Merge branch 'master' of https://whale.am28.uni-tuebingen.de/git/teaching/scientificComputing
This commit is contained in:
commit
0666b5aeb6
@ -1,8 +1,7 @@
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import numpy as np
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import matplotlib.pyplot as plt
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from plotstyle import *
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plt.xkcd()
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fig = plt.figure( figsize=(6,3.5) )
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rng = np.random.RandomState(637281)
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nsamples = 100
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@ -25,11 +24,8 @@ for i in range(nresamples) :
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musrs.append(np.mean(rng.randn(nsamples)))
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hmusrs, _ = np.histogram(musrs, bins, density=True)
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ax = fig.add_subplot(1, 1, 1)
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ax.spines['right'].set_visible(False)
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ax.spines['top'].set_visible(False)
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ax.yaxis.set_ticks_position('left')
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ax.xaxis.set_ticks_position('bottom')
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fig, ax = plt.subplots(figsize=cm_size(figure_width, 1.2*figure_height))
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fig.subplots_adjust(**adjust_fs(left=4.0, bottom=2.7, right=1.5))
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ax.set_xlabel('Mean')
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ax.set_xlim(-0.4, 0.4)
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ax.set_ylabel('Probability density')
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@ -45,9 +41,7 @@ ax.annotate('bootstrap\ndistribution',
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xytext=(0.25, 4), textcoords='data',
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arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
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connectionstyle="angle3,angleA=20,angleB=60") )
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ax.bar(bins[:-1]-0.25*db, hmusrs, 0.5*db, color='r')
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ax.bar(bins[:-1]+0.25*db, hmus, 0.5*db, color='b')
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ax.bar(bins[:-1]-0.25*db, hmusrs, 0.5*db, **fsB)
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ax.bar(bins[:-1]+0.25*db, hmus, 0.5*db, **fsA)
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plt.tight_layout()
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plt.savefig('bootstrapsem.pdf')
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#plt.show();
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@ -1,9 +1,8 @@
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import numpy as np
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import scipy.stats as st
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import matplotlib.pyplot as plt
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from plotstyle import *
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plt.xkcd()
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fig = plt.figure( figsize=(6,3.5) )
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rng = np.random.RandomState(637281)
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# generate correlated data:
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@ -36,33 +35,28 @@ print('Measured correlation coefficient %.2f is at %.4f percentile of bootstrap'
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rp, ra = st.pearsonr(x, y)
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print('Measured correlation coefficient %.2f is at %.4f percentile of test' % (rp, ra))
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ax = fig.add_subplot(1, 1, 1)
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ax.spines['right'].set_visible(False)
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ax.spines['top'].set_visible(False)
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ax.yaxis.set_ticks_position('left')
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ax.xaxis.set_ticks_position('bottom')
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fig, ax = plt.subplots(figsize=cm_size(figure_width, 1.2*figure_height))
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fig.subplots_adjust(**adjust_fs(left=4.0, bottom=2.7, right=0.5, top=1.0))
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ax.annotate('Measured\ncorrelation\nis significant!',
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xy=(rd, 1.1), xycoords='data',
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xytext=(rd, 2.2), textcoords='data', ha='left',
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arrowprops=dict(arrowstyle="->", relpos=(0.2,0.0),
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connectionstyle="angle3,angleA=10,angleB=80") )
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connectionstyle="angle3,angleA=10,angleB=80") )
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ax.annotate('95% percentile',
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xy=(0.14, 0.9), xycoords='data',
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xytext=(0.2, 4.0), textcoords='data', ha='left',
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arrowprops=dict(arrowstyle="->", relpos=(0.1,0.0),
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connectionstyle="angle3,angleA=30,angleB=70") )
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connectionstyle="angle3,angleA=30,angleB=70") )
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ax.annotate('Distribution of\nuncorrelated\nsamples',
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xy=(-0.08, 3.6), xycoords='data',
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xytext=(-0.22, 5.0), textcoords='data', ha='left',
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arrowprops=dict(arrowstyle="->", relpos=(0.5,0.0),
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connectionstyle="angle3,angleA=150,angleB=100") )
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ax.bar(b[:-1], h, width=b[1]-b[0], color='#ffff66')
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ax.bar(b[:-1][b[:-1]>=rq], h[b[:-1]>=rq], width=b[1]-b[0], color='#ff9900')
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ax.plot( [rd, rd], [0, 1], 'b', linewidth=4 )
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connectionstyle="angle3,angleA=150,angleB=100") )
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ax.bar(b[:-1], h, width=b[1]-b[0], **fsC)
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ax.bar(b[:-1][b[:-1]>=rq], h[b[:-1]>=rq], width=b[1]-b[0], **fsB)
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ax.plot( [rd, rd], [0, 1], **lsA)
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ax.set_xlim(-0.25, 0.35)
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ax.set_xlabel('Correlation coefficient')
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ax.set_ylabel('Probability density of H0')
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plt.tight_layout()
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plt.savefig('permutecorrelation.pdf')
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#plt.show();
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@ -1,24 +1,22 @@
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import numpy as np
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import matplotlib.pyplot as plt
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import matplotlib.cm as cm
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import matplotlib.gridspec as gridspec
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from plotstyle import *
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plt.xkcd()
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fig = plt.figure( figsize=(6,6.8) )
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rng = np.random.RandomState(4637281)
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lmarg=0.1
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rmarg=0.1
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ax = fig.add_axes([lmarg, 0.75, 1.0-rmarg, 0.25])
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ax.spines['bottom'].set_position('zero')
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ax.spines['left'].set_visible(False)
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ax.spines['right'].set_visible(False)
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ax.spines['top'].set_visible(False)
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ax.xaxis.set_ticks_position('bottom')
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ax.get_yaxis().set_visible(False)
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fig = plt.figure(figsize=cm_size(figure_width, 2.8*figure_height))
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spec = gridspec.GridSpec(nrows=4, ncols=1, height_ratios=[4, 4, 1, 3], hspace=0.2,
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**adjust_fs(fig, left=4.0))
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ax = fig.add_subplot(spec[0, 0])
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ax.set_xlim(0.0, np.pi)
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ax.set_xticks(np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi))
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ax.set_xticklabels([])
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ax.set_ylim(0.0, 3.5)
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ax.yaxis.set_major_locator(plt.NullLocator())
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ax.text(-0.2, 0.5*3.5, 'Activity', rotation='vertical', va='center')
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ax.annotate('Tuning curve',
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xy=(0.42*np.pi, 2.5), xycoords='data',
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@ -31,55 +29,49 @@ ax.annotate('',
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arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5),
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connectionstyle="angle3,angleA=80,angleB=90") )
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ax.text(0.52*np.pi, 0.7, 'preferred\norientation')
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ax.plot([0, 0], [0.0, 3.5], 'k', zorder=10, clip_on=False)
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xx = np.arange(0.0, 2.0*np.pi, 0.01)
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pp = 0.5*np.pi
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yy = np.exp(np.cos(2.0*(xx+pp)))
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ax.fill_between(xx, yy+0.25*yy, yy-0.25*yy, color=cm.autumn(0.3, 1), alpha=0.5)
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ax.plot(xx, yy, color=cm.autumn(0.0, 1))
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ax.fill_between(xx, yy+0.25*yy, yy-0.25*yy, **fsBa)
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ax.plot(xx, yy, **lsB)
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ax = fig.add_axes([lmarg, 0.34, 1.0-rmarg, 0.38])
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ax.spines['bottom'].set_position('zero')
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ax.spines['left'].set_visible(False)
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ax.spines['right'].set_visible(False)
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ax.spines['top'].set_visible(False)
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ax.xaxis.set_ticks_position('bottom')
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ax.get_yaxis().set_visible(False)
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ax = fig.add_subplot(spec[1, 0])
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ax.set_xlim(0.0, np.pi)
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ax.set_xticks(np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi))
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ax.set_xticklabels([])
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ax.set_ylim(-1.5, 3.0)
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ax.text(0.5*np.pi, -1.8, 'Orientation', ha='center')
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ax.set_ylim(0.0, 3.0)
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ax.yaxis.set_major_locator(plt.NullLocator())
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ax.text(-0.2, 0.5*3.5, 'Activity', rotation='vertical', va='center')
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ax.plot([0, 0], [0.0, 3.0], 'k', zorder=10, clip_on=False)
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xx = np.arange(0.0, 1.0*np.pi, 0.01)
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prefphases = np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi)
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responses = []
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xresponse = 0.475*np.pi
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for pp in prefphases :
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for pp, ls, ps in zip(prefphases, [lsE, lsC, lsD, lsB, lsD, lsC, lsE],
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[psE, psC, psD, psB, psD, psC, psE]) :
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yy = np.exp(np.cos(2.0*(xx+pp)))
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ax.plot(xx, yy, color=cm.autumn(2.0*np.abs(pp/np.pi-0.5), 1))
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#ax.plot(xx, yy, color=cm.autumn(2.0*np.abs(pp/np.pi-0.5), 1))
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ax.plot(xx, yy, **ls)
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y = np.exp(np.cos(2.0*(xresponse+pp)))
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responses.append(y + rng.randn()*0.25*y)
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ax.plot(xresponse, y, '.', markersize=20, color=cm.autumn(2.0*np.abs(pp/np.pi-0.5), 1))
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r=0.3
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y=-0.8
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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)
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ax.plot(xresponse, y, **ps)
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responses = np.array(responses)
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ax = fig.add_axes([lmarg, 0.05, 1.0-rmarg, 0.22])
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ax.spines['left'].set_visible(False)
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ax.spines['right'].set_visible(False)
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ax.spines['top'].set_visible(False)
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ax.xaxis.set_ticks_position('bottom')
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ax.get_yaxis().set_visible(False)
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ax = fig.add_subplot(spec[2, 0])
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ax.show_spines('')
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r = 0.3
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ax.set_ylim(-1.1*r, 1.1*r)
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for pp in prefphases:
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ax.plot([pp-0.5*r*np.cos(pp), pp+0.5*r*np.cos(pp)], [-r*np.sin(pp), r*np.sin(pp)],
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colors['black'], lw=6, clip_on=False)
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ax = fig.add_subplot(spec[3, 0])
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ax.set_xlim(0.0, np.pi)
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ax.set_xticks(np.arange(0.125*np.pi, 1.*np.pi, 0.125*np.pi))
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ax.set_xticklabels([])
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ax.set_ylim(-1600, 0)
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ax.yaxis.set_major_locator(plt.NullLocator())
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ax.set_xlabel('Orientation')
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ax.text(-0.2, -800, 'Log-Likelihood', rotation='vertical', va='center')
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ax.plot([0, 0], [-1600, 0], 'k', zorder=10, clip_on=False)
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phases = np.linspace(0.0, 1.1*np.pi, 100)
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probs = np.zeros((len(responses), len(phases)))
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for k, (pp, r) in enumerate(zip(prefphases, responses)) :
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@ -95,7 +87,6 @@ ax.annotate('',
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arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5),
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connectionstyle="angle3,angleA=80,angleB=90") )
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ax.text(maxp+0.05, -1100, 'most likely\norientation\ngiven the responses')
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ax.plot(phases, loglikelihood, '-b')
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ax.plot(phases, loglikelihood, **lsA)
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plt.savefig('mlecoding.pdf')
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#plt.show();
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@ -1,8 +1,11 @@
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import numpy as np
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import matplotlib.pyplot as plt
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import matplotlib.gridspec as gridspec
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from plotstyle import *
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plt.xkcd()
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fig = plt.figure( figsize=(6,5) )
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fig = plt.figure(figsize=cm_size(figure_width, 1.8*figure_height))
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spec = gridspec.GridSpec(nrows=2, ncols=2, hspace=0.6,
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**adjust_fs(fig, left=5.5))
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# the data:
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n = 40
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@ -11,21 +14,17 @@ sigma = 0.5
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rmu = 2.0
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xd = rng.randn(n)*sigma+rmu
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# and possible pdfs:
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x = np.arange( 0.0, 4.0, 0.01 )
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x = np.arange(0.0, 4.0, 0.01)
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mus = [1.5, 2.0, 2.5]
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g=np.zeros((len(x), len(mus)))
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for k, mu in enumerate(mus) :
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g[:,k] = np.exp(-0.5*((x-mu)/sigma)**2.0)/np.sqrt(2.0*np.pi)/sigma
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# plot it:
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ax = fig.add_subplot( 2, 1, 1 )
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ax.spines['right'].set_visible(False)
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ax.spines['top'].set_visible(False)
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ax.yaxis.set_ticks_position('left')
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ax.xaxis.set_ticks_position('bottom')
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ax = fig.add_subplot(spec[0, :])
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ax.set_xlim(0.5, 3.5)
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ax.set_ylim(-0.02, 0.85)
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ax.set_xticks( np.arange(0, 5))
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ax.set_yticks( np.arange(0, 0.9, 0.2))
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ax.set_xticks(np.arange(0, 5))
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ax.set_yticks(np.arange(0, 0.9, 0.2))
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ax.set_xlabel('x')
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ax.set_ylabel('Probability density')
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s = 1
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@ -36,14 +35,14 @@ for mu in mus :
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ax.annotate('', xy=(mu, 0.02), xycoords='data',
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xytext=(mu, 0.75), textcoords='data',
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arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5),
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connectionstyle=cs), zorder=1 )
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connectionstyle=cs), zorder=1)
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if mu > rmu :
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ax.text(mu-0.1, 0.04, '?', zorder=1, ha='right')
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else :
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ax.text(mu+0.1, 0.04, '?', zorder=1)
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for k in range(len(mus)) :
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ax.plot(x, g[:,k], zorder=5)
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ax.scatter(xd, 0.05*rng.rand(len(xd))+0.2, s=30, zorder=10)
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for k, ls in enumerate([lsCm, lsBm, lsDm]) :
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ax.plot(x, g[:,k], zorder=5, **ls)
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ax.plot(xd, 0.05*rng.rand(len(xd))+0.2, zorder=10, **psAm)
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# likelihood:
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thetas=np.arange(1.5, 2.6, 0.01)
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@ -52,48 +51,38 @@ for i, theta in enumerate(thetas) :
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ps[:,i]=np.exp(-0.5*((xd-theta)/sigma)**2.0)/np.sqrt(2.0*np.pi)/sigma
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p=np.prod(ps,axis=0)
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# plot it:
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ax = fig.add_subplot( 2, 2, 3 )
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ax.spines['right'].set_visible(False)
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ax.spines['top'].set_visible(False)
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ax.yaxis.set_ticks_position('left')
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ax.xaxis.set_ticks_position('bottom')
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ax = fig.add_subplot(spec[1, 0])
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ax.set_xlabel(r'Parameter $\theta$')
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ax.set_ylabel('Likelihood')
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ax.set_xticks( np.arange(1.6, 2.5, 0.4))
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ax.set_xticks(np.arange(1.6, 2.5, 0.4))
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ax.annotate('Maximum',
|
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xy=(2.0, 5.5e-11), xycoords='data',
|
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xytext=(1.0, 1.1), textcoords='axes fraction', ha='right',
|
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arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
|
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connectionstyle="angle3,angleA=10,angleB=70") )
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connectionstyle="angle3,angleA=10,angleB=70"))
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ax.annotate('',
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xy=(2.0, 0), xycoords='data',
|
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xytext=(2.0, 5e-11), textcoords='data',
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arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5),
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connectionstyle="angle3,angleA=90,angleB=80") )
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ax.plot(thetas,p)
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connectionstyle="angle3,angleA=90,angleB=80"))
|
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ax.plot(thetas, p, **lsAm)
|
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ax = fig.add_subplot( 2, 2, 4 )
|
||||
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 = fig.add_subplot(spec[1, 1])
|
||||
ax.set_xlabel(r'Parameter $\theta$')
|
||||
ax.set_ylabel('Log-Likelihood')
|
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ax.set_ylim(-50,-20)
|
||||
ax.set_xticks( np.arange(1.6, 2.5, 0.4))
|
||||
ax.set_yticks( np.arange(-50, -19, 10.0))
|
||||
ax.set_xticks(np.arange(1.6, 2.5, 0.4))
|
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ax.set_yticks(np.arange(-50, -19, 10.0))
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ax.annotate('Maximum',
|
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xy=(2.0, -23), xycoords='data',
|
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xytext=(1.0, 1.1), textcoords='axes fraction', ha='right',
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arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
|
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connectionstyle="angle3,angleA=10,angleB=70") )
|
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connectionstyle="angle3,angleA=10,angleB=70"))
|
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ax.annotate('',
|
||||
xy=(2.0, -50), xycoords='data',
|
||||
xytext=(2.0, -26), textcoords='data',
|
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arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5),
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connectionstyle="angle3,angleA=80,angleB=100") )
|
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ax.plot(thetas,np.log(p))
|
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connectionstyle="angle3,angleA=80,angleB=100"))
|
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ax.plot(thetas,np.log(p), **lsAm)
|
||||
|
||||
plt.tight_layout();
|
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plt.savefig('mlemean.pdf')
|
||||
#plt.show();
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||||
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@ -2,9 +2,10 @@ import numpy as np
|
||||
import scipy.stats as st
|
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import scipy.optimize as opt
|
||||
import matplotlib.pyplot as plt
|
||||
from plotstyle import *
|
||||
|
||||
plt.xkcd()
|
||||
fig = plt.figure( figsize=(6,3) )
|
||||
fig, (ax1, ax2) = plt.subplots(1, 2)
|
||||
fig.subplots_adjust(**adjust_fs(fig, right=1.0))
|
||||
|
||||
# the data:
|
||||
n = 100
|
||||
@ -23,27 +24,23 @@ a = st.gamma.fit(xd, 5.0)
|
||||
yf = st.gamma.pdf(xx, *a)
|
||||
|
||||
# plot it:
|
||||
ax = fig.add_subplot( 1, 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_xlim(0, 10.0)
|
||||
ax.set_ylim(0.0, 0.42)
|
||||
ax.set_xticks( np.arange(0, 11, 2))
|
||||
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')
|
||||
ax1.set_xlim(0, 10.0)
|
||||
ax1.set_ylim(0.0, 0.42)
|
||||
ax1.set_xticks(np.arange(0, 11, 2))
|
||||
ax1.set_yticks(np.arange(0, 0.42, 0.1))
|
||||
ax1.set_xlabel('x')
|
||||
ax1.set_ylabel('Probability density')
|
||||
ax1.plot(xx, yy, label='pdf', **lsB)
|
||||
ax1.plot(xx, yf, label='mle', **lsCm)
|
||||
kernel = st.gaussian_kde(xd)
|
||||
x = kernel(xd)
|
||||
x /= np.max(x)
|
||||
ax.scatter(xd, 0.05*x*(rng.rand(len(xd))-0.5)+0.05, s=30, zorder=10)
|
||||
ax.legend(loc='upper right', frameon=False)
|
||||
sigma = 0.07
|
||||
ax1.plot(xd, sigma*x*(rng.rand(len(xd))-0.5)+sigma, zorder=10, **psAm)
|
||||
ax1.legend(loc='upper right')
|
||||
|
||||
# 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:
|
||||
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)
|
||||
|
||||
# plot it:
|
||||
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, 10.0)
|
||||
ax.set_xticks( np.arange(0, 11, 2))
|
||||
ax.set_xlabel('x')
|
||||
ax.set_ylim(0.0, 0.42)
|
||||
ax.set_yticks( np.arange(0, 0.42, 0.1))
|
||||
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)
|
||||
ax2.set_xlim(0, 10.0)
|
||||
ax2.set_xticks(np.arange(0, 11, 2))
|
||||
ax2.set_xlabel('x')
|
||||
ax2.set_ylim(0.0, 0.42)
|
||||
ax2.set_yticks(np.arange(0, 0.42, 0.1))
|
||||
ax2.set_ylabel('Probability density')
|
||||
ax2.plot(xx, yy, label='pdf', **lsB)
|
||||
ax2.plot(xx, yc, label='fit', **lsCm)
|
||||
ax2.bar(b[:-1], h, np.diff(b), **fsA)
|
||||
ax2.legend(loc='upper right')
|
||||
|
||||
plt.tight_layout();
|
||||
plt.savefig('mlepdf.pdf')
|
||||
#plt.show();
|
||||
|
||||
@ -1,9 +1,14 @@
|
||||
import numpy as np
|
||||
import scipy.stats as st
|
||||
import matplotlib.pyplot as plt
|
||||
import matplotlib.gridspec as gridspec
|
||||
from plotstyle import *
|
||||
|
||||
plt.xkcd()
|
||||
fig = plt.figure(figsize=(6, 3))
|
||||
fig = plt.figure()
|
||||
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:
|
||||
slope = 2.0
|
||||
@ -20,71 +25,44 @@ slopef = np.sum(x*y)/np.sum(x*x)
|
||||
yf = slopef*xx
|
||||
|
||||
# plot it:
|
||||
ax = fig.add_axes([0.09, 0.02, 0.33, 0.9])
|
||||
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 = fig.add_subplot(spec1[0, 0])
|
||||
ax.set_xticks(np.arange(0.0, 4.1))
|
||||
ax.set_xlim(0.0, 4.2)
|
||||
ax.set_ylim(-4.0, 12.0)
|
||||
ax.set_xlabel('x')
|
||||
ax.set_ylabel('y')
|
||||
ax.scatter(x, y, label='data', s=40, zorder=10)
|
||||
ax.plot(xx, yy, 'r', lw=5.0, color='#ff0000', label='original', zorder=5)
|
||||
ax.plot(xx, yf, '--', lw=1.0, color='#ffcc00', label='fit', zorder=7)
|
||||
ax.legend(loc='upper left', bbox_to_anchor=(0.0, 1.15), frameon=False)
|
||||
ax.plot(x, y, label='data', zorder=10, **psAm)
|
||||
ax.plot(xx, yy, label='original', zorder=5, **lsB)
|
||||
ax.plot(xx, yf, label='fit', zorder=7, **lsCm)
|
||||
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.spines['left'].set_position('zero')
|
||||
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 = fig.add_subplot(spec1[0, 1])
|
||||
ax.show_spines('l')
|
||||
ax.set_xticks([])
|
||||
ax.set_ylim(-4.0, 12.0)
|
||||
ax.set_yticks([])
|
||||
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.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 = fig.add_subplot(spec2[0, 0])
|
||||
ax.set_xticks(np.arange(0.0, 4.1))
|
||||
ax.set_xlim(0.0, 4.2)
|
||||
ax.set_ylim(-4.0, 12.0)
|
||||
ax.set_xlabel('x')
|
||||
ax.set_ylabel('y - mx')
|
||||
ax.scatter(x, y - slopef*x, label='residuals', s=40, zorder=10)
|
||||
#ax.legend(loc='upper left', bbox_to_anchor=(0.0, 1.0), frameon=False)
|
||||
ax.plot(x, y - slopef*x, label='residuals', zorder=10, **psAm)
|
||||
#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.spines['left'].set_position('zero')
|
||||
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 = fig.add_subplot(spec2[0, 1])
|
||||
ax.show_spines('l')
|
||||
ax.set_xlim(0.0, 11.0)
|
||||
ax.set_xticks([])
|
||||
ax.set_ylim(-4.0, 12.0)
|
||||
ax.set_yticks([])
|
||||
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)
|
||||
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.show();
|
||||
|
||||
225
plotstyle.py
225
plotstyle.py
@ -1,6 +1,5 @@
|
||||
import matplotlib as mpl
|
||||
import matplotlib.pyplot as plt
|
||||
from cycler import cycler
|
||||
from mpl_toolkits.mplot3d import Axes3D
|
||||
|
||||
xkcd_style = False
|
||||
@ -13,125 +12,103 @@ figure_height = 6.0 # cm, for a 1 x 2 figure
|
||||
ppi = 72.0
|
||||
|
||||
# colors:
|
||||
|
||||
def lighter(color, lightness):
|
||||
""" Make a color lighter.
|
||||
|
||||
Parameters
|
||||
----------
|
||||
color: string
|
||||
An RGB color as a hexadecimal string (e.g. '#rrggbb').
|
||||
lightness: float
|
||||
The smaller the lightness, the lighter the returned color.
|
||||
A lightness of 1 leaves the color untouched.
|
||||
A lightness of 0 returns white.
|
||||
|
||||
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'
|
||||
colors = {}
|
||||
colors['red'] = '#DD1000'
|
||||
colors['orange'] = '#FF9900'
|
||||
colors['lightorange'] = '#FFCC00'
|
||||
colors['yellow'] = '#FFF720'
|
||||
colors['green'] = '#99FF00'
|
||||
colors['blue'] = '#0010CC'
|
||||
colors['gray'] = '#A7A7A7'
|
||||
colors['black'] = '#000000'
|
||||
colors['white'] = '#FFFFFF'
|
||||
|
||||
#colors_bendalab_vivid['green'] = '#30D700'
|
||||
#colors_bendalab_vivid['blue'] = '#0020C0'
|
||||
|
||||
# line styles for plot():
|
||||
lsSpine = {'c': colors['black'], 'linestyle': '-', 'linewidth': 1}
|
||||
lsGrid = {'c': colors['grey'], 'linestyle': '--', 'linewidth': 1}
|
||||
|
||||
# 'B1': prominent line with first color and style from color group 'B'
|
||||
# 'C2m': minor line with second color and style from color group 'C'
|
||||
ls = {
|
||||
'A1': {'c': colors['red'], 'linestyle': '-', 'linewidth': 3},
|
||||
'A2': {'c': colors['orange'], 'linestyle': '-', 'linewidth': 3},
|
||||
'A3': {'c': colors['lightorange'], 'linestyle': '-', 'linewidth': 3},
|
||||
'B1': {'c': colors['orange'], 'linestyle': '-', 'linewidth': 3},
|
||||
'B2': {'c': colors['lightorange'], 'linestyle': '-', 'linewidth': 3},
|
||||
'B3': {'c': colors['yellow'], 'linestyle': '-', 'linewidth': 3},
|
||||
'C1': {'c': colors['green'], 'linestyle': '-', 'linewidth': 3},
|
||||
'D1': {'c': colors['blue'], 'linestyle': '-', 'linewidth': 3},
|
||||
'A1m': {'c': colors['red'], 'linestyle': '-', 'linewidth': 2},
|
||||
'A2m': {'c': colors['orange'], 'linestyle': '-', 'linewidth': 2},
|
||||
'A3m': {'c': colors['lightorange'], 'linestyle': '-', 'linewidth': 2},
|
||||
'B1m': {'c': colors['orange'], 'linestyle': '-', 'linewidth': 2},
|
||||
'B2m': {'c': colors['lightorange'], 'linestyle': '-', 'linewidth': 2},
|
||||
'B3m': {'c': colors['yellow'], 'linestyle': '-', 'linewidth': 2},
|
||||
'C1m': {'c': colors['green'], 'linestyle': '-', 'linewidth': 2},
|
||||
'D1m': {'c': colors['blue'], 'linestyle': '-', 'linewidth': 2},
|
||||
}
|
||||
lwthick = 3.0
|
||||
lwthin = 1.8
|
||||
mainline = {'linestyle': '-', 'linewidth': lwthick}
|
||||
minorline = {'linestyle': '-', 'linewidth': lwthin}
|
||||
largemarker = {'marker': 'o', 'markersize': 9, 'markeredgecolor': colors['white'], 'markeredgewidth': 1}
|
||||
smallmarker = {'marker': 'o', 'markersize': 6, 'markeredgecolor': colors['white'], 'markeredgewidth': 1}
|
||||
largelinepoints = {'linestyle': '-', 'linewidth': lwthick, 'marker': 'o', 'markersize': 10, 'markeredgecolor': colors['white'], 'markeredgewidth': 1}
|
||||
smalllinepoints = {'linestyle': '-', 'linewidth': 1.4, 'marker': 'o', 'markersize': 7, 'markeredgecolor': colors['white'], 'markeredgewidth': 1}
|
||||
filllw = 1
|
||||
fillec = colors['white']
|
||||
fillalpha = 0.4
|
||||
|
||||
# helper lines:
|
||||
lsSpine = {'c': colors['black'], 'linestyle': '-', 'linewidth': 1, 'clip_on': False}
|
||||
lsGrid = {'c': colors['gray'], 'linestyle': '--', 'linewidth': 1}
|
||||
lsMarker = {'c': colors['black'], 'linestyle': '-', 'linewidth': 2}
|
||||
|
||||
# line (ls), point (ps), and fill styles (fs).
|
||||
|
||||
# Each style is derived from a main color as indicated by the capital letter.
|
||||
|
||||
# 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:
|
||||
bar_fac = 1.0
|
||||
@ -364,19 +341,19 @@ def common_format():
|
||||
mpl.rcParams['grid.color'] = lsGrid['c']
|
||||
mpl.rcParams['grid.linestyle'] = lsGrid['linestyle']
|
||||
mpl.rcParams['grid.linewidth'] = lsGrid['linewidth']
|
||||
mpl.rcParams['legend.frameon'] = False
|
||||
mpl.rcParams['axes.facecolor'] = 'none'
|
||||
mpl.rcParams['axes.edgecolor'] = lsSpine['c']
|
||||
mpl.rcParams['axes.linewidth'] = lsSpine['linewidth']
|
||||
if 'axes.prop_cycle' in mpl.rcParams:
|
||||
from cycler import cycler
|
||||
mpl.rcParams['axes.prop_cycle'] = cycler(color=[colors['blue'], colors['red'],
|
||||
colors['orange'], colors['green'],
|
||||
colors['purple'], colors['yellow'],
|
||||
colors['cyan'], colors['pink']])
|
||||
colors['lightorange'], colors['orange'],
|
||||
colors['yellow'], colors['green']])
|
||||
else:
|
||||
mpl.rcParams['axes.color_cycle'] = [colors['blue'], colors['red'],
|
||||
colors['orange'], colors['green'],
|
||||
colors['purple'], colors['yellow'],
|
||||
colors['cyan'], colors['pink']]
|
||||
colors['lightorange'], colors['orange'],
|
||||
colors['yellow'], colors['green']]
|
||||
# overwrite axes constructor:
|
||||
if not hasattr(mpl.axes.Subplot, '__init__orig'):
|
||||
mpl.axes.Subplot.__init__orig = mpl.axes.Subplot.__init__
|
||||
|
||||
@ -1,94 +1,104 @@
|
||||
For new projects:
|
||||
Copy project_template/ and adapt according to your needs
|
||||
|
||||
All projects:
|
||||
|
||||
check for time information
|
||||
|
||||
|
||||
1) project_activation_curve
|
||||
medium
|
||||
Write questions
|
||||
|
||||
project_adaptation_fit
|
||||
2) project_adaptation_fit
|
||||
OK, medium
|
||||
Add plotting of cost function
|
||||
|
||||
project_eod
|
||||
3) project_eod
|
||||
OK, medium - difficult
|
||||
b_0 is not defined
|
||||
|
||||
project_eyetracker
|
||||
4) project_eyetracker
|
||||
OK, difficult
|
||||
no statistics, but kmeans
|
||||
|
||||
project_fano_slope
|
||||
5) project_face_selectivity
|
||||
medium-difficult
|
||||
(Marius monkey data)
|
||||
|
||||
6) project_fano_slope
|
||||
OK, difficult
|
||||
|
||||
project_fano_test
|
||||
7) project_fano_test
|
||||
OK -
|
||||
|
||||
project_fano_time
|
||||
8) project_fano_time
|
||||
OK, medium-difficult
|
||||
|
||||
project_ficurves
|
||||
9) project_ficurves
|
||||
OK, medium
|
||||
Maybe add correlation test or fit statistics
|
||||
|
||||
project_input_resistance
|
||||
10) project_input_resistance
|
||||
medium
|
||||
What is the problem with this project? --> No difference between segments
|
||||
Improve questions
|
||||
|
||||
project_isicorrelations
|
||||
11) project_isicorrelations
|
||||
medium-difficult
|
||||
Need to finish solution
|
||||
|
||||
project_isipdffit
|
||||
12) project_isipdffit
|
||||
Too technical
|
||||
|
||||
project_lif
|
||||
13) project_lif
|
||||
OK, difficult
|
||||
no statistics
|
||||
|
||||
project_mutualinfo
|
||||
14) project_mutualinfo
|
||||
OK, medium
|
||||
|
||||
project_noiseficurves
|
||||
15) project_noiseficurves
|
||||
OK, simple-medium
|
||||
no statistics
|
||||
|
||||
project_numbers
|
||||
16) project_numbers
|
||||
simple
|
||||
We might add some more involved statistical analysis
|
||||
|
||||
project_pca_natural_images
|
||||
17) project_pca_natural_images
|
||||
medium
|
||||
Make a solution (->Lukas)
|
||||
|
||||
project_photoreceptor
|
||||
18) project_photoreceptor
|
||||
OK, simple
|
||||
|
||||
project_populationvector
|
||||
19) project_populationvector
|
||||
difficult
|
||||
OK
|
||||
|
||||
project_qvalues
|
||||
20) project_power_analysis
|
||||
medium
|
||||
|
||||
21) project_qvalues
|
||||
-
|
||||
Interesting! But needs solution.
|
||||
|
||||
project_random_walk
|
||||
22) project_random_walk
|
||||
simple-medium
|
||||
|
||||
project_serialcorrelation
|
||||
23) project_serialcorrelation
|
||||
OK, simple-medium
|
||||
|
||||
project_spectra
|
||||
24) project_shorttermpotentiation
|
||||
Write questions
|
||||
|
||||
25) project_spectra
|
||||
-
|
||||
Needs improvements and a solution
|
||||
|
||||
project_stimulus_reconstruction
|
||||
26) project_stimulus_reconstruction
|
||||
OK, difficult
|
||||
|
||||
project_vector_strength
|
||||
27) project_vector_strength
|
||||
OK, medium-difficult
|
||||
|
||||
project_power_analysis
|
||||
medium
|
||||
|
||||
Marius monkey data:
|
||||
medium-difficult
|
||||
47
projects/project_activation_curve/activation_curve.tex
Normal file
47
projects/project_activation_curve/activation_curve.tex
Normal file
@ -0,0 +1,47 @@
|
||||
\documentclass[a4paper,12pt,pdftex]{exam}
|
||||
|
||||
\newcommand{\ptitle}{Activation curve}
|
||||
\input{../header.tex}
|
||||
\firstpagefooter{Supervisor: Lukas Sonnenberg}{}%
|
||||
{email: lukas.sonnenberg@student.uni-tuebingen.de}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\input{../instructions.tex}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\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.
|
||||
|
||||
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.
|
||||
|
||||
|
||||
|
||||
\begin{questions}
|
||||
\question In the accompanying datasets you find recordings of cells with WT or A1622D transfections. The cells were all clamped to -70mV for some time to bring all ion channels in the same closed states. They are activated by a step change in the command voltage to a value described in the "steps" vector. The corresponding recorded current (in pA) and time (in ms) traces are also saved in the files.
|
||||
|
||||
\begin{parts}
|
||||
\part Plot the current traces of a WT and a A1622D cell. Because the number of transfected channels can vary the peak values have little value. Normalize the curves accordingly (what kind of normalization would be appropriate?). Can you already spot differences between the cells?
|
||||
|
||||
\part \textbf{IV curve}: Find the peak values for each voltage step and plot them against the steps.
|
||||
|
||||
\part \textbf{Reversal potential}: Use the IV-curve to estimate the reversal potential of the sodium current. Consider a linear interpolation to increase the accuracy of your estimation.
|
||||
|
||||
\part \textbf{Activation curve}: The activation curve is a representation of the voltage dependence of the sodium conductivity. It is computed with a variation of Ohm's law:
|
||||
\begin{equation}
|
||||
g_{Na}(V) = \frac{I_{peak}}{V - V_{reversal}}
|
||||
\end{equation}
|
||||
|
||||
\part \textbf{Compare the two variants}: To compare WT and A1622D activation curves you should first parameterise your data. Fit a sigmoid curve
|
||||
\begin{equation}
|
||||
g_{Na}(V) = g_{max,Na} / ( 1 + e^{ - \frac{V-V_{1/2}}{k}} )
|
||||
\end{equation}
|
||||
to the activation curves. With $g_{max,Na}$ being the maximum conductivity, $V_{1/2}$ the half activation voltage and $k$ a slope factor. Now you can compare the two variants with a few simple parameters. What do the differences mean?
|
||||
|
||||
\part \textbf{BONUS question}: Take a good look at your raw data. What other differences can you see? How could you analyse these?
|
||||
|
||||
\end{parts}
|
||||
\end{questions}
|
||||
|
||||
\end{document}
|
||||
@ -16,11 +16,11 @@ def create_data():
|
||||
|
||||
|
||||
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)
|
||||
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]:
|
||||
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_ylabel('Weight y', 'kg')
|
||||
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',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(0.9,1.0),
|
||||
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]
|
||||
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)
|
||||
ax.plot(xx, c*xx**3.0, color=colors['red'], lw=2)
|
||||
ax.plot(xx, c*xx**3.0, **lsBm)
|
||||
for i in inxs :
|
||||
xx = [x[i], x[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):
|
||||
ax.set_xlabel('Squared error')
|
||||
@ -67,7 +67,7 @@ def plot_error_hist(ax, x, y, c):
|
||||
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=colors['orange'])
|
||||
ax.hist(errors, bins, **fsC)
|
||||
|
||||
|
||||
|
||||
|
||||
@ -16,11 +16,11 @@ if __name__ == "__main__":
|
||||
fig, ax = plt.subplots(figsize=cm_size(figure_width, 1.4*figure_height))
|
||||
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)
|
||||
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]:
|
||||
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_ylabel('Weight y', 'kg')
|
||||
ax.set_xlim(2, 4)
|
||||
|
||||
@ -39,9 +39,9 @@ def plot_mse(ax, x, y, c, cs):
|
||||
for i, cc in enumerate(ccs):
|
||||
mses[i] = np.mean((y-(cc*x**3.0))**2.0)
|
||||
|
||||
ax.plot(ccs, mses, colors['blue'], lw=2, zorder=10)
|
||||
ax.scatter(cs, ms, color=colors['red'], s=40, zorder=20)
|
||||
ax.scatter(cs[-1], ms[-1], color=colors['orange'], s=60, zorder=30)
|
||||
ax.plot(ccs, mses, zorder=10, **lsAm)
|
||||
ax.plot(cs[:12], ms[:12], zorder=20, **psB)
|
||||
ax.plot(cs[-1], ms[-1], zorder=30, **psC)
|
||||
for i in range(4):
|
||||
ax.annotate('',
|
||||
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))
|
||||
|
||||
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_ylabel('Mean squared error')
|
||||
ax.set_xlim(0, 10.5)
|
||||
ax.set_xlim(0, 12.5)
|
||||
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_yticklabels([])
|
||||
|
||||
|
||||
@ -4,8 +4,7 @@ from plotstyle import *
|
||||
|
||||
plain_style()
|
||||
|
||||
fig = plt.figure( figsize=(2.5,3.4) )
|
||||
ax = fig.add_subplot(1, 1, 1)
|
||||
fig, ax = plt.subplots(figsize=(2.5,3.4))
|
||||
|
||||
# parabula:
|
||||
x1 = -0.2
|
||||
@ -14,7 +13,7 @@ x = np.linspace(x1, x2, 200)
|
||||
y = x*x
|
||||
ax.set_xlim(x1, x2)
|
||||
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:
|
||||
x = np.asarray([0.1, 0.7])
|
||||
y = x*x
|
||||
@ -22,33 +21,33 @@ ax.set_xticks(x)
|
||||
ax.set_yticks(y)
|
||||
ax.set_xticklabels(['$x$','$x+\Delta x$'])
|
||||
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:
|
||||
ax.plot([x[0], x[0], x1],[-0.2, y[0], y[0]], '--k', lw=1, zorder=6)
|
||||
ax.plot([x[1], x[1], x1],[-0.2, y[1], y[1]], '--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]], zorder=6, **lsGrid)
|
||||
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)
|
||||
# slope triangle:
|
||||
ax.plot([x[0], x[1], x[1]],[y[0], y[0], y[1]], '-k', lw=2, zorder=7)
|
||||
ax.text(np.mean(x), y[0]-0.08, '$\Delta x$', ha='center', 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]-0.07, '$\Delta x$', 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)
|
||||
# secant line:
|
||||
m = np.diff(y)/np.diff(x)
|
||||
xl = [x1, x2]
|
||||
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:
|
||||
md = 2.0*x[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:
|
||||
for ml in np.linspace(md, m, 5)[1:] :
|
||||
yl = ml*(xl-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(xl, yl, c=colors['orange'], lw=2, zorder=3)
|
||||
ax.plot([xs], [xs*xs], zorder=3, **psC)
|
||||
ax.plot(xl, yl, zorder=3, **lsCm)
|
||||
|
||||
fig.subplots_adjust(**adjust_fs(fig, 0.5, 0.5, 1.4, 0.5))
|
||||
plt.savefig('derivative.pdf')
|
||||
|
||||
@ -14,7 +14,7 @@ def create_data():
|
||||
|
||||
|
||||
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_ylabel('Output y')
|
||||
ax.set_xlim(0, 120)
|
||||
@ -24,10 +24,10 @@ def plot_data(ax, x, y):
|
||||
|
||||
|
||||
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])
|
||||
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_ylabel('Output y')
|
||||
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):
|
||||
ax.scatter(x, y, marker='o', color=colors['blue'], s=40)
|
||||
ax.plot(x, y, **psA)
|
||||
xx = np.asarray([2, 118])
|
||||
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_ylabel('Output y')
|
||||
ax.set_xlim(0, 120)
|
||||
|
||||
@ -25,15 +25,15 @@ def plot_data(ax, x, y, m, n):
|
||||
xytext=(80, -50), 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=colors['blue'], s=10, zorder=0)
|
||||
ax.plot(x[:40], y[:40], zorder=0, **psAm)
|
||||
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])
|
||||
ax.plot(xx, m*xx+n, color=colors['red'], lw=2)
|
||||
ax.plot(xx, m*xx+n, **lsBm)
|
||||
for i in inxs :
|
||||
xx = [x[i], x[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):
|
||||
@ -51,7 +51,7 @@ def plot_error_hist(ax, x, y, m, n):
|
||||
xytext=(350, 20), textcoords='data', ha='left',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.2),
|
||||
connectionstyle="angle3,angleA=10,angleB=90") )
|
||||
ax.hist(errors, bins, color=colors['orange'])
|
||||
ax.hist(errors, bins, **fsD)
|
||||
|
||||
|
||||
|
||||
|
||||
@ -19,7 +19,7 @@ if __name__ == "__main__":
|
||||
spec = gridspec.GridSpec(nrows=1, ncols=2, width_ratios=[3, 1], wspace=0.08,
|
||||
**adjust_fs(fig, left=6.0))
|
||||
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_ylabel('Weight', 'kg')
|
||||
ax1.set_xlim(-10, 310)
|
||||
@ -29,10 +29,10 @@ if __name__ == "__main__":
|
||||
ax2 = fig.add_subplot(spec[0, 1])
|
||||
xx = np.arange(0.0, 350.0, 0.5)
|
||||
yy = st.norm.pdf(xx, mu, sigma)
|
||||
ax2.plot(yy, xx, color=colors['red'], lw=2)
|
||||
ax2.plot(yy, xx, **lsBm)
|
||||
bw = 20.0
|
||||
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_xlim(0, 0.012)
|
||||
ax2.set_xticks([0, 0.005, 0.01])
|
||||
|
||||
@ -24,32 +24,28 @@ if __name__ == "__main__":
|
||||
spec = gridspec.GridSpec(nrows=2, ncols=2, **adjust_fs(fig))
|
||||
|
||||
ax = fig.add_subplot(spec[0, 0])
|
||||
ax.plot(indices, x1, c=colors['blue'], lw=1, zorder=10)
|
||||
ax.scatter(indices, x1, c=colors['blue'], edgecolor='white', s=50, zorder=20)
|
||||
ax.plot(indices, x1, zorder=10, **lpsAm)
|
||||
ax.set_xlabel('Index')
|
||||
ax.set_ylabel('Random number')
|
||||
ax.set_xlim(-1.0, n+1.0)
|
||||
ax.set_ylim(-0.05, 1.05)
|
||||
|
||||
ax = fig.add_subplot(spec[0, 1])
|
||||
ax.plot(indices, x2, c=colors['blue'], lw=1, zorder=10)
|
||||
ax.scatter(indices, x2, c=colors['blue'], edgecolor='white', s=50, zorder=20)
|
||||
ax.plot(indices, x2, zorder=10, **lpsAm)
|
||||
ax.set_xlabel('Index')
|
||||
ax.set_ylabel('Random number')
|
||||
ax.set_xlim(-1.0, n+1.0)
|
||||
ax.set_ylim(-0.05, 1.05)
|
||||
|
||||
ax = fig.add_subplot(spec[1, 1])
|
||||
ax.plot(indices, x3, c=colors['blue'], lw=1, zorder=10)
|
||||
ax.scatter(indices, x3, c=colors['blue'], edgecolor='white', s=50, zorder=20)
|
||||
ax.plot(indices, x3, zorder=10, **lpsAm)
|
||||
ax.set_xlabel('Index')
|
||||
ax.set_ylabel('Random number')
|
||||
ax.set_xlim(-1.0, n+1.0)
|
||||
ax.set_ylim(-0.05, 1.05)
|
||||
|
||||
ax = fig.add_subplot(spec[1, 0])
|
||||
ax.plot(lags, corrs, c=colors['red'], lw=1, zorder=10)
|
||||
ax.scatter(lags, corrs, c=colors['red'], edgecolor='white', s=50, zorder=20)
|
||||
ax.plot(lags, corrs, zorder=10, **lpsBm)
|
||||
ax.set_xlabel('Lag')
|
||||
ax.set_ylabel('Serial correlation')
|
||||
ax.set_xlim(-maxl-0.5, maxl+0.5)
|
||||
|
||||
@ -23,8 +23,8 @@ if __name__ == "__main__":
|
||||
fig = plt.figure()
|
||||
spec = gridspec.GridSpec(nrows=1, ncols=2, **adjust_fs(fig, left=4.5))
|
||||
ax1 = fig.add_subplot(spec[0, 0])
|
||||
ax1.plot(xx, yy, **ls['A1'])
|
||||
ax1.scatter(x, y, c=colors['blue'], edgecolor='white', s=50)
|
||||
ax1.plot(x, y, **psAm)
|
||||
ax1.plot(xx, yy, **lsB)
|
||||
ax1.set_xlabel('Hair deflection', 'nm')
|
||||
ax1.set_ylabel('Conductance', 'nS')
|
||||
ax1.set_xlim(-20, 20)
|
||||
@ -35,10 +35,10 @@ if __name__ == "__main__":
|
||||
ax2 = fig.add_subplot(spec[0, 1])
|
||||
xg = np.linspace(-3.0, 3.01, 200)
|
||||
yg = st.norm.pdf(xg, 0.0, sigma)
|
||||
ax2.plot(xg, yg, **ls['A1'])
|
||||
ax2.plot(xg, yg, **lsB)
|
||||
bw = 0.25
|
||||
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_ylabel('Pdf', '1/nS')
|
||||
ax2.set_xlim(-2.5, 2.5)
|
||||
|
||||
@ -1,19 +1,14 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
from plotstyle import *
|
||||
|
||||
rng = np.random.RandomState(981)
|
||||
x = rng.randn( 40, 10 )
|
||||
|
||||
plt.xkcd()
|
||||
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')
|
||||
fig, ax = plt.subplots()
|
||||
ax.set_xlabel('Experiment')
|
||||
ax.set_ylabel('x')
|
||||
ax.set_ylim( -4.0, 4.0)
|
||||
ax.set_ylim(-4.0, 4.0)
|
||||
ax.annotate('Median',
|
||||
xy=(3.9, 0.0), xycoords='data',
|
||||
xytext=(3.5, -2.7), textcoords='data', ha='right',
|
||||
@ -39,8 +34,6 @@ ax.annotate('maximum',
|
||||
xytext=(4.9, 3.5), textcoords='data', ha='right',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(1.0,0.5),
|
||||
connectionstyle="angle3,angleA=0,angleB=120") )
|
||||
ax.boxplot( x, whis=100.0 )
|
||||
plt.tight_layout()
|
||||
ax.boxplot(x, whis=100.0)
|
||||
plt.savefig('boxwhisker.pdf')
|
||||
#plt.show()
|
||||
|
||||
|
||||
@ -1,33 +1,30 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
import matplotlib.gridspec as gridspec
|
||||
from plotstyle import *
|
||||
|
||||
plt.xkcd()
|
||||
fig = plt.figure( figsize=(6,4.6) )
|
||||
fig = plt.figure(figsize=cm_size(figure_width, 2.0*figure_height))
|
||||
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)
|
||||
n = 200
|
||||
for k, r in enumerate( [ 1.0, 0.6, 0.0, -0.9 ] ) :
|
||||
x = rng.randn( n )
|
||||
y = r*x + np.sqrt(1.0-r*r)*rng.randn( n )
|
||||
ax = fig.add_subplot( 2, 2, k+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.text( -2, 2.5, 'r=%.1f' % r )
|
||||
for k, r in enumerate([ 1.0, 0.6, 0.0, -0.9 ]) :
|
||||
x = rng.randn(n)
|
||||
y = r*x + np.sqrt(1.0-r*r)*rng.randn(n)
|
||||
ax = fig.add_subplot(spec[k//2, k%2])
|
||||
ax.text(-2, 2.5, 'r=%.1f' % r)
|
||||
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 :
|
||||
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 :
|
||||
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 :
|
||||
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_ylabel('y')
|
||||
ax.set_xlim( -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.set_xlim(-3.0, 3.0)
|
||||
ax.set_ylim(-3.0, 3.0)
|
||||
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.show()
|
||||
|
||||
@ -1,5 +1,6 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
from plotstyle import *
|
||||
|
||||
# data:
|
||||
rng = np.random.RandomState(981)
|
||||
@ -14,39 +15,31 @@ gauss = np.exp(-0.5*xx*xx)/np.sqrt(2.0*np.pi)
|
||||
gausscdf = np.cumsum(gauss)*dx
|
||||
|
||||
# plot:
|
||||
plt.xkcd()
|
||||
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' )
|
||||
fig, ax = plt.subplots(figsize=cm_size(figure_width, 1.2*figure_height))
|
||||
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( 'F(x)' )
|
||||
ax.set_xticks(np.arange(-3.0, 3.1, 1.0))
|
||||
ax.set_ylabel('F(x)')
|
||||
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]
|
||||
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(0.15, 0.25, 'median at %.2f' % med)
|
||||
|
||||
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(0.8, 0.5, '3. quartile at %.2f' % q3)
|
||||
|
||||
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(-0.9, 0.05, '-1')
|
||||
|
||||
ax.plot(xx, gausscdf, '-', color='#0000ff', lw=2, zorder=-1)
|
||||
ax.plot(xs, cdf, '-', color='#cc0000', lw=4, zorder=-1)
|
||||
ax.plot([-3.2, 3.2], [1.0, 1.0], '--', color='k', lw=2, zorder=-10)
|
||||
ax.plot(xx, gausscdf, zorder=-1, **lsAm)
|
||||
ax.plot(xs, cdf, zorder=-1, **lsB)
|
||||
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' )
|
||||
#plt.show()
|
||||
fig.savefig('cumulative.pdf')
|
||||
|
||||
@ -1,39 +1,30 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
from plotstyle import *
|
||||
|
||||
# roll the die:
|
||||
rng = np.random.RandomState(57281)
|
||||
x1 = rng.random_integers( 1, 6, 100 )
|
||||
x2 = rng.random_integers( 1, 6, 500 )
|
||||
x1 = rng.random_integers(1, 6, 100)
|
||||
x2 = rng.random_integers(1, 6, 500)
|
||||
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) )
|
||||
ax = fig.add_subplot( 1, 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_xlim(0, 7)
|
||||
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' )
|
||||
#plt.show()
|
||||
ax2.set_xlim(0, 7)
|
||||
ax2.set_xticks(range(1, 7))
|
||||
ax2.set_xlabel('x')
|
||||
ax2.set_ylim(0, 0.23)
|
||||
ax2.set_ylabel('Probability')
|
||||
ax2.plot([0.2, 6.8], [1.0/6.0, 1.0/6.0], zorder=-10, **lsAm)
|
||||
ax2.hist([x2, x1], bins, normed=True, zorder=-5, **fs)
|
||||
fig.savefig('diehistograms.pdf')
|
||||
|
||||
@ -1,33 +1,24 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
import matplotlib.gridspec as gridspec
|
||||
from scipy.stats import gaussian_kde
|
||||
from plotstyle import *
|
||||
|
||||
#rng = np.random.RandomState(981)
|
||||
#data = rng.randn(40, 1) + 4.0
|
||||
rng = np.random.RandomState(1981)
|
||||
data = rng.gamma(1.0, 1.5, 40) + 1.0
|
||||
data = data[data<7.5]
|
||||
xpos = 0.08
|
||||
ypos = 0.15
|
||||
width = 0.65
|
||||
height = 0.8
|
||||
barwidth = 0.8
|
||||
scatterpos = 1.0
|
||||
barpos = 2.5
|
||||
boxpos = 4.0
|
||||
|
||||
plt.xkcd()
|
||||
fig = plt.figure( figsize=(6,3.4) )
|
||||
fig = plt.figure(figsize=cm_size(figure_width, 1.1*figure_height))
|
||||
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.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([])
|
||||
ax = fig.add_subplot(spec[0, 0])
|
||||
wh = ax.boxplot( data, positions=[boxpos], widths=[barwidth], whis=100.0, patch_artist=True)
|
||||
wh['medians'][0].set_linewidth(4)
|
||||
wh['whiskers'][0].set_linewidth(2)
|
||||
@ -49,7 +40,7 @@ ax.annotate('maximum',
|
||||
connectionstyle="angle3,angleA=0,angleB=120") )
|
||||
ax.annotate('3. quartile',
|
||||
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),
|
||||
connectionstyle="angle3,angleA=0,angleB=120") )
|
||||
ax.annotate('median',
|
||||
@ -57,34 +48,28 @@ ax.annotate('median',
|
||||
xytext=(boxpos+0.1*barwidth, 4.2), textcoords='data', ha='left',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(0.8,0.0),
|
||||
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 = fig.add_subplot(spec[0, 0])
|
||||
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_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:
|
||||
kernel = gaussian_kde(data)
|
||||
x = kernel(data)
|
||||
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)
|
||||
barstd = np.std(data)
|
||||
ew = 0.2
|
||||
ax.bar([barpos-0.5*barwidth], [barmean], barwidth, color='#FFCC00')
|
||||
eargs = {'color': 'k', 'lw': 2}
|
||||
ax.plot([barpos, barpos], [barmean-barstd, barmean+barstd], **eargs)
|
||||
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], **eargs)
|
||||
ax.bar([barpos-0.5*barwidth], [barmean], barwidth, **fsC)
|
||||
ax.plot([barpos, barpos], [barmean-barstd, barmean+barstd], **lsMarker)
|
||||
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], **lsMarker)
|
||||
ax.annotate('mean',
|
||||
xy=(barpos-0.4*barwidth, 2.7), xycoords='data',
|
||||
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),
|
||||
connectionstyle="angle3,angleA=-60,angleB=80") )
|
||||
|
||||
ax = fig.add_axes([xpos+width+0.03, ypos, 0.98-(xpos+width+0.03), height])
|
||||
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 = fig.add_subplot(spec[0, 1])
|
||||
ax.set_yticklabels([])
|
||||
ax.set_ylim( 0.0, 8.0)
|
||||
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
|
||||
bins = np.arange(0, 8.0+bw, bw)
|
||||
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.show()
|
||||
|
||||
|
||||
@ -1,9 +1,11 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
import matplotlib.gridspec as gridspec
|
||||
from plotstyle import *
|
||||
|
||||
# normal distribution:
|
||||
rng = np.random.RandomState(6281)
|
||||
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)
|
||||
r = rng.randn(100)
|
||||
|
||||
@ -30,54 +32,40 @@ def kerneldensity(data, xmin, xmax, sigma=1.0) :
|
||||
kd /= len(data)
|
||||
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()
|
||||
|
||||
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 = fig.add_subplot(spec[0, 0])
|
||||
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( 'p(x)' )
|
||||
ax.set_xticks(np.arange(-3.0, 3.1, 1.0))
|
||||
ax.set_ylabel('p(x)')
|
||||
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.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.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 = fig.add_subplot(spec[1, 0])
|
||||
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( 'p(x)' )
|
||||
ax.set_xticks(np.arange(-3.0, 3.1, 1.0))
|
||||
ax.set_ylabel('p(x)')
|
||||
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.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.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 = fig.add_subplot(spec[:, 1])
|
||||
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_xticks(np.arange(-3.0, 3.1, 1.0))
|
||||
ax.set_ylabel('Prob. density p(x)')
|
||||
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)
|
||||
ax.fill_between(xx, 0.0, kd, color='#FF9900', zorder=-5)
|
||||
ax.plot(xx, kd, '-', lw=3, color='#CC0000', zorder=-1)
|
||||
ax.fill_between(xx, 0.0, kd, zorder=-5, **fsDs)
|
||||
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' )
|
||||
#plt.show()
|
||||
fig.savefig('kerneldensity.pdf')
|
||||
|
||||
|
||||
@ -1,78 +1,66 @@
|
||||
import numpy as np
|
||||
import scipy.stats as st
|
||||
import matplotlib.pyplot as plt
|
||||
from plotstyle import *
|
||||
|
||||
# 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)
|
||||
|
||||
plt.xkcd()
|
||||
fig = plt.figure( figsize=(6, 2.8) )
|
||||
ax = fig.add_subplot(1, 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_ylabel( 'Prob. density p(x)' )
|
||||
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',
|
||||
fig, (ax1, ax2) = plt.subplots(1, 2)
|
||||
fig.subplots_adjust(**adjust_fs(bottom=2.7, top=0.1))
|
||||
ax1.set_xlabel('x')
|
||||
ax1.set_ylabel('Prob. density p(x)')
|
||||
ax1.set_ylim(0.0, 0.46)
|
||||
ax1.set_yticks(np.arange(0.0, 0.45, 0.1))
|
||||
ax1.text(-1.0, 0.06, '50%', ha='center')
|
||||
ax1.text(+1.0, 0.06, '50%', ha='center')
|
||||
ax1.annotate('Median\n= mean',
|
||||
xy=(0.1, 0.3), xycoords='data',
|
||||
xytext=(1.2, 0.35), textcoords='data', ha='left',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.2),
|
||||
connectionstyle="angle3,angleA=10,angleB=40") )
|
||||
ax.annotate('Mode',
|
||||
connectionstyle="angle3,angleA=10,angleB=40"))
|
||||
ax1.annotate('Mode',
|
||||
xy=(-0.1, 0.4), xycoords='data',
|
||||
xytext=(-2.5, 0.43), textcoords='data', ha='left',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.2),
|
||||
connectionstyle="angle3,angleA=10,angleB=120") )
|
||||
ax.fill_between( x[x<0], 0.0, g[x<0], color='#ffcc00' )
|
||||
ax.fill_between( x[x>0], 0.0, g[x>0], color='#99ff00' )
|
||||
ax.plot(x, g, 'b', lw=4)
|
||||
ax.plot([0.0, 0.0], [0.0, 0.45], 'k', lw=2 )
|
||||
connectionstyle="angle3,angleA=10,angleB=120"))
|
||||
ax1.fill_between(x[x<0], 0.0, g[x<0], **fsCs)
|
||||
ax1.fill_between(x[x>0], 0.0, g[x>0], **fsFs)
|
||||
ax1.plot(x, g, **lsA)
|
||||
ax1.plot([0.0, 0.0], [0.0, 0.45], **lsMarker)
|
||||
|
||||
# normal distribution:
|
||||
x = np.arange( 0.0, 6.0, 0.01 )
|
||||
# gamma distribution:
|
||||
x = np.arange(0.0, 6.0, 0.01)
|
||||
shape = 2.0
|
||||
g = st.gamma.pdf(x, shape)
|
||||
m = st.gamma.median(shape)
|
||||
gm = st.gamma.mean(shape)
|
||||
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_ylabel( 'Prob. density p(x)' )
|
||||
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',
|
||||
ax2.set_xlabel('x')
|
||||
ax2.set_ylabel('Prob. density p(x)')
|
||||
ax2.set_ylim(0.0, 0.46)
|
||||
ax2.set_yticks(np.arange(0.0, 0.45, 0.1))
|
||||
ax2.text(m-0.8, 0.06, '50%', ha='center')
|
||||
ax2.text(m+1.2, 0.06, '50%', ha='center')
|
||||
ax2.annotate('Median',
|
||||
xy=(m+0.1, 0.2), xycoords='data',
|
||||
xytext=(m+1.6, 0.25), textcoords='data', ha='left',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
|
||||
connectionstyle="angle3,angleA=30,angleB=70") )
|
||||
ax.annotate('Mean',
|
||||
connectionstyle="angle3,angleA=30,angleB=70"))
|
||||
ax2.annotate('Mean',
|
||||
xy=(gm, 0.01), xycoords='data',
|
||||
xytext=(gm+1.8, 0.15), textcoords='data', ha='left',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
|
||||
connectionstyle="angle3,angleA=0,angleB=90") )
|
||||
ax.annotate('Mode',
|
||||
connectionstyle="angle3,angleA=0,angleB=90"))
|
||||
ax2.annotate('Mode',
|
||||
xy=(1.0, 0.38), xycoords='data',
|
||||
xytext=(1.8, 0.42), textcoords='data', ha='left',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
|
||||
connectionstyle="angle3,angleA=0,angleB=70") )
|
||||
ax.fill_between( x[x<m], 0.0, g[x<m], color='#ffcc00' )
|
||||
ax.fill_between( x[x>m], 0.0, g[x>m], color='#99ff00' )
|
||||
ax.plot(x, g, 'b', lw=4)
|
||||
ax.plot([m, m], [0.0, 0.38], 'k', lw=2 )
|
||||
#ax.plot([gm, gm], [0.0, 0.42], 'k', lw=2 )
|
||||
connectionstyle="angle3,angleA=0,angleB=70"))
|
||||
ax2.fill_between(x[x<m], 0.0, g[x<m], **fsCs)
|
||||
ax2.fill_between(x[x>m], 0.0, g[x>m], **fsFs)
|
||||
ax2.plot(x, g, **lsA)
|
||||
ax2.plot([m, m], [0.0, 0.38], **lsMarker)
|
||||
#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' )
|
||||
#plt.show()
|
||||
fig.savefig('median.pdf')
|
||||
|
||||
@ -1,42 +1,35 @@
|
||||
import numpy as np
|
||||
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
|
||||
x = np.random.randn( n )
|
||||
y = np.random.randn( n )
|
||||
rng = np.random.RandomState(3981)
|
||||
x = rng.randn(n)
|
||||
y = rng.randn(n)
|
||||
|
||||
z = x*x+0.2*y
|
||||
r =np.corrcoef(x,z)[0,1]
|
||||
ax = fig.add_subplot( 1, 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.text( 0, 4.0, 'r=%.1f' % r, ha='center' )
|
||||
ax.text( 0, 6, r'$y = x^2+\xi/5$', ha='center' )
|
||||
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 )
|
||||
ax1.text(0, 4.0, 'r=%.1f' % r, ha='center')
|
||||
ax1.text(0, 5.6, r'$y = x^2+\xi/5$', ha='center')
|
||||
ax1.set_xlabel('x')
|
||||
ax1.set_ylabel('y')
|
||||
ax1.set_xlim(-3.0, 3.0)
|
||||
ax1.set_ylim(-0.5, 6.0)
|
||||
ax1.plot(x, z, **psAm)
|
||||
|
||||
z = 0.5*x*y
|
||||
r =np.corrcoef(x,z)[0,1]
|
||||
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.text( 0, 1.5, 'r=%.1f' % r, ha='center' )
|
||||
ax.text( 0, 3, r'$y = x \cdot \xi/2$', ha='center' )
|
||||
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 )
|
||||
ax2.text(0, 1.5, 'r=%.1f' % r, ha='center')
|
||||
ax2.text(0, 2.8, r'$y = x \cdot \xi/2$', ha='center')
|
||||
ax2.set_xlabel('x')
|
||||
ax2.set_ylabel('y')
|
||||
ax2.set_xlim(-3.0, 3.0)
|
||||
ax2.set_ylim(-3.0, 3.0)
|
||||
ax2.plot(x, z, **psAm)
|
||||
|
||||
plt.tight_layout()
|
||||
plt.savefig('nonlincorrelation.pdf')
|
||||
#plt.show()
|
||||
|
||||
@ -1,44 +1,31 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
from plotstyle import *
|
||||
|
||||
# normal distribution:
|
||||
rng = np.random.RandomState(6281)
|
||||
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)
|
||||
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) )
|
||||
ax = fig.add_subplot( 1, 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_xlim(-3.2, 3.2)
|
||||
ax.set_xticks( np.arange( -3.0, 3.1, 1.0 ) )
|
||||
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')
|
||||
ax2.set_xlabel('x')
|
||||
ax2.set_xlim(-3.2, 3.2)
|
||||
ax2.set_xticks(np.arange(-3.0, 3.1, 1.0))
|
||||
ax2.set_ylabel('Probab. density p(x)')
|
||||
ax2.set_ylim(0.0, 0.44)
|
||||
ax2.set_yticks(np.arange(0.0, 0.41, 0.1))
|
||||
ax2.plot(x, g, zorder=-1, **lsA)
|
||||
ax2.hist(r, 5, normed=True, zorder=-10, **fsB)
|
||||
ax2.hist(r, 20, normed=True, zorder=-5, **fsC)
|
||||
|
||||
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' )
|
||||
#plt.show()
|
||||
fig.savefig('pdfhistogram.pdf')
|
||||
|
||||
|
||||
@ -1,36 +1,28 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
from plotstyle import *
|
||||
|
||||
# normal distribution:
|
||||
x = np.arange( -3.0, 5.0, 0.01 )
|
||||
x = np.arange(-3.0, 5.0, 0.01)
|
||||
g = np.exp(-0.5*x*x)/np.sqrt(2.0*np.pi)
|
||||
x1=0.0
|
||||
x2=1.0
|
||||
|
||||
plt.xkcd()
|
||||
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_ylabel( 'Probability density p(x)' )
|
||||
ax.set_ylim( 0.0, 0.46 )
|
||||
ax.set_yticks( np.arange( 0.0, 0.45, 0.1 ) )
|
||||
fig, ax = plt.subplots(figsize=cm_size(figure_width, 1.2*figure_height))
|
||||
ax.set_xlabel('x')
|
||||
ax.set_ylabel('Probability density p(x)')
|
||||
ax.set_ylim(0.0, 0.46)
|
||||
ax.set_yticks(np.arange(0.0, 0.45, 0.1))
|
||||
ax.annotate('Gaussian',
|
||||
xy=(-1.0, 0.28), xycoords='data',
|
||||
xytext=(-2.5, 0.35), textcoords='data', ha='left',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(0.5,0.0),
|
||||
connectionstyle="angle3,angleA=10,angleB=110") )
|
||||
connectionstyle="angle3,angleA=10,angleB=110"))
|
||||
ax.annotate('$P(0<x<1) = \int_0^1 p(x) \, dx$',
|
||||
xy=(0.6, 0.28), xycoords='data',
|
||||
xy=(0.5, 0.24), xycoords='data',
|
||||
xytext=(1.2, 0.4), textcoords='data', ha='left',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
|
||||
connectionstyle="angle3,angleA=10,angleB=80") )
|
||||
ax.fill_between( x[(x>x1)&(x<x2)], 0.0, g[(x>x1)&(x<x2)], color='#cc0000' )
|
||||
ax.plot(x,g, 'b', lw=4)
|
||||
#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' )
|
||||
#plt.show()
|
||||
connectionstyle="angle3,angleA=10,angleB=80"))
|
||||
ax.fill_between(x[(x>x1)&(x<x2)], 0.0, g[(x>x1)&(x<x2)], **fsBs)
|
||||
ax.plot(x,g, **lsA)
|
||||
fig.savefig('pdfprobabilities.pdf')
|
||||
|
||||
@ -1,22 +1,18 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
from plotstyle import *
|
||||
|
||||
# normal distribution:
|
||||
x = np.arange( -4.0, 4.0, 0.01 )
|
||||
g = np.exp(-0.5*x*x)/np.sqrt(2.0*np.pi)
|
||||
q = [ -0.67488, 0.0, 0.67488 ]
|
||||
|
||||
plt.xkcd()
|
||||
fig = plt.figure( figsize=(6,3.2) )
|
||||
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_ylabel( 'Probability density p(x)' )
|
||||
ax.set_ylim( 0.0, 0.46 )
|
||||
ax.set_yticks( np.arange( 0.0, 0.45, 0.1 ) )
|
||||
fig, ax = plt.subplots(figsize=cm_size(figure_width, 1.0*figure_height))
|
||||
fig.subplots_adjust(**adjust_fs(bottom=2.7, top=0.1))
|
||||
ax.set_xlabel('x')
|
||||
ax.set_ylabel('Probability density p(x)')
|
||||
ax.set_ylim(0.0, 0.46)
|
||||
ax.set_yticks(np.arange(0.0, 0.45, 0.1))
|
||||
ax.text(-1.2, 0.1, '25%', ha='center' )
|
||||
ax.text(-0.35, 0.1, '25%', ha='center' )
|
||||
ax.text(+0.35, 0.1, '25%', ha='center' )
|
||||
@ -36,15 +32,12 @@ ax.annotate('Median',
|
||||
xytext=(1.6, 0.35), textcoords='data', ha='left',
|
||||
arrowprops=dict(arrowstyle="->", relpos=(0.0,0.5),
|
||||
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])&(x<q[1])], 0.0, g[(x>q[0])&(x<q[1])], color='#ff0000' )
|
||||
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[2]], 0.0, g[x>q[2]], color='#ffff66' )
|
||||
ax.plot(x,g, 'b', lw=4)
|
||||
ax.plot([0.0, 0.0], [0.0, 0.45], 'k', lw=2 )
|
||||
ax.plot([q[0], q[0]], [0.0, 0.4], 'k', lw=2 )
|
||||
ax.plot([q[2], q[2]], [0.0, 0.4], 'k', lw=2 )
|
||||
plt.subplots_adjust(left=0.1, right=0.98, bottom=0.15, top=0.98, wspace=0.4, hspace=0.0)
|
||||
#plt.tight_layout()
|
||||
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])], **fsBs)
|
||||
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]], **fsEs)
|
||||
ax.plot(x,g, **lsA)
|
||||
ax.plot([0.0, 0.0], [0.0, 0.45], **lsMarker)
|
||||
ax.plot([q[0], q[0]], [0.0, 0.4], **lsMarker)
|
||||
ax.plot([q[2], q[2]], [0.0, 0.4], **lsMarker)
|
||||
fig.savefig( 'quartile.pdf' )
|
||||
#plt.show()
|
||||
|
||||
@ -115,17 +115,9 @@ function \mcode{median()} computes the median.
|
||||
writing reliable code!
|
||||
\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
|
||||
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
|
||||
distribution of the data in more detail. The 3$^{\rm rd}$ quartile
|
||||
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
|
||||
set (\figref{displayunivariatedatafig} (3)).
|
||||
|
||||
\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}
|
||||
% \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}
|
||||
|
||||
% \begin{exercise}{boxwhisker.m}{}
|
||||
% 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}
|
||||
(\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}
|
||||
In the frequentist interpretation of probability, the
|
||||
\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.
|
||||
|
||||
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
|
||||
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
|
||||
@ -280,7 +280,7 @@ inverse of the unit of the data values --- hence the name ``density''.
|
||||
\end{figure}
|
||||
|
||||
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:
|
||||
\[ P(x_1 < x < x2) = \int\limits_{x_1}^{x_2} p(x) \, dx \; . \]
|
||||
Because the probability to get any value $x$ is one, the integral of
|
||||
the probability density over the whole real axis must be one:
|
||||
@ -329,7 +329,7 @@ values fall within each bin (\figref{pdfhistogramfig} left).
|
||||
observe?
|
||||
\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
|
||||
integral equals one. While histograms of categorical data are
|
||||
normalized such that their sum equals one, here we need to integrate
|
||||
@ -343,7 +343,7 @@ and the
|
||||
\[ p(x_i) = \frac{n_i}{A} = \frac{n_i}{\Delta x \sum_{i=1}^N n_i} =
|
||||
\frac{n_i}{N \Delta x} \; .\]
|
||||
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
|
||||
compared with other distributions and in particular with theoretical
|
||||
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
|
||||
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
|
||||
shape histogram depends on the exact position of its bins
|
||||
(\figref{kerneldensityfig} left).
|
||||
shape of the resulting histogram depends on the exact position of its
|
||||
bins (\figref{kerneldensityfig} left).
|
||||
|
||||
\begin{figure}[t]
|
||||
\includegraphics[width=1\textwidth]{kerneldensity}
|
||||
\titlecaption{\label{kerneldensityfig} Kernel densities.}{Left: The
|
||||
histogram-based estimation of the probability density is dependent
|
||||
on the position of the bins. In the bottom plot the bins have
|
||||
\titlecaption{\label{kerneldensityfig} Kernel densities.}{The
|
||||
histogram-based estimation of the probability density depends on
|
||||
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
|
||||
result details of the probability density look different. Look,
|
||||
for example, at the height of the largest bin. Right: In contrast,
|
||||
a kernel density is uniquely defined for a given kernel width
|
||||
(here Gaussian kernels with standard deviation of $\sigma=0.2$).}
|
||||
for example, at the height of the largest bin. In contrast, a
|
||||
kernel density is uniquely defined for a given kernel width
|
||||
(right, Gaussian kernels with standard deviation of
|
||||
$\sigma=0.2$).}
|
||||
\end{figure}
|
||||
|
||||
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.
|
||||
\end{exercise}
|
||||
|
||||
\newpage
|
||||
\section{Correlations}
|
||||
|
||||
Until now we described properties of univariate data sets. In
|
||||
|
||||
Reference in New Issue
Block a user