = '#FFF720'
---------- colors['green'] = '#99FF00'
color: string colors['blue'] = '#0010CC'
An RGB color as a hexadecimal string (e.g. '#rrggbb'). colors['gray'] = '#A7A7A7'
lightness: float colors['black'] = '#000000'
The smaller the lightness, the lighter the returned color. colors['white'] = '#FFFFFF'
A lightness of 1 leaves the color untouched.
A lightness of 0 returns white. #colors_bendalab_vivid['green'] = '#30D700'
#colors_bendalab_vivid['blue'] = '#0020C0'
Returns
-------
color: string
The lighter color as a hexadecimal RGB string (e.g. '#rrggbb').
"""
r = int(color[1:3], 16)
g = int(color[3:5], 16)
b = int(color[5:7], 16)
rl = r + (1.0-lightness)*(0xff - r)
gl = g + (1.0-lightness)*(0xff - g)
bl = b + (1.0-lightness)*(0xff - b)
return '#%02X%02X%02X' % (rl, gl, bl)
def darker(color, saturation):
""" Make a color darker.
Parameters
----------
color: string
An RGB color as a hexadecimal string (e.g. '#rrggbb').
saturation: float
The smaller the saturation, the darker the returned color.
A saturation of 1 leaves the color untouched.
A saturation of 0 returns black.
Returns
-------
color: string
The darker color as a hexadecimal RGB string (e.g. '#rrggbb').
"""
r = int(color[1:3], 16)
g = int(color[3:5], 16)
b = int(color[5:7], 16)
rd = r * saturation
gd = g * saturation
bd = b * saturation
return '#%02X%02X%02X' % (rd, gd, bd)
# colors:
colors = {
'red': '#CC0000',
'orange': '#FF9900',
'lightorange': '#FFCC00',
'yellow': '#FFFF66',
'green': '#99FF00',
'blue': '#0000CC'
}
""" Muted colors used by the Benda-lab. """
colors_bendalab = {}
colors_bendalab['red'] = '#C02010'
colors_bendalab['orange'] = '#F78010'
colors_bendalab['yellow'] = '#F0D730'
colors_bendalab['green'] = '#A0B717'
colors_bendalab['cyan'] = '#40A787'
colors_bendalab['blue'] = '#2757A0'
colors_bendalab['purple'] = '#573790'
colors_bendalab['pink'] = '#C72750'
colors_bendalab['grey'] = '#A0A0A0'
colors_bendalab['black'] = '#000000'
""" Vivid colors used by the Benda-lab. """
colors_bendalab_vivid = {}
colors_bendalab_vivid['red'] = '#D71000'
colors_bendalab_vivid['orange'] = '#FF9000'
colors_bendalab_vivid['yellow'] = '#FFF700'
colors_bendalab_vivid['green'] = '#30D700'
colors_bendalab_vivid['cyan'] = '#00F0B0'
colors_bendalab_vivid['blue'] = '#0020C0'
colors_bendalab_vivid['purple'] = '#B000B0'
colors_bendalab_vivid['pink'] = '#F00080'
colors_bendalab_vivid['grey'] = '#A7A7A7'
colors_bendalab_vivid['black'] = '#000000'
# colors for the plots of the script:
colors = colors_bendalab_vivid
colors['lightorange'] = colors['yellow']
#colors['yellow'] = lighter(colors['yellow'], 0.65)
colors['yellow'] = '#FFFF55'
# line styles for plot(): # line styles for plot():
lsSpine = {'c': colors['black'], 'linestyle': '-', 'linewidth': 1} lwthick = 3.0
lsGrid = {'c': colors['grey'], 'linestyle': '--', 'linewidth': 1} lwthin = 1.8
mainline = {'linestyle': '-', 'linewidth': lwthick}
# 'B1': prominent line with first color and style from color group 'B' minorline = {'linestyle': '-', 'linewidth': lwthin}
# 'C2m': minor line with second color and style from color group 'C' largemarker = {'marker': 'o', 'markersize': 9, 'markeredgecolor': colors['white'], 'markeredgewidth': 1}
ls = { smallmarker = {'marker': 'o', 'markersize': 6, 'markeredgecolor': colors['white'], 'markeredgewidth': 1}
'A1': {'c': colors['red'], 'linestyle': '-', 'linewidth': 3}, largelinepoints = {'linestyle': '-', 'linewidth': lwthick, 'marker': 'o', 'markersize': 10, 'markeredgecolor': colors['white'], 'markeredgewidth': 1}
'A2': {'c': colors['orange'], 'linestyle': '-', 'linewidth': 3}, smalllinepoints = {'linestyle': '-', 'linewidth': 1.4, 'marker': 'o', 'markersize': 7, 'markeredgecolor': colors['white'], 'markeredgewidth': 1}
'A3': {'c': colors['lightorange'], 'linestyle': '-', 'linewidth': 3}, filllw = 1
'B1': {'c': colors['orange'], 'linestyle': '-', 'linewidth': 3}, fillec = colors['white']
'B2': {'c': colors['lightorange'], 'linestyle': '-', 'linewidth': 3}, fillalpha = 0.4
'B3': {'c': colors['yellow'], 'linestyle': '-', 'linewidth': 3},
'C1': {'c': colors['green'], 'linestyle': '-', 'linewidth': 3}, # helper lines:
'D1': {'c': colors['blue'], 'linestyle': '-', 'linewidth': 3}, lsSpine = {'c': colors['black'], 'linestyle': '-', 'linewidth': 1, 'clip_on': False}
'A1m': {'c': colors['red'], 'linestyle': '-', 'linewidth': 2}, lsGrid = {'c': colors['gray'], 'linestyle': '--', 'linewidth': 1}
'A2m': {'c': colors['orange'], 'linestyle': '-', 'linewidth': 2}, lsMarker = {'c': colors['black'], 'linestyle': '-', 'linewidth': 2}
'A3m': {'c': colors['lightorange'], 'linestyle': '-', 'linewidth': 2},
'B1m': {'c': colors['orange'], 'linestyle': '-', 'linewidth': 2}, # line (ls), point (ps), and fill styles (fs).
'B2m': {'c': colors['lightorange'], 'linestyle': '-', 'linewidth': 2},
'B3m': {'c': colors['yellow'], 'linestyle': '-', 'linewidth': 2}, # Each style is derived from a main color as indicated by the capital letter.
'C1m': {'c': colors['green'], 'linestyle': '-', 'linewidth': 2},
'D1m': {'c': colors['blue'], 'linestyle': '-', 'linewidth': 2}, # Line styles come in two variants:
} # - plain style with a thick/solid line (e.g. lsA), and
# - minor style with a thinner or dashed line (e.g. lsAm).
# Point (marker) styles come in two variants:
# - plain style with large solid markers (e.g. psB), and
# - minor style with smaller markers (e.g. psBm).
# Linepoint styles (markers connected by lines) come in two variants:
# - plain style with large solid markers (e.g. lpsA), and
# - minor style with smaller markers (e.g. lpsAm).
# Fill styles come in three variants:
# - plain (e.g. fsB) for a solid fill color and a darker edge color,
# - solid (e.g. fsBs) for a solid fill color and without edge color, and
# - alpha (e.g. fsBa) for a transparent fill color without edge color.
lsA = dict({'color': colors['blue']}, **mainline)
lsAm = dict({'color': colors['blue']}, **minorline)
psA = dict({'color': colors['blue'], 'linestyle': 'none'}, **largemarker)
psAm = dict({'color': colors['blue'], 'linestyle': 'none'}, **smallmarker)
lpsA = dict({'color': colors['blue']}, **largelinepoints)
lpsAm = dict({'color': colors['blue']}, **smalllinepoints)
fsA = {'facecolor': colors['blue'], 'edgecolor': fillec, 'linewidth': filllw}
fsAs = {'facecolor': colors['blue'], 'edgecolor': 'none'}
fsAa = {'facecolor': colors['blue'], 'edgecolor': 'none', 'alpha': fillalpha}
lsB = dict({'color': colors['red']}, **mainline)
lsBm = dict({'color': colors['red']}, **minorline)
psB = dict({'color': colors['red'], 'linestyle': 'none'}, **largemarker)
psBm = dict({'color': colors['red'], 'linestyle': 'none'}, **smallmarker)
lpsB = dict({'color': colors['red']}, **largelinepoints)
lpsBm = dict({'color': colors['red']}, **smalllinepoints)
fsB = {'facecolor': colors['red'], 'edgecolor': fillec, 'linewidth': filllw}
fsBs = {'facecolor': colors['red'], 'edgecolor': 'none'}
fsBa = {'facecolor': colors['red'], 'edgecolor': 'none', 'alpha': fillalpha}
lsC = dict({'color': colors['lightorange']}, **mainline)
lsCm = dict({'color': colors['lightorange']}, **minorline)
psC = dict({'color': colors['lightorange'], 'linestyle': 'none'}, **largemarker)
psCm = dict({'color': colors['lightorange'], 'linestyle': 'none'}, **smallmarker)
fsC = {'facecolor': colors['lightorange'], 'edgecolor': fillec, 'linewidth': filllw}
fsCs = {'facecolor': colors['lightorange'], 'edgecolor': 'none'}
fsCa = {'facecolor': colors['lightorange'], 'edgecolor': 'none', 'alpha': fillalpha}
lsD = dict({'color': colors['orange']}, **mainline)
lsDm = dict({'color': colors['orange']}, **minorline)
psD = dict({'color': colors['orange'], 'linestyle': 'none'}, **largemarker)
psDm = dict({'color': colors['orange'], 'linestyle': 'none'}, **smallmarker)
fsD = {'facecolor': colors['orange'], 'edgecolor': fillec, 'linewidth': filllw}
fsDs = {'facecolor': colors['orange'], 'edgecolor': 'none'}
lsE = dict({'color': colors['yellow']}, **mainline)
lsEm = dict({'color': colors['yellow']}, **minorline)
psE = dict({'color': colors['yellow'], 'linestyle': 'none'}, **largemarker)
psEm = dict({'color': colors['yellow'], 'linestyle': 'none'}, **smallmarker)
fsE = {'facecolor': colors['yellow'], 'edgecolor': fillec, 'linewidth': filllw}
fsEs = {'facecolor': colors['yellow'], 'edgecolor': 'none'}
fsF = {'facecolor': colors['green'], 'edgecolor': fillec, 'linewidth': filllw}
fsFs = {'facecolor': colors['green'], 'edgecolor': 'none'}
# factor for scaling widths of bars in a bar plot: # factor for scaling widths of bars in a bar plot:
bar_fac = 1.0 bar_fac = 1.0
@ -364,19 +341,19 @@ def common_format():
mpl.rcParams['grid.color'] = lsGrid['c'] mpl.rcParams['grid.color'] = lsGrid['c']
mpl.rcParams['grid.linestyle'] = lsGrid['linestyle'] mpl.rcParams['grid.linestyle'] = lsGrid['linestyle']
mpl.rcParams['grid.linewidth'] = lsGrid['linewidth'] mpl.rcParams['grid.linewidth'] = lsGrid['linewidth']
mpl.rcParams['legend.frameon'] = False
mpl.rcParams['axes.facecolor'] = 'none' mpl.rcParams['axes.facecolor'] = 'none'
mpl.rcParams['axes.edgecolor'] = lsSpine['c'] mpl.rcParams['axes.edgecolor'] = lsSpine['c']
mpl.rcParams['axes.linewidth'] = lsSpine['linewidth'] mpl.rcParams['axes.linewidth'] = lsSpine['linewidth']
if 'axes.prop_cycle' in mpl.rcParams: if 'axes.prop_cycle' in mpl.rcParams:
from cycler import cycler
mpl.rcParams['axes.prop_cycle'] = cycler(color=[colors['blue'], colors['red'], mpl.rcParams['axes.prop_cycle'] = cycler(color=[colors['blue'], colors['red'],
colors['orange'], colors['green'], colors['lightorange'], colors['orange'],
colors['purple'], colors['yellow'], colors['yellow'], colors['green']])
colors['cyan'], colors['pink']])
else: else:
mpl.rcParams['axes.color_cycle'] = [colors['blue'], colors['red'], mpl.rcParams['axes.color_cycle'] = [colors['blue'], colors['red'],
colors['orange'], colors['green'], colors['lightorange'], colors['orange'],
colors['purple'], colors['yellow'], colors['yellow'], colors['green']]
colors['cyan'], colors['pink']]
# overwrite axes constructor: # overwrite axes constructor:
if not hasattr(mpl.axes.Subplot, '__init__orig'): if not hasattr(mpl.axes.Subplot, '__init__orig'):
mpl.axes.Subplot.__init__orig = mpl.axes.Subplot.__init__ mpl.axes.Subplot.__init__orig = mpl.axes.Subplot.__init__

68
projects/README
View File

@ -1,94 +1,104 @@
For new projects:
Copy project_template/ and adapt according to your needs
All projects: All projects:
check for time information check for time information
1) project_activation_curve
medium
Write questions
project_adaptation_fit 2) project_adaptation_fit
OK, medium OK, medium
Add plotting of cost function Add plotting of cost function
project_eod 3) project_eod
OK, medium - difficult OK, medium - difficult
b_0 is not defined b_0 is not defined
project_eyetracker 4) project_eyetracker
OK, difficult OK, difficult
no statistics, but kmeans no statistics, but kmeans
project_fano_slope 5) project_face_selectivity
medium-difficult
(Marius monkey data)
6) project_fano_slope
OK, difficult OK, difficult
project_fano_test 7) project_fano_test
OK - OK -
project_fano_time 8) project_fano_time
OK, medium-difficult OK, medium-difficult
project_ficurves 9) project_ficurves
OK, medium OK, medium
Maybe add correlation test or fit statistics Maybe add correlation test or fit statistics
project_input_resistance 10) project_input_resistance
medium medium
What is the problem with this project? --> No difference between segments What is the problem with this project? --> No difference between segments
Improve questions Improve questions
project_isicorrelations 11) project_isicorrelations
medium-difficult medium-difficult
Need to finish solution Need to finish solution
project_isipdffit 12) project_isipdffit
Too technical Too technical
project_lif 13) project_lif
OK, difficult OK, difficult
no statistics no statistics
project_mutualinfo 14) project_mutualinfo
OK, medium OK, medium
project_noiseficurves 15) project_noiseficurves
OK, simple-medium OK, simple-medium
no statistics no statistics
project_numbers 16) project_numbers
simple simple
We might add some more involved statistical analysis We might add some more involved statistical analysis
project_pca_natural_images 17) project_pca_natural_images
medium medium
Make a solution (->Lukas) Make a solution (->Lukas)
project_photoreceptor 18) project_photoreceptor
OK, simple OK, simple
project_populationvector 19) project_populationvector
difficult difficult
OK OK
project_qvalues 20) project_power_analysis
medium
21) project_qvalues
- -
Interesting! But needs solution. Interesting! But needs solution.
project_random_walk 22) project_random_walk
simple-medium simple-medium
project_serialcorrelation 23) project_serialcorrelation
OK, simple-medium OK, simple-medium
project_spectra 24) project_shorttermpotentiation
Write questions
25) project_spectra
- -
Needs improvements and a solution Needs improvements and a solution
project_stimulus_reconstruction 26) project_stimulus_reconstruction
OK, difficult OK, difficult
project_vector_strength 27) project_vector_strength
OK, medium-difficult OK, medium-difficult
project_power_analysis
medium
Marius monkey data:
medium-difficult

2
projects/project_activation_curve/activation_curve.tex
View File

@ -14,7 +14,7 @@
\section{Estimation of the activation curve} \section{Estimation of the activation curve}
Mutations in genes, encoding for ion channels, can result in a variety of neurological diseases like epilepsy, autism and intellectual disability. One way to find a possible treatment is to compare the voltage dependent kinetics of the mutated channel with its corresponding wild-type. These kinetics are described in voltage-clamp experiments and the subsequent data analysis. Mutations in genes, encoding for ion channels, can result in a variety of neurological diseases like epilepsy, autism and intellectual disability. One way to find a possible treatment is to compare the voltage dependent kinetics of the mutated channel with its corresponding wild-type. These kinetics are described in voltage-clamp experiments and the subsequent data analysis.
In this task you will compute and compare the activation curves of the Nav1.6 wild-type channel and one with a mutation named A1622D (the amino acid Alanine (A) at the 1622nd position is replaced by Aspartic acid (D)) that causes intellectual disability in humans. In this task you will compute and compare the activation curves of the Nav1.6 wild-type channel and a variation named A1622D (the amino acid Alanine (A) at the 1622nd position is replaced by Aspartic acid (D)) that causes intellectual disability in humans.

16
regression/lecture/cubicerrors.py
View File

@ -16,11 +16,11 @@ def create_data():
def plot_data(ax, x, y, c): def plot_data(ax, x, y, c):
ax.scatter(x, y, marker='o', color=colors['blue'], s=40, zorder=10) ax.plot(x, y, zorder=10, **psAm)
xx = np.linspace(2.1, 3.9, 100) xx = np.linspace(2.1, 3.9, 100)
ax.plot(xx, c*xx**3.0, color=colors['red'], lw=2, zorder=5) 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]: for cc in [0.25*c, 0.5*c, 2.0*c, 4.0*c]:
ax.plot(xx, cc*xx**3.0, color=colors['orange'], lw=1.5, zorder=5) ax.plot(xx, cc*xx**3.0, zorder=5, **lsDm)
ax.set_xlabel('Size x', 'm') 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_xlim(2, 4)
@ -42,15 +42,15 @@ def plot_data_errors(ax, x, y, c):
xytext=(3.4, 70), textcoords='data', ha='left', xytext=(3.4, 70), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.9,1.0), arrowprops=dict(arrowstyle="->", relpos=(0.9,1.0),
connectionstyle="angle3,angleA=50,angleB=-30") ) connectionstyle="angle3,angleA=50,angleB=-30") )
ax.scatter(x[:40], y[:40], color=colors['blue'], s=10, zorder=0) ax.plot(x[:40], y[:40], zorder=0, **psAm)
inxs = [3, 10, 11, 17, 18, 21, 28, 30, 33] inxs = [3, 10, 11, 17, 18, 21, 28, 30, 33]
ax.scatter(x[inxs], y[inxs], color=colors['blue'], s=40, zorder=10) ax.plot(x[inxs], y[inxs], zorder=10, **psA)
xx = np.linspace(2.1, 3.9, 100) xx = np.linspace(2.1, 3.9, 100)
ax.plot(xx, c*xx**3.0, color=colors['red'], lw=2) ax.plot(xx, c*xx**3.0, **lsBm)
for i in inxs : for i in inxs :
xx = [x[i], x[i]] xx = [x[i], x[i]]
yy = [c*x[i]**3.0, y[i]] yy = [c*x[i]**3.0, y[i]]
ax.plot(xx, yy, color=colors['orange'], lw=2, zorder=5) ax.plot(xx, yy, zorder=5, **lsDm)
def plot_error_hist(ax, x, y, c): def plot_error_hist(ax, x, y, c):
ax.set_xlabel('Squared error') ax.set_xlabel('Squared error')
@ -67,7 +67,7 @@ def plot_error_hist(ax, x, y, c):
xytext=(800, 3), textcoords='data', ha='left', xytext=(800, 3), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.2), arrowprops=dict(arrowstyle="->", relpos=(0.0,0.2),
connectionstyle="angle3,angleA=10,angleB=90") ) connectionstyle="angle3,angleA=10,angleB=90") )
ax.hist(errors, bins, color=colors['orange']) ax.hist(errors, bins, **fsC)

6
regression/lecture/cubicfunc.py
View File

@ -16,11 +16,11 @@ if __name__ == "__main__":
fig, ax = plt.subplots(figsize=cm_size(figure_width, 1.4*figure_height)) fig, ax = plt.subplots(figsize=cm_size(figure_width, 1.4*figure_height))
fig.subplots_adjust(**adjust_fs(left=6.0, right=1.2)) fig.subplots_adjust(**adjust_fs(left=6.0, right=1.2))
ax.scatter(x, y, marker='o', color=colors['blue'], s=40, zorder=10) ax.plot(x, y, zorder=10, **psA)
xx = np.linspace(2.1, 3.9, 100) xx = np.linspace(2.1, 3.9, 100)
ax.plot(xx, c*xx**3.0, color=colors['red'], lw=3, zorder=5) ax.plot(xx, c*xx**3.0, zorder=5, **lsB)
for cc in [0.25*c, 0.5*c, 2.0*c, 4.0*c]: for cc in [0.25*c, 0.5*c, 2.0*c, 4.0*c]:
ax.plot(xx, cc*xx**3.0, color=colors['orange'], lw=2, zorder=5) ax.plot(xx, cc*xx**3.0, zorder=5, **lsDm)
ax.set_xlabel('Size x', 'm') 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_xlim(2, 4)

12
regression/lecture/cubicmse.py
View File

@ -39,9 +39,9 @@ def plot_mse(ax, x, y, c, cs):
for i, cc in enumerate(ccs): for i, cc in enumerate(ccs):
mses[i] = np.mean((y-(cc*x**3.0))**2.0) mses[i] = np.mean((y-(cc*x**3.0))**2.0)
ax.plot(ccs, mses, colors['blue'], lw=2, zorder=10) ax.plot(ccs, mses, zorder=10, **lsAm)
ax.scatter(cs, ms, color=colors['red'], s=40, zorder=20) ax.plot(cs[:12], ms[:12], zorder=20, **psB)
ax.scatter(cs[-1], ms[-1], color=colors['orange'], s=60, zorder=30) ax.plot(cs[-1], ms[-1], zorder=30, **psC)
for i in range(4): for i in range(4):
ax.annotate('', ax.annotate('',
xy=(cs[i+1]+0.2, ms[i+1]), xycoords='data', xy=(cs[i+1]+0.2, ms[i+1]), xycoords='data',
@ -56,12 +56,12 @@ def plot_mse(ax, x, y, c, cs):
ax.set_yticks(np.arange(0, 30001, 10000)) ax.set_yticks(np.arange(0, 30001, 10000))
def plot_descent(ax, cs, mses): def plot_descent(ax, cs, mses):
ax.plot(np.arange(len(mses))+1, mses, '-o', c=colors['red'], mew=0, ms=8) ax.plot(np.arange(len(mses))+1, mses, **lpsBm)
ax.set_xlabel('Iteration') ax.set_xlabel('Iteration')
#ax.set_ylabel('Mean squared error') #ax.set_ylabel('Mean squared error')
ax.set_xlim(0, 10.5) ax.set_xlim(0, 12.5)
ax.set_ylim(0, 25000) ax.set_ylim(0, 25000)
ax.set_xticks(np.arange(0.0, 10.1, 2.0)) ax.set_xticks(np.arange(0.0, 12.1, 2.0))
ax.set_yticks(np.arange(0, 30001, 10000)) ax.set_yticks(np.arange(0, 30001, 10000))
ax.set_yticklabels([]) ax.set_yticklabels([])

23
regression/lecture/derivative.py
View File

@ -4,8 +4,7 @@ from plotstyle import *
plain_style() plain_style()
fig = plt.figure( figsize=(2.5,3.4) ) fig, ax = plt.subplots(figsize=(2.5,3.4))
ax = fig.add_subplot(1, 1, 1)
# parabula: # parabula:
x1 = -0.2 x1 = -0.2
@ -14,7 +13,7 @@ x = np.linspace(x1, x2, 200)
y = x*x y = x*x
ax.set_xlim(x1, x2) ax.set_xlim(x1, x2)
ax.set_ylim(-0.2, 0.7) ax.set_ylim(-0.2, 0.7)
ax.plot(x, y, c=colors['blue'], lw=4, zorder=0) ax.plot(x, y, zorder=0, **lsA)
# secant: # secant:
x = np.asarray([0.1, 0.7]) x = np.asarray([0.1, 0.7])
y = x*x y = x*x
@ -22,33 +21,33 @@ ax.set_xticks(x)
ax.set_yticks(y) ax.set_yticks(y)
ax.set_xticklabels(['$x$','$x+\Delta x$']) ax.set_xticklabels(['$x$','$x+\Delta x$'])
ax.set_yticklabels(['','']) ax.set_yticklabels(['',''])
ax.scatter(x, y, c=colors['red'], edgecolor='none', s=150, zorder=10) ax.plot(x, y, zorder=10, **psB)
# function values: # function values:
ax.plot([x[0], x[0], x1],[-0.2, y[0], y[0]], '--k', lw=1, zorder=6) ax.plot([x[0], x[0], x1],[-0.2, y[0], y[0]], zorder=6, **lsGrid)
ax.plot([x[1], x[1], x1],[-0.2, y[1], y[1]], '--k', lw=1, zorder=6) ax.plot([x[1], x[1], x1],[-0.2, y[1], y[1]], zorder=6, **lsGrid)
ax.text(x1+0.05, y[0]+0.05, '$f(x)$', zorder=6) ax.text(x1+0.05, y[0]+0.05, '$f(x)$', zorder=6)
ax.text(x1+0.05, y[1]+0.05, '$f(x+\Delta x)$', zorder=6) ax.text(x1+0.05, y[1]+0.05, '$f(x+\Delta x)$', zorder=6)
# slope triangle: # slope triangle:
ax.plot([x[0], x[1], x[1]],[y[0], y[0], y[1]], '-k', lw=2, zorder=7) ax.plot([x[0], x[1], x[1]],[y[0], y[0], y[1]], zorder=7, **lsMarker)
ax.text(np.mean(x), y[0]-0.08, '$\Delta x$', ha='center', zorder=7) ax.text(np.mean(x), y[0]-0.07, '$\Delta x$', ha='center', zorder=7)
ax.text(x[1]+0.05, np.mean(y), '$f(x+\Delta x)-f(x)$', va='center', rotation='vertical', zorder=7) ax.text(x[1]+0.05, np.mean(y), '$f(x+\Delta x)-f(x)$', va='center', rotation='vertical', zorder=7)
# secant line: # secant line:
m = np.diff(y)/np.diff(x) m = np.diff(y)/np.diff(x)
xl = [x1, x2] xl = [x1, x2]
yl = m*(xl-x[0])+y[0] yl = m*(xl-x[0])+y[0]
ax.plot(xl, yl, c=colors['red'], lw=3, zorder=7) ax.plot(xl, yl, zorder=7, **lsBm)
# derivative: # derivative:
md = 2.0*x[0] md = 2.0*x[0]
yl = md*(xl-x[0])+y[0] yl = md*(xl-x[0])+y[0]
ax.plot(xl, yl, c=colors['yellow'], lw=3, zorder=5) ax.plot(xl, yl, zorder=5, **lsDm)
# limit: # limit:
for ml in np.linspace(md, m, 5)[1:] : for ml in np.linspace(md, m, 5)[1:] :
yl = ml*(xl-x[0])+y[0] yl = ml*(xl-x[0])+y[0]
xs = 0.5*(ml+np.sqrt(ml*ml-4.0*(ml*x[0]-y[0]))) xs = 0.5*(ml+np.sqrt(ml*ml-4.0*(ml*x[0]-y[0])))
ax.scatter([xs], [xs*xs], c=colors['orange'], edgecolor='none', s=80, zorder=3) ax.plot([xs], [xs*xs], zorder=3, **psC)
ax.plot(xl, yl, c=colors['orange'], lw=2, zorder=3) ax.plot(xl, yl, zorder=3, **lsCm)
fig.subplots_adjust(**adjust_fs(fig, 0.5, 0.5, 1.4, 0.5)) fig.subplots_adjust(**adjust_fs(fig, 0.5, 0.5, 1.4, 0.5))
plt.savefig('derivative.pdf') plt.savefig('derivative.pdf')

10
regression/lecture/lin_regress.py
View File

@ -14,7 +14,7 @@ def create_data():
def plot_data(ax, x, y): def plot_data(ax, x, y):
ax.scatter(x, y, marker='o', color=colors['blue'], s=40) ax.plot(x, y, **psA)
ax.set_xlabel('Input x') ax.set_xlabel('Input x')
ax.set_ylabel('Output y') ax.set_ylabel('Output y')
ax.set_xlim(0, 120) ax.set_xlim(0, 120)
@ -24,10 +24,10 @@ def plot_data(ax, x, y):
def plot_data_slopes(ax, x, y, m, n): def plot_data_slopes(ax, x, y, m, n):
ax.scatter(x, y, marker='o', color=colors['blue'], s=40) ax.plot(x, y, **psA)
xx = np.asarray([2, 118]) xx = np.asarray([2, 118])
for i in np.linspace(0.3*m, 2.0*m, 5): for i in np.linspace(0.3*m, 2.0*m, 5):
ax.plot(xx, i*xx+n, color=colors['red'], lw=2) ax.plot(xx, i*xx+n, **lsBm)
ax.set_xlabel('Input x') ax.set_xlabel('Input x')
#ax.set_ylabel('Output y') #ax.set_ylabel('Output y')
ax.set_xlim(0, 120) ax.set_xlim(0, 120)
@ -37,10 +37,10 @@ def plot_data_slopes(ax, x, y, m, n):
def plot_data_intercepts(ax, x, y, m, n): def plot_data_intercepts(ax, x, y, m, n):
ax.scatter(x, y, marker='o', color=colors['blue'], s=40) ax.plot(x, y, **psA)
xx = np.asarray([2, 118]) xx = np.asarray([2, 118])
for i in np.linspace(n-1*n, n+1*n, 5): for i in np.linspace(n-1*n, n+1*n, 5):
ax.plot(xx, m*xx + i, color=colors['red'], lw=2) ax.plot(xx, m*xx + i, **lsBm)
ax.set_xlabel('Input x') ax.set_xlabel('Input x')
#ax.set_ylabel('Output y') #ax.set_ylabel('Output y')
ax.set_xlim(0, 120) ax.set_xlim(0, 120)

10
regression/lecture/linear_least_squares.py
View File

@ -25,15 +25,15 @@ def plot_data(ax, x, y, m, n):
xytext=(80, -50), textcoords='data', ha='left', xytext=(80, -50), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.9,1.0), arrowprops=dict(arrowstyle="->", relpos=(0.9,1.0),
connectionstyle="angle3,angleA=50,angleB=-30") ) connectionstyle="angle3,angleA=50,angleB=-30") )
ax.scatter(x[:40], y[:40], color=colors['blue'], s=10, zorder=0) ax.plot(x[:40], y[:40], zorder=0, **psAm)
inxs = [3, 13, 16, 19, 25, 34, 36] inxs = [3, 13, 16, 19, 25, 34, 36]
ax.scatter(x[inxs], y[inxs], color=colors['blue'], s=40, zorder=10) ax.plot(x[inxs], y[inxs], zorder=10, **psA)
xx = np.asarray([2, 118]) xx = np.asarray([2, 118])
ax.plot(xx, m*xx+n, color=colors['red'], lw=2) ax.plot(xx, m*xx+n, **lsBm)
for i in inxs : for i in inxs :
xx = [x[i], x[i]] xx = [x[i], x[i]]
yy = [m*x[i]+n, y[i]] yy = [m*x[i]+n, y[i]]
ax.plot(xx, yy, color=colors['orange'], lw=2, zorder=5) ax.plot(xx, yy, zorder=5, **lsDm)
def plot_error_hist(ax, x, y, m, n): def plot_error_hist(ax, x, y, m, n):
@ -51,7 +51,7 @@ def plot_error_hist(ax, x, y, m, n):
xytext=(350, 20), textcoords='data', ha='left', xytext=(350, 20), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.2), arrowprops=dict(arrowstyle="->", relpos=(0.0,0.2),
connectionstyle="angle3,angleA=10,angleB=90") ) connectionstyle="angle3,angleA=10,angleB=90") )
ax.hist(errors, bins, color=colors['orange']) ax.hist(errors, bins, **fsD)

6
simulations/lecture/normaldata.py
View File

@ -19,7 +19,7 @@ if __name__ == "__main__":
spec = gridspec.GridSpec(nrows=1, ncols=2, width_ratios=[3, 1], wspace=0.08, spec = gridspec.GridSpec(nrows=1, ncols=2, width_ratios=[3, 1], wspace=0.08,
**adjust_fs(fig, left=6.0)) **adjust_fs(fig, left=6.0))
ax1 = fig.add_subplot(spec[0, 0]) ax1 = fig.add_subplot(spec[0, 0])
ax1.scatter(indices, data, c=colors['blue'], edgecolor='white', s=50) ax1.plot(indices, data, **psAm)
ax1.set_xlabel('Index') ax1.set_xlabel('Index')
ax1.set_ylabel('Weight', 'kg') ax1.set_ylabel('Weight', 'kg')
ax1.set_xlim(-10, 310) ax1.set_xlim(-10, 310)
@ -29,10 +29,10 @@ if __name__ == "__main__":
ax2 = fig.add_subplot(spec[0, 1]) ax2 = fig.add_subplot(spec[0, 1])
xx = np.arange(0.0, 350.0, 0.5) xx = np.arange(0.0, 350.0, 0.5)
yy = st.norm.pdf(xx, mu, sigma) yy = st.norm.pdf(xx, mu, sigma)
ax2.plot(yy, xx, color=colors['red'], lw=2) ax2.plot(yy, xx, **lsBm)
bw = 20.0 bw = 20.0
h, b = np.histogram(data, np.arange(0, 401, bw)) h, b = np.histogram(data, np.arange(0, 401, bw))
ax2.barh(b[:-1], h/np.sum(h)/(b[1]-b[0]), fc=colors['yellow'], height=bar_fac*bw, align='edge') ax2.barh(b[:-1], h/np.sum(h)/(b[1]-b[0]), height=bar_fac*bw, align='edge', **fsC)
ax2.set_xlabel('Pdf', '1/kg') ax2.set_xlabel('Pdf', '1/kg')
ax2.set_xlim(0, 0.012) ax2.set_xlim(0, 0.012)
ax2.set_xticks([0, 0.005, 0.01]) ax2.set_xticks([0, 0.005, 0.01])

12
simulations/lecture/randomnumbers.py
View File

@ -24,32 +24,28 @@ if __name__ == "__main__":
spec = gridspec.GridSpec(nrows=2, ncols=2, **adjust_fs(fig)) spec = gridspec.GridSpec(nrows=2, ncols=2, **adjust_fs(fig))
ax = fig.add_subplot(spec[0, 0]) ax = fig.add_subplot(spec[0, 0])
ax.plot(indices, x1, c=colors['blue'], lw=1, zorder=10) ax.plot(indices, x1, zorder=10, **lpsAm)
ax.scatter(indices, x1, c=colors['blue'], edgecolor='white', s=50, zorder=20)
ax.set_xlabel('Index') ax.set_xlabel('Index')
ax.set_ylabel('Random number') ax.set_ylabel('Random number')
ax.set_xlim(-1.0, n+1.0) ax.set_xlim(-1.0, n+1.0)
ax.set_ylim(-0.05, 1.05) ax.set_ylim(-0.05, 1.05)
ax = fig.add_subplot(spec[0, 1]) ax = fig.add_subplot(spec[0, 1])
ax.plot(indices, x2, c=colors['blue'], lw=1, zorder=10) ax.plot(indices, x2, zorder=10, **lpsAm)
ax.scatter(indices, x2, c=colors['blue'], edgecolor='white', s=50, zorder=20)
ax.set_xlabel('Index') ax.set_xlabel('Index')
ax.set_ylabel('Random number') ax.set_ylabel('Random number')
ax.set_xlim(-1.0, n+1.0) ax.set_xlim(-1.0, n+1.0)
ax.set_ylim(-0.05, 1.05) ax.set_ylim(-0.05, 1.05)
ax = fig.add_subplot(spec[1, 1]) ax = fig.add_subplot(spec[1, 1])
ax.plot(indices, x3, c=colors['blue'], lw=1, zorder=10) ax.plot(indices, x3, zorder=10, **lpsAm)
ax.scatter(indices, x3, c=colors['blue'], edgecolor='white', s=50, zorder=20)
ax.set_xlabel('Index') ax.set_xlabel('Index')
ax.set_ylabel('Random number') ax.set_ylabel('Random number')
ax.set_xlim(-1.0, n+1.0) ax.set_xlim(-1.0, n+1.0)
ax.set_ylim(-0.05, 1.05) ax.set_ylim(-0.05, 1.05)
ax = fig.add_subplot(spec[1, 0]) ax = fig.add_subplot(spec[1, 0])
ax.plot(lags, corrs, c=colors['red'], lw=1, zorder=10) ax.plot(lags, corrs, zorder=10, **lpsBm)
ax.scatter(lags, corrs, c=colors['red'], edgecolor='white', s=50, zorder=20)
ax.set_xlabel('Lag') ax.set_xlabel('Lag')
ax.set_ylabel('Serial correlation') ax.set_ylabel('Serial correlation')
ax.set_xlim(-maxl-0.5, maxl+0.5) ax.set_xlim(-maxl-0.5, maxl+0.5)

8
simulations/lecture/staticnonlinearity.py
View File

@ -23,8 +23,8 @@ if __name__ == "__main__":
fig = plt.figure() fig = plt.figure()
spec = gridspec.GridSpec(nrows=1, ncols=2, **adjust_fs(fig, left=4.5)) spec = gridspec.GridSpec(nrows=1, ncols=2, **adjust_fs(fig, left=4.5))
ax1 = fig.add_subplot(spec[0, 0]) ax1 = fig.add_subplot(spec[0, 0])
ax1.plot(xx, yy, **ls['A1']) ax1.plot(x, y, **psAm)
ax1.scatter(x, y, c=colors['blue'], edgecolor='white', s=50) ax1.plot(xx, yy, **lsB)
ax1.set_xlabel('Hair deflection', 'nm') ax1.set_xlabel('Hair deflection', 'nm')
ax1.set_ylabel('Conductance', 'nS') ax1.set_ylabel('Conductance', 'nS')
ax1.set_xlim(-20, 20) ax1.set_xlim(-20, 20)
@ -35,10 +35,10 @@ if __name__ == "__main__":
ax2 = fig.add_subplot(spec[0, 1]) ax2 = fig.add_subplot(spec[0, 1])
xg = np.linspace(-3.0, 3.01, 200) xg = np.linspace(-3.0, 3.01, 200)
yg = st.norm.pdf(xg, 0.0, sigma) yg = st.norm.pdf(xg, 0.0, sigma)
ax2.plot(xg, yg, **ls['A1']) ax2.plot(xg, yg, **lsB)
bw = 0.25 bw = 0.25
h, b = np.histogram(y-boltzmann(x, x0, k), np.arange(-3.0, 3.01, bw)) h, b = np.histogram(y-boltzmann(x, x0, k), np.arange(-3.0, 3.01, bw))
ax2.bar(b[:-1], h/np.sum(h)/(b[1]-b[0]), fc=colors['yellow'], width=bar_fac*bw, align='edge') ax2.bar(b[:-1], h/np.sum(h)/(b[1]-b[0]), width=bar_fac*bw, align='edge', **fsC)
ax2.set_xlabel('Residuals', 'nS') ax2.set_xlabel('Residuals', 'nS')
ax2.set_ylabel('Pdf', '1/nS') ax2.set_ylabel('Pdf', '1/nS')
ax2.set_xlim(-2.5, 2.5) ax2.set_xlim(-2.5, 2.5)

11
statistics/lecture/boxwhisker.py
View File

@ -1,16 +1,11 @@
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
from plotstyle import *
rng = np.random.RandomState(981) rng = np.random.RandomState(981)
x = rng.randn( 40, 10 ) x = rng.randn( 40, 10 )
plt.xkcd() fig, ax = plt.subplots()
fig = plt.figure( figsize=(6,3.4) )
ax = fig.add_subplot( 1, 1, 1 )
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.set_xlabel('Experiment') ax.set_xlabel('Experiment')
ax.set_ylabel('x') ax.set_ylabel('x')
ax.set_ylim(-4.0, 4.0) ax.set_ylim(-4.0, 4.0)
@ -40,7 +35,5 @@ ax.annotate('maximum',
arrowprops=dict(arrowstyle="->", relpos=(1.0,0.5), arrowprops=dict(arrowstyle="->", relpos=(1.0,0.5),
connectionstyle="angle3,angleA=0,angleB=120") ) connectionstyle="angle3,angleA=0,angleB=120") )
ax.boxplot(x, whis=100.0) ax.boxplot(x, whis=100.0)
plt.tight_layout()
plt.savefig('boxwhisker.pdf') plt.savefig('boxwhisker.pdf')
#plt.show()

25
statistics/lecture/correlation.py
View File

@ -1,33 +1,30 @@
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec
from plotstyle import *
plt.xkcd() fig = plt.figure(figsize=cm_size(figure_width, 2.0*figure_height))
fig = plt.figure( figsize=(6,4.6) ) spec = gridspec.GridSpec(nrows=2, ncols=2, wspace=0.35, hspace=0.5,
**adjust_fs(fig, left=5.5, top=0.5, bottom=2.7))
rng = np.random.RandomState(2981) rng = np.random.RandomState(2981)
n = 200 n = 200
for k, r in enumerate([ 1.0, 0.6, 0.0, -0.9 ]) : for k, r in enumerate([ 1.0, 0.6, 0.0, -0.9 ]) :
x = rng.randn(n) x = rng.randn(n)
y = r*x + np.sqrt(1.0-r*r)*rng.randn(n) y = r*x + np.sqrt(1.0-r*r)*rng.randn(n)
ax = fig.add_subplot( 2, 2, k+1 ) ax = fig.add_subplot(spec[k//2, k%2])
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.text(-2, 2.5, 'r=%.1f' % r) ax.text(-2, 2.5, 'r=%.1f' % r)
if k == 0 : if k == 0 :
ax.text( 2.8, -2, 'positively\ncorrelated', ha='right' ) ax.text(2.8, -2.8, 'positively\ncorrelated', ha='right', va='bottom')
elif k == 1 : elif k == 1 :
ax.text( 2.8, -2.5, 'weakly\ncorrelated', ha='right' ) ax.text(2.8, -2.8, 'weakly\ncorrelated', ha='right', va='bottom')
elif k == 2 : elif k == 2 :
ax.text( 2.8, -2.5, 'not\ncorrelated', ha='right' ) ax.text(2.8, -2.8, 'not\ncorrelated', ha='right', va='bottom')
elif k == 3 : elif k == 3 :
ax.text( -2.5, -2, 'negatively\ncorrelated', ha='left' ) ax.text(-2.8, -2.8, 'negatively\ncorrelated', ha='left', va='bottom')
ax.set_xlabel('x') ax.set_xlabel('x')
ax.set_ylabel('y') ax.set_ylabel('y')
ax.set_xlim(-3.0, 3.0) ax.set_xlim(-3.0, 3.0)
ax.set_ylim(-3.0, 3.0) ax.set_ylim(-3.0, 3.0)
ax.scatter( x[(np.abs(x)<2.8)&(np.abs(y)<2.8)], y[(np.abs(x)<2.8)&(np.abs(y)<2.8)] ) ax.plot(x[(np.abs(x)<2.8)&(np.abs(y)<2.8)], y[(np.abs(x)<2.8)&(np.abs(y)<2.8)], **psAm)
plt.tight_layout()
plt.savefig('correlation.pdf') plt.savefig('correlation.pdf')
#plt.show()

23
statistics/lecture/cumulative.py
View File

@ -1,5 +1,6 @@
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
from plotstyle import *
# data: # data:
rng = np.random.RandomState(981) rng = np.random.RandomState(981)
@ -14,13 +15,7 @@ gauss = np.exp(-0.5*xx*xx)/np.sqrt(2.0*np.pi)
gausscdf = np.cumsum(gauss)*dx gausscdf = np.cumsum(gauss)*dx
# plot: # plot:
plt.xkcd() fig, ax = plt.subplots(figsize=cm_size(figure_width, 1.2*figure_height))
fig = plt.figure( figsize=(6, 2.4) )
ax = fig.add_subplot(1, 1, 1)
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.set_xlabel('x') ax.set_xlabel('x')
ax.set_xlim(-3.2, 3.2) ax.set_xlim(-3.2, 3.2)
ax.set_xticks(np.arange(-3.0, 3.1, 1.0)) ax.set_xticks(np.arange(-3.0, 3.1, 1.0))
@ -29,24 +24,22 @@ ax.set_ylim(-0.05, 1.05)
ax.set_yticks(np.arange(0.0, 1.1, 0.2)) ax.set_yticks(np.arange(0.0, 1.1, 0.2))
med = xs[cdf>=0.5][0] med = xs[cdf>=0.5][0]
ax.plot([-3.2, med, med], [0.5, 0.5, 0.0], 'k', lw=1, zorder=-5) ax.plot([-3.2, med, med], [0.5, 0.5, 0.0], zorder=-5, **lsSpine)
ax.text(-2.8, 0.55, 'F=0.5') ax.text(-2.8, 0.55, 'F=0.5')
ax.text(0.15, 0.25, 'median at %.2f' % med) ax.text(0.15, 0.25, 'median at %.2f' % med)
q3 = xs[cdf>=0.75][0] q3 = xs[cdf>=0.75][0]
ax.plot([-3.2, q3, q3], [0.75, 0.75, 0.0], 'k', lw=1, zorder=-5) ax.plot([-3.2, q3, q3], [0.75, 0.75, 0.0], zorder=-5, **lsSpine)
ax.text(-2.8, 0.8, 'F=0.75') ax.text(-2.8, 0.8, 'F=0.75')
ax.text(0.8, 0.5, '3. quartile at %.2f' % q3) ax.text(0.8, 0.5, '3. quartile at %.2f' % q3)
p = cdf[xs>=-1.0][0] p = cdf[xs>=-1.0][0]
ax.plot([-3.2, -1.0, -1.0], [p, p, 0.0], 'k', lw=1, zorder=-5) ax.plot([-3.2, -1.0, -1.0], [p, p, 0.0], zorder=-5, **lsSpine)
ax.text(-2.8, 0.2, 'F=%.2f' % p) ax.text(-2.8, 0.2, 'F=%.2f' % p)
ax.text(-0.9, 0.05, '-1') ax.text(-0.9, 0.05, '-1')
ax.plot(xx, gausscdf, '-', color='#0000ff', lw=2, zorder=-1) ax.plot(xx, gausscdf, zorder=-1, **lsAm)
ax.plot(xs, cdf, '-', color='#cc0000', lw=4, zorder=-1) ax.plot(xs, cdf, zorder=-1, **lsB)
ax.plot([-3.2, 3.2], [1.0, 1.0], '--', color='k', lw=2, zorder=-10) ax.plot([-3.2, 3.2], [1.0, 1.0], zorder=-10, **lsGrid)
plt.subplots_adjust(left=0.1, right=0.98, bottom=0.15, top=0.98, wspace=0.35, hspace=0.3)
fig.savefig('cumulative.pdf') fig.savefig('cumulative.pdf')
#plt.show()

47
statistics/lecture/diehistograms.py
View File

@ -1,5 +1,6 @@
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
from plotstyle import *
# roll the die: # roll the die:
rng = np.random.RandomState(57281) rng = np.random.RandomState(57281)
@ -7,33 +8,23 @@ x1 = rng.random_integers( 1, 6, 100 )
x2 = rng.random_integers(1, 6, 500) x2 = rng.random_integers(1, 6, 500)
bins = np.arange(0.5, 7, 1.0) bins = np.arange(0.5, 7, 1.0)
plt.xkcd() fig, (ax1, ax2) = plt.subplots(1, 2)
fig.subplots_adjust(**adjust_fs(bottom=2.7, top=0.1))
ax1.set_xlim(0, 7)
ax1.set_xticks(range(1, 7))
ax1.set_xlabel('x')
ax1.set_ylim(0, 98)
ax1.set_ylabel('Frequency')
fs = fsC
fs['color'] = [fsC['facecolor'], fsE['facecolor']]
del fs['facecolor']
ax1.hist([x2, x1], bins, **fs)
fig = plt.figure( figsize=(6,3) ) ax2.set_xlim(0, 7)
ax = fig.add_subplot( 1, 2, 1 ) ax2.set_xticks(range(1, 7))
ax.spines['right'].set_visible(False) ax2.set_xlabel('x')
ax.spines['top'].set_visible(False) ax2.set_ylim(0, 0.23)
ax.yaxis.set_ticks_position('left') ax2.set_ylabel('Probability')
ax.xaxis.set_ticks_position('bottom') ax2.plot([0.2, 6.8], [1.0/6.0, 1.0/6.0], zorder=-10, **lsAm)
ax.set_xlim(0, 7) ax2.hist([x2, x1], bins, normed=True, zorder=-5, **fs)
ax.set_xticks( range(1, 7) )
ax.set_xlabel( 'x' )
ax.set_ylim(0, 98)
ax.set_ylabel( 'Frequency' )
ax.hist([x2, x1], bins, color=['#FFCC00', '#FFFF66' ])
ax = fig.add_subplot( 1, 2, 2 )
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.set_xlim(0, 7)
ax.set_xticks( range(1, 7) )
ax.set_xlabel( 'x' )
ax.set_ylim(0, 0.23)
ax.set_ylabel( 'Probability' )
ax.plot([0.2, 6.8], [1.0/6.0, 1.0/6.0], '-b', lw=2, zorder=1)
ax.hist([x2, x1], bins, normed=True, color=['#FFCC00', '#FFFF66' ], zorder=10)
plt.subplots_adjust(left=0.1, right=0.98, bottom=0.15, top=0.98, wspace=0.4, hspace=0.0)
fig.savefig('diehistograms.pdf') fig.savefig('diehistograms.pdf')
#plt.show()

58
statistics/lecture/displayunivariatedata.py
View File

@ -1,33 +1,24 @@
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec
from scipy.stats import gaussian_kde from scipy.stats import gaussian_kde
from plotstyle import *
#rng = np.random.RandomState(981) #rng = np.random.RandomState(981)
#data = rng.randn(40, 1) + 4.0 #data = rng.randn(40, 1) + 4.0
rng = np.random.RandomState(1981) rng = np.random.RandomState(1981)
data = rng.gamma(1.0, 1.5, 40) + 1.0 data = rng.gamma(1.0, 1.5, 40) + 1.0
data = data[data<7.5] data = data[data<7.5]
xpos = 0.08
ypos = 0.15
width = 0.65
height = 0.8
barwidth = 0.8 barwidth = 0.8
scatterpos = 1.0 scatterpos = 1.0
barpos = 2.5 barpos = 2.5
boxpos = 4.0 boxpos = 4.0
plt.xkcd() fig = plt.figure(figsize=cm_size(figure_width, 1.1*figure_height))
fig = plt.figure( figsize=(6,3.4) ) spec = gridspec.GridSpec(nrows=1, ncols=2, width_ratios=[3, 1], wspace=0.1,
**adjust_fs(fig, left=4.0))
ax = fig.add_axes([xpos, ypos, width, height]) ax = fig.add_subplot(spec[0, 0])
ax.spines['right'].set_visible(False)
ax.spines['top'].set_visible(False)
#ax.spines['left'].set_visible(False)
#ax.spines['bottom'].set_visible(False)
#ax.xaxis.set_ticks_position('none')
#ax.yaxis.set_ticks_position('none')
#ax.set_xticklabels([])
#ax.set_yticklabels([])
wh = ax.boxplot( data, positions=[boxpos], widths=[barwidth], whis=100.0, patch_artist=True) wh = ax.boxplot( data, positions=[boxpos], widths=[barwidth], whis=100.0, patch_artist=True)
wh['medians'][0].set_linewidth(4) wh['medians'][0].set_linewidth(4)
wh['whiskers'][0].set_linewidth(2) wh['whiskers'][0].set_linewidth(2)
@ -49,7 +40,7 @@ ax.annotate('maximum',
connectionstyle="angle3,angleA=0,angleB=120") ) connectionstyle="angle3,angleA=0,angleB=120") )
ax.annotate('3. quartile', ax.annotate('3. quartile',
xy=(boxpos-0.3*barwidth, 3.7), xycoords='data', xy=(boxpos-0.3*barwidth, 3.7), xycoords='data',
xytext=(boxpos-1.3*barwidth, 5.5), textcoords='data', ha='left', xytext=(boxpos-0.1*barwidth, 5.5), textcoords='data', ha='right',
arrowprops=dict(arrowstyle="->", relpos=(0.4,0.0), arrowprops=dict(arrowstyle="->", relpos=(0.4,0.0),
connectionstyle="angle3,angleA=0,angleB=120") ) connectionstyle="angle3,angleA=0,angleB=120") )
ax.annotate('median', ax.annotate('median',
@ -57,34 +48,28 @@ ax.annotate('median',
xytext=(boxpos+0.1*barwidth, 4.2), textcoords='data', ha='left', xytext=(boxpos+0.1*barwidth, 4.2), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.8,0.0), arrowprops=dict(arrowstyle="->", relpos=(0.8,0.0),
connectionstyle="angle3,angleA=-60,angleB=20") ) connectionstyle="angle3,angleA=-60,angleB=20") )
ax = fig.add_axes([xpos, ypos, width, height])
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.set_xticklabels([]) ax.set_xticklabels([])
ax = fig.add_subplot(spec[0, 0])
ax.set_xlim(0.0, 4.8) ax.set_xlim(0.0, 4.8)
ax.set_xticks([scatterpos, barpos, boxpos])
ax.set_xticklabels(['(1) data', '(2) bar\n plot', '(3) box-\nwhisker'], fontsize='medium')
ax.set_ylabel('x') ax.set_ylabel('x')
ax.set_ylim( 0.0, 8.0) ax.set_ylim( 0.0, 8.0)
ax.set_xticks([scatterpos, barpos, boxpos])
ax.set_xticklabels(['(1) data', '(2) bar\n plot', '(3) box-\nwhisker'])
# scatter data points according to their density: # scatter data points according to their density:
kernel = gaussian_kde(data) kernel = gaussian_kde(data)
x = kernel(data) x = kernel(data)
x /= np.max(x) x /= np.max(x)
ax.scatter(scatterpos+barwidth*x*(rng.rand(len(data))-0.5), data, s=50) ax.plot(scatterpos+barwidth*x*(rng.rand(len(data))-0.5), data, **psA)
barmean = np.mean(data) barmean = np.mean(data)
barstd = np.std(data) barstd = np.std(data)
ew = 0.2 ew = 0.2
ax.bar([barpos-0.5*barwidth], [barmean], barwidth, color='#FFCC00') ax.bar([barpos-0.5*barwidth], [barmean], barwidth, **fsC)
eargs = {'color': 'k', 'lw': 2} ax.plot([barpos, barpos], [barmean-barstd, barmean+barstd], **lsMarker)
ax.plot([barpos, barpos], [barmean-barstd, barmean+barstd], **eargs) ax.plot([barpos-0.5*ew, barpos+0.5*ew], [barmean-barstd, barmean-barstd], **lsMarker)
ax.plot([barpos-0.5*ew, barpos+0.5*ew], [barmean-barstd, barmean-barstd], **eargs) ax.plot([barpos-0.5*ew, barpos+0.5*ew], [barmean+barstd, barmean+barstd], **lsMarker)
ax.plot([barpos-0.5*ew, barpos+0.5*ew], [barmean+barstd, barmean+barstd], **eargs)
ax.annotate('mean', ax.annotate('mean',
xy=(barpos-0.4*barwidth, 2.7), xycoords='data', xy=(barpos-0.4*barwidth, 2.7), xycoords='data',
xytext=(barpos-1*barwidth, 5.5), textcoords='data', ha='left', xytext=(barpos-1*barwidth, 5.5), textcoords='data', ha='left',
@ -96,20 +81,15 @@ ax.annotate('mean plus\nstd. dev.',
arrowprops=dict(arrowstyle="->", relpos=(0.5,0.0), arrowprops=dict(arrowstyle="->", relpos=(0.5,0.0),
connectionstyle="angle3,angleA=-60,angleB=80") ) connectionstyle="angle3,angleA=-60,angleB=80") )
ax = fig.add_axes([xpos+width+0.03, ypos, 0.98-(xpos+width+0.03), height]) ax = fig.add_subplot(spec[0, 1])
ax.spines['right'].set_visible(False)
ax.spines['top'].set_visible(False)
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
ax.set_yticklabels([]) ax.set_yticklabels([])
ax.set_ylim( 0.0, 8.0) ax.set_ylim( 0.0, 8.0)
ax.set_xticks(np.arange(0.0, 0.4, 0.1)) ax.set_xticks(np.arange(0.0, 0.4, 0.1))
ax.set_xlabel('(4) p(x)') ax.set_xlabel('(4) pdf')
bw = 0.75 bw = 0.75
bins = np.arange(0, 8.0+bw, bw) bins = np.arange(0, 8.0+bw, bw)
h, b = np.histogram(data, bins) h, b = np.histogram(data, bins)
ax.barh(b[:-1], h/bw/np.sum(h), bw, color='#CC0000') ax.barh(b[:-1], h/bw/np.sum(h), bw, **fsB)
plt.savefig('displayunivariatedata.pdf') plt.savefig('displayunivariatedata.pdf')
#plt.show()

38
statistics/lecture/kerneldensity.py
View File

@ -1,5 +1,7 @@
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec
from plotstyle import *
# normal distribution: # normal distribution:
rng = np.random.RandomState(6281) rng = np.random.RandomState(6281)
@ -30,15 +32,11 @@ def kerneldensity(data, xmin, xmax, sigma=1.0) :
kd /= len(data) kd /= len(data)
return kd, x return kd, x
fig = plt.figure()
spec = gridspec.GridSpec(nrows=2, ncols=2, wspace=0.35, hspace=0.3,
**adjust_fs(fig, left=5.5, top=0.2, bottom=2.7))
plt.xkcd() ax = fig.add_subplot(spec[0, 0])
fig = plt.figure( figsize=(6,3) )
ax = fig.add_subplot(2, 2, 1)
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.set_xlabel('x') ax.set_xlabel('x')
ax.set_xlim(-3.2, 3.2) ax.set_xlim(-3.2, 3.2)
ax.set_xticks(np.arange(-3.0, 3.1, 1.0)) ax.set_xticks(np.arange(-3.0, 3.1, 1.0))
@ -46,13 +44,9 @@ ax.set_ylabel( 'p(x)' )
ax.set_ylim(0.0, 0.49) ax.set_ylim(0.0, 0.49)
ax.set_yticks(np.arange(0.0, 0.41, 0.1)) ax.set_yticks(np.arange(0.0, 0.41, 0.1))
#ax.plot(x, g, '-b', lw=2, zorder=-1) #ax.plot(x, g, '-b', lw=2, zorder=-1)
ax.hist(r, np.arange(-4.1, 4, 0.4), normed=True, color='#FFCC00', zorder=-5) ax.hist(r, np.arange(-4.1, 4, 0.4), normed=True, zorder=-5, **fsC)
ax = fig.add_subplot(2, 2, 3) ax = fig.add_subplot(spec[1, 0])
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.set_xlabel('x') ax.set_xlabel('x')
ax.set_xlim(-3.2, 3.2) ax.set_xlim(-3.2, 3.2)
ax.set_xticks(np.arange(-3.0, 3.1, 1.0)) ax.set_xticks(np.arange(-3.0, 3.1, 1.0))
@ -60,24 +54,18 @@ ax.set_ylabel( 'p(x)' )
ax.set_ylim(0.0, 0.49) ax.set_ylim(0.0, 0.49)
ax.set_yticks(np.arange(0.0, 0.41, 0.1)) ax.set_yticks(np.arange(0.0, 0.41, 0.1))
#ax.plot(x, g, '-b', lw=2, zorder=-1) #ax.plot(x, g, '-b', lw=2, zorder=-1)
ax.hist(r, np.arange(-4.3, 4, 0.4), normed=True, color='#FFCC00', zorder=-5) ax.hist(r, np.arange(-4.3, 4, 0.4), normed=True, zorder=-5, **fsC)
ax = fig.add_subplot(1, 2, 2) ax = fig.add_subplot(spec[:, 1])
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.set_xlabel('x') ax.set_xlabel('x')
ax.set_xlim(-3.2, 3.2) ax.set_xlim(-3.2, 3.2)
ax.set_xticks(np.arange(-3.0, 3.1, 1.0)) ax.set_xticks(np.arange(-3.0, 3.1, 1.0))
ax.set_ylabel( 'Probab. density p(x)' ) ax.set_ylabel('Prob. density p(x)')
ax.set_ylim(0.0, 0.49) ax.set_ylim(0.0, 0.49)
ax.set_yticks(np.arange(0.0, 0.41, 0.1)) ax.set_yticks(np.arange(0.0, 0.41, 0.1))
kd, xx = kerneldensity(r, -3.2, 3.2, 0.2) kd, xx = kerneldensity(r, -3.2, 3.2, 0.2)
ax.fill_between(xx, 0.0, kd, color='#FF9900', zorder=-5) ax.fill_between(xx, 0.0, kd, zorder=-5, **fsDs)
ax.plot(xx, kd, '-', lw=3, color='#CC0000', zorder=-1) ax.plot(xx, kd, '-', zorder=-1, **lsB)
plt.subplots_adjust(left=0.1, right=0.98, bottom=0.15, top=0.98, wspace=0.35, hspace=0.3)
fig.savefig('kerneldensity.pdf') fig.savefig('kerneldensity.pdf')
#plt.show()

72
statistics/lecture/median.py
View File

@ -1,78 +1,66 @@
import numpy as np import numpy as np
import scipy.stats as st import scipy.stats as st
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
from plotstyle import *
# normal distribution: # normal distribution:
x = np.arange(-3.0, 3.0, 0.01) x = np.arange(-3.0, 3.0, 0.01)
g = np.exp(-0.5*x*x)/np.sqrt(2.0*np.pi) g = np.exp(-0.5*x*x)/np.sqrt(2.0*np.pi)
plt.xkcd() fig, (ax1, ax2) = plt.subplots(1, 2)
fig = plt.figure( figsize=(6, 2.8) ) fig.subplots_adjust(**adjust_fs(bottom=2.7, top=0.1))
ax = fig.add_subplot(1, 2, 1) ax1.set_xlabel('x')
ax.spines['right'].set_visible(False) ax1.set_ylabel('Prob. density p(x)')
ax.spines['top'].set_visible(False) ax1.set_ylim(0.0, 0.46)
ax.yaxis.set_ticks_position('left') ax1.set_yticks(np.arange(0.0, 0.45, 0.1))
ax.xaxis.set_ticks_position('bottom') ax1.text(-1.0, 0.06, '50%', ha='center')
ax.set_xlabel( 'x' ) ax1.text(+1.0, 0.06, '50%', ha='center')
ax.set_ylabel( 'Prob. density p(x)' ) ax1.annotate('Median\n= mean',
ax.set_ylim( 0.0, 0.46 )
ax.set_yticks( np.arange( 0.0, 0.45, 0.1 ) )
ax.text(-1.0, 0.06, '50%', ha='center' )
ax.text(+1.0, 0.06, '50%', ha='center' )
ax.annotate('Median\n= mean',
xy=(0.1, 0.3), xycoords='data', xy=(0.1, 0.3), xycoords='data',
xytext=(1.2, 0.35), textcoords='data', ha='left', xytext=(1.2, 0.35), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.2), arrowprops=dict(arrowstyle="->", relpos=(0.0,0.2),
connectionstyle="angle3,angleA=10,angleB=40")) connectionstyle="angle3,angleA=10,angleB=40"))
ax.annotate('Mode', ax1.annotate('Mode',
xy=(-0.1, 0.4), xycoords='data', xy=(-0.1, 0.4), xycoords='data',
xytext=(-2.5, 0.43), textcoords='data', ha='left', xytext=(-2.5, 0.43), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.2), arrowprops=dict(arrowstyle="->", relpos=(0.0,0.2),
connectionstyle="angle3,angleA=10,angleB=120")) connectionstyle="angle3,angleA=10,angleB=120"))
ax.fill_between( x[x<0], 0.0, g[x<0], color='#ffcc00' ) ax1.fill_between(x[x<0], 0.0, g[x<0], **fsCs)
ax.fill_between( x[x>0], 0.0, g[x>0], color='#99ff00' ) ax1.fill_between(x[x>0], 0.0, g[x>0], **fsFs)
ax.plot(x, g, 'b', lw=4) ax1.plot(x, g, **lsA)
ax.plot([0.0, 0.0], [0.0, 0.45], 'k', lw=2 ) ax1.plot([0.0, 0.0], [0.0, 0.45], **lsMarker)
# normal distribution: # gamma distribution:
x = np.arange(0.0, 6.0, 0.01) x = np.arange(0.0, 6.0, 0.01)
shape = 2.0 shape = 2.0
g = st.gamma.pdf(x, shape) g = st.gamma.pdf(x, shape)
m = st.gamma.median(shape) m = st.gamma.median(shape)
gm = st.gamma.mean(shape) gm = st.gamma.mean(shape)
ax = fig.add_subplot(1, 2, 2) ax2.set_xlabel('x')
ax.spines['right'].set_visible(False) ax2.set_ylabel('Prob. density p(x)')
ax.spines['top'].set_visible(False) ax2.set_ylim(0.0, 0.46)
ax.yaxis.set_ticks_position('left') ax2.set_yticks(np.arange(0.0, 0.45, 0.1))
ax.xaxis.set_ticks_position('bottom') ax2.text(m-0.8, 0.06, '50%', ha='center')
ax.set_xlabel( 'x' ) ax2.text(m+1.2, 0.06, '50%', ha='center')
ax.set_ylabel( 'Prob. density p(x)' ) ax2.annotate('Median',
ax.set_ylim( 0.0, 0.46 )
ax.set_yticks( np.arange( 0.0, 0.45, 0.1 ) )
ax.text(m-0.8, 0.06, '50%', ha='center' )
ax.text(m+1.2, 0.06, '50%', ha='center' )
ax.annotate('Median',
xy=(m+0.1, 0.2), xycoords='data', xy=(m+0.1, 0.2), xycoords='data',
xytext=(m+1.6, 0.25), textcoords='data', ha='left', xytext=(m+1.6, 0.25), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5), arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
connectionstyle="angle3,angleA=30,angleB=70")) connectionstyle="angle3,angleA=30,angleB=70"))
ax.annotate('Mean', ax2.annotate('Mean',
xy=(gm, 0.01), xycoords='data', xy=(gm, 0.01), xycoords='data',
xytext=(gm+1.8, 0.15), textcoords='data', ha='left', xytext=(gm+1.8, 0.15), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5), arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
connectionstyle="angle3,angleA=0,angleB=90")) connectionstyle="angle3,angleA=0,angleB=90"))
ax.annotate('Mode', ax2.annotate('Mode',
xy=(1.0, 0.38), xycoords='data', xy=(1.0, 0.38), xycoords='data',
xytext=(1.8, 0.42), textcoords='data', ha='left', xytext=(1.8, 0.42), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5), arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
connectionstyle="angle3,angleA=0,angleB=70")) connectionstyle="angle3,angleA=0,angleB=70"))
ax.fill_between( x[x<m], 0.0, g[x<m], color='#ffcc00' ) ax2.fill_between(x[x<m], 0.0, g[x<m], **fsCs)
ax.fill_between( x[x>m], 0.0, g[x>m], color='#99ff00' ) ax2.fill_between(x[x>m], 0.0, g[x>m], **fsFs)
ax.plot(x, g, 'b', lw=4) ax2.plot(x, g, **lsA)
ax.plot([m, m], [0.0, 0.38], 'k', lw=2 ) ax2.plot([m, m], [0.0, 0.38], **lsMarker)
#ax.plot([gm, gm], [0.0, 0.42], 'k', lw=2 ) #ax2.plot([gm, gm], [0.0, 0.38], **lsMarker)
#plt.tight_layout()
plt.subplots_adjust(left=0.1, right=0.98, bottom=0.15, top=0.98, wspace=0.4, hspace=0.0)
fig.savefig('median.pdf') fig.savefig('median.pdf')
#plt.show()

53
statistics/lecture/nonlincorrelation.py
View File

@ -1,42 +1,35 @@
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec
from plotstyle import *
fig, (ax1, ax2) = plt.subplots(1, 2)
fig.subplots_adjust(wspace=0.35, hspace=0.5,
**adjust_fs(fig, left=4, top=0.5, bottom=2.7))
plt.xkcd()
fig = plt.figure( figsize=(6,2.2) )
n = 200 n = 200
x = np.random.randn( n ) rng = np.random.RandomState(3981)
y = np.random.randn( n ) x = rng.randn(n)
y = rng.randn(n)
z = x*x+0.2*y z = x*x+0.2*y
r =np.corrcoef(x,z)[0,1] r =np.corrcoef(x,z)[0,1]
ax = fig.add_subplot( 1, 2, 1 ) ax1.text(0, 4.0, 'r=%.1f' % r, ha='center')
ax.spines['right'].set_visible(False) ax1.text(0, 5.6, r'$y = x^2+\xi/5$', ha='center')
ax.spines['top'].set_visible(False) ax1.set_xlabel('x')
ax.yaxis.set_ticks_position('left') ax1.set_ylabel('y')
ax.xaxis.set_ticks_position('bottom') ax1.set_xlim(-3.0, 3.0)
ax.text( 0, 4.0, 'r=%.1f' % r, ha='center' ) ax1.set_ylim(-0.5, 6.0)
ax.text( 0, 6, r'$y = x^2+\xi/5$', ha='center' ) ax1.plot(x, z, **psAm)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_xlim( -3.0, 3.0)
ax.set_ylim( -0.5, 6.0)
ax.scatter( x, z )
z = 0.5*x*y z = 0.5*x*y
r =np.corrcoef(x,z)[0,1] r =np.corrcoef(x,z)[0,1]
ax = fig.add_subplot( 1, 2, 2 ) ax2.text(0, 1.5, 'r=%.1f' % r, ha='center')
ax.spines['right'].set_visible(False) ax2.text(0, 2.8, r'$y = x \cdot \xi/2$', ha='center')
ax.spines['top'].set_visible(False) ax2.set_xlabel('x')
ax.yaxis.set_ticks_position('left') ax2.set_ylabel('y')
ax.xaxis.set_ticks_position('bottom') ax2.set_xlim(-3.0, 3.0)
ax.text( 0, 1.5, 'r=%.1f' % r, ha='center' ) ax2.set_ylim(-3.0, 3.0)
ax.text( 0, 3, r'$y = x \cdot \xi/2$', ha='center' ) ax2.plot(x, z, **psAm)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_xlim( -3.0, 3.0)
ax.set_ylim( -3.0, 3.0)
ax.scatter( x, z )
plt.tight_layout()
plt.savefig('nonlincorrelation.pdf') plt.savefig('nonlincorrelation.pdf')
#plt.show()

49
statistics/lecture/pdfhistogram.py
View File

@ -1,5 +1,6 @@
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
from plotstyle import *
# normal distribution: # normal distribution:
rng = np.random.RandomState(6281) rng = np.random.RandomState(6281)
@ -7,38 +8,24 @@ x = np.arange( -4.0, 4.0, 0.01 )
g = np.exp(-0.5*x*x)/np.sqrt(2.0*np.pi) g = np.exp(-0.5*x*x)/np.sqrt(2.0*np.pi)
r = rng.randn(100) r = rng.randn(100)
plt.xkcd() fig, (ax1, ax2) = plt.subplots(1, 2)
ax1.set_xlabel('x')
ax1.set_xlim(-3.2, 3.2)
ax1.set_xticks(np.arange(-3.0, 3.1, 1.0))
ax1.set_ylabel('Frequency')
ax1.set_yticks(np.arange(0.0, 41.0, 10.0))
ax1.hist(r, 5, zorder=-10, **fsB)
ax1.hist(r, 20, zorder=-5, **fsC)
fig = plt.figure( figsize=(6,3) ) ax2.set_xlabel('x')
ax = fig.add_subplot( 1, 2, 1 ) ax2.set_xlim(-3.2, 3.2)
ax.spines['right'].set_visible(False) ax2.set_xticks(np.arange(-3.0, 3.1, 1.0))
ax.spines['top'].set_visible(False) ax2.set_ylabel('Probab. density p(x)')
ax.yaxis.set_ticks_position('left') ax2.set_ylim(0.0, 0.44)
ax.xaxis.set_ticks_position('bottom') ax2.set_yticks(np.arange(0.0, 0.41, 0.1))
ax.set_xlabel( 'x' ) ax2.plot(x, g, zorder=-1, **lsA)
ax.set_xlim(-3.2, 3.2) ax2.hist(r, 5, normed=True, zorder=-10, **fsB)
ax.set_xticks( np.arange( -3.0, 3.1, 1.0 ) ) ax2.hist(r, 20, normed=True, zorder=-5, **fsC)
ax.set_ylabel( 'Frequency' )
ax.set_yticks( np.arange( 0.0, 41.0, 10.0 ) )
ax.hist(r, 5, color='#CC0000')
ax.hist(r, 20, color='#FFCC00')
ax = fig.add_subplot( 1, 2, 2 )
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.set_xlabel( 'x' )
ax.set_xlim(-3.2, 3.2)
ax.set_xticks( np.arange( -3.0, 3.1, 1.0 ) )
ax.set_ylabel( 'Probab. density p(x)' )
ax.set_ylim(0.0, 0.44)
ax.set_yticks( np.arange( 0.0, 0.41, 0.1 ) )
ax.plot(x, g, '-b', lw=2, zorder=-1)
ax.hist(r, 5, normed=True, color='#CC0000', zorder=-10)
ax.hist(r, 20, normed=True, color='#FFCC00', zorder=-5)
plt.subplots_adjust(left=0.1, right=0.98, bottom=0.15, top=0.98, wspace=0.4, hspace=0.0)
fig.savefig('pdfhistogram.pdf') fig.savefig('pdfhistogram.pdf')
#plt.show()

18
statistics/lecture/pdfprobabilities.py
View File

@ -1,5 +1,6 @@
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
from plotstyle import *
# normal distribution: # normal distribution:
x = np.arange(-3.0, 5.0, 0.01) x = np.arange(-3.0, 5.0, 0.01)
@ -7,13 +8,7 @@ g = np.exp(-0.5*x*x)/np.sqrt(2.0*np.pi)
x1=0.0 x1=0.0
x2=1.0 x2=1.0
plt.xkcd() fig, ax = plt.subplots(figsize=cm_size(figure_width, 1.2*figure_height))
fig = plt.figure( figsize=(6,3.4) )
ax = fig.add_subplot( 1, 1, 1 )
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.set_xlabel('x') ax.set_xlabel('x')
ax.set_ylabel('Probability density p(x)') ax.set_ylabel('Probability density p(x)')
ax.set_ylim(0.0, 0.46) ax.set_ylim(0.0, 0.46)
@ -24,13 +19,10 @@ ax.annotate('Gaussian',
arrowprops=dict(arrowstyle="->", relpos=(0.5,0.0), arrowprops=dict(arrowstyle="->", relpos=(0.5,0.0),
connectionstyle="angle3,angleA=10,angleB=110")) connectionstyle="angle3,angleA=10,angleB=110"))
ax.annotate('$P(0<x<1) = \int_0^1 p(x) \, dx$', ax.annotate('$P(0<x<1) = \int_0^1 p(x) \, dx$',
xy=(0.6, 0.28), xycoords='data', xy=(0.5, 0.24), xycoords='data',
xytext=(1.2, 0.4), textcoords='data', ha='left', xytext=(1.2, 0.4), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5), arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
connectionstyle="angle3,angleA=10,angleB=80")) connectionstyle="angle3,angleA=10,angleB=80"))
ax.fill_between( x[(x>x1)&(x<x2)], 0.0, g[(x>x1)&(x<x2)], color='#cc0000' ) ax.fill_between(x[(x>x1)&(x<x2)], 0.0, g[(x>x1)&(x<x2)], **fsBs)
ax.plot(x,g, 'b', lw=4) ax.plot(x,g, **lsA)
#plt.tight_layout()
plt.subplots_adjust(left=0.1, right=0.98, bottom=0.15, top=0.98, wspace=0.4, hspace=0.0)
fig.savefig('pdfprobabilities.pdf') fig.savefig('pdfprobabilities.pdf')
#plt.show()

29
statistics/lecture/quartile.py
View File

@ -1,18 +1,14 @@
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
from plotstyle import *
# normal distribution: # normal distribution:
x = np.arange( -4.0, 4.0, 0.01 ) x = np.arange( -4.0, 4.0, 0.01 )
g = np.exp(-0.5*x*x)/np.sqrt(2.0*np.pi) g = np.exp(-0.5*x*x)/np.sqrt(2.0*np.pi)
q = [ -0.67488, 0.0, 0.67488 ] q = [ -0.67488, 0.0, 0.67488 ]
plt.xkcd() fig, ax = plt.subplots(figsize=cm_size(figure_width, 1.0*figure_height))
fig = plt.figure( figsize=(6,3.2) ) fig.subplots_adjust(**adjust_fs(bottom=2.7, top=0.1))
ax = fig.add_subplot( 1, 1, 1 )
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.set_xlabel('x') ax.set_xlabel('x')
ax.set_ylabel('Probability density p(x)') ax.set_ylabel('Probability density p(x)')
ax.set_ylim(0.0, 0.46) ax.set_ylim(0.0, 0.46)
@ -36,15 +32,12 @@ ax.annotate('Median',
xytext=(1.6, 0.35), textcoords='data', ha='left', xytext=(1.6, 0.35), textcoords='data', ha='left',
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5), arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
connectionstyle="angle3,angleA=10,angleB=40") ) connectionstyle="angle3,angleA=10,angleB=40") )
ax.fill_between( x[x<q[0]], 0.0, g[x<q[0]], color='#ffcc00' ) ax.fill_between( x[x<q[0]], 0.0, g[x<q[0]], **fsCs)
ax.fill_between( x[(x>q[0])&(x<q[1])], 0.0, g[(x>q[0])&(x<q[1])], color='#ff0000' ) ax.fill_between( x[(x>q[0])&(x<q[1])], 0.0, g[(x>q[0])&(x<q[1])], **fsBs)
ax.fill_between( x[(x>q[1])&(x<q[2])], 0.0, g[(x>q[1])&(x<q[2])], color='#ff9900' ) ax.fill_between( x[(x>q[1])&(x<q[2])], 0.0, g[(x>q[1])&(x<q[2])], **fsDs)
ax.fill_between( x[x>q[2]], 0.0, g[x>q[2]], color='#ffff66' ) ax.fill_between( x[x>q[2]], 0.0, g[x>q[2]], **fsEs)
ax.plot(x,g, 'b', lw=4) ax.plot(x,g, **lsA)
ax.plot([0.0, 0.0], [0.0, 0.45], 'k', lw=2 ) ax.plot([0.0, 0.0], [0.0, 0.45], **lsMarker)
ax.plot([q[0], q[0]], [0.0, 0.4], 'k', lw=2 ) ax.plot([q[0], q[0]], [0.0, 0.4], **lsMarker)
ax.plot([q[2], q[2]], [0.0, 0.4], 'k', lw=2 ) ax.plot([q[2], q[2]], [0.0, 0.4], **lsMarker)
plt.subplots_adjust(left=0.1, right=0.98, bottom=0.15, top=0.98, wspace=0.4, hspace=0.0)
#plt.tight_layout()
fig.savefig( 'quartile.pdf' ) fig.savefig( 'quartile.pdf' )
#plt.show()

62
statistics/lecture/statistics.tex
View File

@ -115,17 +115,9 @@ function \mcode{median()} computes the median.
writing reliable code! writing reliable code!
\end{exercise} \end{exercise}
\begin{figure}[t]
\includegraphics[width=1\textwidth]{quartile}
\titlecaption{\label{quartilefig} Median and quartiles of a normal
distribution.}{ The interquartile range between the first and the
third quartile contains 50\,\% of the data and contains the
median.}
\end{figure}
The distribution of data can be further characterized by the position The distribution of data can be further characterized by the position
of its \entermde[quartile]{Quartil}{quartiles}. Neighboring quartiles are of its \entermde[quartile]{Quartil}{quartiles}. Neighboring quartiles are
separated by 25\,\% of the data (\figref{quartilefig}). separated by 25\,\% of the data.% (\figref{quartilefig}).
\entermde[percentile]{Perzentil}{Percentiles} allow to characterize the \entermde[percentile]{Perzentil}{Percentiles} allow to characterize the
distribution of the data in more detail. The 3$^{\rm rd}$ quartile distribution of the data in more detail. The 3$^{\rm rd}$ quartile
corresponds to the 75$^{\rm th}$ percentile, because 75\,\% of the corresponds to the 75$^{\rm th}$ percentile, because 75\,\% of the
@ -156,15 +148,13 @@ median that extends from the 1$^{\rm st}$ to the 3$^{\rm rd}$
quartile. The whiskers mark the minimum and maximum value of the data quartile. The whiskers mark the minimum and maximum value of the data
set (\figref{displayunivariatedatafig} (3)). set (\figref{displayunivariatedatafig} (3)).
\begin{exercise}{univariatedata.m}{} % \begin{figure}[t]
Generate 40 normally distributed random numbers with a mean of 2 and % \includegraphics[width=1\textwidth]{quartile}
illustrate their distribution in a box-whisker plot % \titlecaption{\label{quartilefig} Median and quartiles of a normal
(\code{boxplot()} function), with a bar and errorbar illustrating % distribution.}{ The interquartile range between the first and the
the mean and standard deviation (\code{bar()}, \code{errorbar()}), % third quartile contains 50\,\% of the data and contains the
and the data themselves jittered randomly (as in % median.}
\figref{displayunivariatedatafig}). How to interpret the different % \end{figure}
plots?
\end{exercise}
% \begin{exercise}{boxwhisker.m}{} % \begin{exercise}{boxwhisker.m}{}
% Generate a $40 \times 10$ matrix of random numbers and % Generate a $40 \times 10$ matrix of random numbers and
@ -201,6 +191,16 @@ Histograms are often used to estimate the
\enterm[probability!distribution]{probability distribution} \enterm[probability!distribution]{probability distribution}
(\determ[Wahrscheinlichkeits!-verteilung]{Wahrscheinlichkeitsverteilung}) of the data values. (\determ[Wahrscheinlichkeits!-verteilung]{Wahrscheinlichkeitsverteilung}) of the data values.
\begin{exercise}{univariatedata.m}{}
Generate 40 normally distributed random numbers with a mean of 2 and
illustrate their distribution in a box-whisker plot
(\code{boxplot()} function), with a bar and errorbar illustrating
the mean and standard deviation (\code{bar()}, \code{errorbar()}),
and the data themselves jittered randomly (as in
\figref{displayunivariatedatafig}). How to interpret the different
plots?
\end{exercise}
\subsection{Probabilities} \subsection{Probabilities}
In the frequentist interpretation of probability, the In the frequentist interpretation of probability, the
\enterm{probability} (\determ{Wahrscheinlichkeit}) of an event \enterm{probability} (\determ{Wahrscheinlichkeit}) of an event
@ -252,7 +252,7 @@ real number like, e.g., 0.123456789 is zero, because there are
uncountable many real numbers. uncountable many real numbers.
We can only ask for the probability to get a measurement value in some We can only ask for the probability to get a measurement value in some
range. For example, we can ask for the probability $P(1.2<x<1.3)$ to range. For example, we can ask for the probability $P(0<x<1)$ to
get a measurement between 0 and 1 (\figref{pdfprobabilitiesfig}). More get a measurement between 0 and 1 (\figref{pdfprobabilitiesfig}). More
generally, we want to know the probability $P(x_0<x<x_1)$ to obtain a generally, we want to know the probability $P(x_0<x<x_1)$ to obtain a
measurement between $x_0$ and $x_1$. If we define the width of the measurement between $x_0$ and $x_1$. If we define the width of the
@ -280,7 +280,7 @@ inverse of the unit of the data values --- hence the name ``density''.
\end{figure} \end{figure}
The probability to get a value $x$ between $x_1$ and $x_2$ is The probability to get a value $x$ between $x_1$ and $x_2$ is
given by the integral of the probability density: given by an integral over the probability density:
\[ P(x_1 < x < x2) = \int\limits_{x_1}^{x_2} p(x) \, dx \; . \] \[ P(x_1 < x < x2) = \int\limits_{x_1}^{x_2} p(x) \, dx \; . \]
Because the probability to get any value $x$ is one, the integral of Because the probability to get any value $x$ is one, the integral of
the probability density over the whole real axis must be one: the probability density over the whole real axis must be one:
@ -329,7 +329,7 @@ values fall within each bin (\figref{pdfhistogramfig} left).
observe? observe?
\end{exercise} \end{exercise}
To turn such histograms to estimates of probability densities they To turn such histograms into estimates of probability densities they
need to be normalized such that according to \eqnref{pdfnorm} their need to be normalized such that according to \eqnref{pdfnorm} their
integral equals one. While histograms of categorical data are integral equals one. While histograms of categorical data are
normalized such that their sum equals one, here we need to integrate normalized such that their sum equals one, here we need to integrate
@ -343,7 +343,7 @@ and the
\[ p(x_i) = \frac{n_i}{A} = \frac{n_i}{\Delta x \sum_{i=1}^N n_i} = \[ p(x_i) = \frac{n_i}{A} = \frac{n_i}{\Delta x \sum_{i=1}^N n_i} =
\frac{n_i}{N \Delta x} \; .\] \frac{n_i}{N \Delta x} \; .\]
A histogram needs to be divided by both the sum of the frequencies A histogram needs to be divided by both the sum of the frequencies
$n_i$ and the bin width $\Delta x$ to results in an estimate of the $n_i$ and the bin width $\Delta x$ to result in an estimate of the
corresponding probability density. Only then can the distribution be corresponding probability density. Only then can the distribution be
compared with other distributions and in particular with theoretical compared with other distributions and in particular with theoretical
probability density functions like the one of the normal distribution probability density functions like the one of the normal distribution
@ -371,19 +371,20 @@ probability density functions like the one of the normal distribution
A problem of using histograms for estimating probability densities is A problem of using histograms for estimating probability densities is
that they have hard bin edges. Depending on where the bin edges are that they have hard bin edges. Depending on where the bin edges are
placed a data value falls in one or the other bin. As a result the placed a data value falls in one or the other bin. As a result the
shape histogram depends on the exact position of its bins shape of the resulting histogram depends on the exact position of its
(\figref{kerneldensityfig} left). bins (\figref{kerneldensityfig} left).
\begin{figure}[t] \begin{figure}[t]
\includegraphics[width=1\textwidth]{kerneldensity} \includegraphics[width=1\textwidth]{kerneldensity}
\titlecaption{\label{kerneldensityfig} Kernel densities.}{Left: The \titlecaption{\label{kerneldensityfig} Kernel densities.}{The
histogram-based estimation of the probability density is dependent histogram-based estimation of the probability density depends on
on the position of the bins. In the bottom plot the bins have the position of the bins (left). In the bottom plot the bins have
been shifted by half a bin width (here $\Delta x=0.4$) and as a been shifted by half a bin width (here $\Delta x=0.4$) and as a
result details of the probability density look different. Look, result details of the probability density look different. Look,
for example, at the height of the largest bin. Right: In contrast, for example, at the height of the largest bin. In contrast, a
a kernel density is uniquely defined for a given kernel width kernel density is uniquely defined for a given kernel width
(here Gaussian kernels with standard deviation of $\sigma=0.2$).} (right, Gaussian kernels with standard deviation of
$\sigma=0.2$).}
\end{figure} \end{figure}
To avoid this problem so called \entermde[kernel To avoid this problem so called \entermde[kernel
@ -460,7 +461,6 @@ and percentiles can be determined from the inverse cumulative function.
Use the estimate to compute the value of the 5\,\% percentile. Use the estimate to compute the value of the 5\,\% percentile.
\end{exercise} \end{exercise}
\newpage
\section{Correlations} \section{Correlations}
Until now we described properties of univariate data sets. In Until now we described properties of univariate data sets. In