Set up

In [1]:
import pylab
pylab.rcParams['xtick.major.pad']='8'
pylab.rcParams['ytick.major.pad']='8'
#import matplotlib.gridspec as gridspec
from matplotlib import rc
rc('text', usetex=False)
rc('font', family='serif')
In [2]:
from os import listdir
files = listdir('.')
if 'blackouts.txt' not in files:
    import urllib
    urllib.urlretrieve('https://raw.github.com/jeffalstott/powerlaw/master/manuscript/blackouts.txt', 'blackouts.txt')
if 'words.txt' not in files:
    import urllib
    urllib.urlretrieve('https://raw.github.com/jeffalstott/powerlaw/master/manuscript/words.txt', 'words.txt')
if 'worm.txt' not in files:
    import urllib
    urllib.urlretrieve('https://raw.github.com/jeffalstott/powerlaw/master/manuscript/worm.txt', 'worm.txt')
In [3]:
from numpy import genfromtxt
blackouts = genfromtxt('blackouts.txt')#/10**3
words = genfromtxt('words.txt')
worm = genfromtxt('worm.txt')
worm = worm[worm>0]
In [5]:
def plot_basics(data, data_inst, fig, units):
    from powerlaw import plot_pdf, Fit, pdf
    annotate_coord = (-.4, .95)
    ax1 = fig.add_subplot(n_graphs,n_data,data_inst)
    plot_pdf(data[data>0], ax=ax1, linear_bins=True, color='r', linewidth=.5)
    x, y = pdf(data, linear_bins=True)
    ind = y>0
    y = y[ind]
    x = x[:-1]
    x = x[ind]
    ax1.scatter(x, y, color='r', s=.5)
    plot_pdf(data[data>0], ax=ax1, color='b', linewidth=2)
    from pylab import setp
    setp( ax1.get_xticklabels(), visible=False)
    #ax1.set_xticks(ax1.get_xticks()[::2])
    ax1.set_yticks(ax1.get_yticks()[::2])
    locs,labels = yticks()
    #yticks(locs, map(lambda x: "%.0f" % x, log10(locs)))
    if data_inst==1:
        ax1.annotate("A", annotate_coord, xycoords="axes fraction", fontsize=14)

    
    from mpl_toolkits.axes_grid.inset_locator import inset_axes
    ax1in = inset_axes(ax1, width = "30%", height = "30%", loc=3)
    ax1in.hist(data, normed=True, color='b')
    ax1in.set_xticks([])
    ax1in.set_yticks([])

    
    ax2 = fig.add_subplot(n_graphs,n_data,n_data+data_inst, sharex=ax1)
    plot_pdf(data, ax=ax2, color='b', linewidth=2)
    fit = Fit(data, xmin=1, discrete=True)
    fit.power_law.plot_pdf(ax=ax2, linestyle=':', color='g')
    p = fit.power_law.pdf()
    #ax2.set_ylim(min(p), max(p))
    ax2.set_xlim(ax1.get_xlim())
    
    fit = Fit(data, discrete=True)
    fit.power_law.plot_pdf(ax=ax2, linestyle='--', color='g')
    from pylab import setp
    setp( ax2.get_xticklabels(), visible=False)
    #ax2.set_xticks(ax2.get_xticks()[::2])
    if ax2.get_ylim()[1] >1:
        ax2.set_ylim(ax2.get_ylim()[0], 1)
    
    ax2.set_yticks(ax2.get_yticks()[::2])
    #locs,labels = yticks()
    #yticks(locs, map(lambda x: "%.0f" % x, log10(locs)))
    if data_inst==1:
       ax2.annotate("B", annotate_coord, xycoords="axes fraction", fontsize=14)        
       ax2.set_ylabel(r"$p(X)$")# (10^n)")
        
    ax3 = fig.add_subplot(n_graphs,n_data,n_data*2+data_inst)#, sharex=ax1)#, sharey=ax2)
    fit.power_law.plot_pdf(ax=ax3, linestyle='--', color='g')
    fit.exponential.plot_pdf(ax=ax3, linestyle='--', color='r')
    fit.plot_pdf(ax=ax3, color='b', linewidth=2)
    
    #p = fit.power_law.pdf()
    ax3.set_ylim(ax2.get_ylim())
    ax3.set_yticks(ax3.get_yticks()[::2])
    ax3.set_xlim(ax1.get_xlim())
    
    #locs,labels = yticks()
    #yticks(locs, map(lambda x: "%.0f" % x, log10(locs)))
    if data_inst==1:
        ax3.annotate("C", annotate_coord, xycoords="axes fraction", fontsize=14)

    #if ax2.get_xlim()!=ax3.get_xlim():
    #    zoom_effect01(ax2, ax3, ax3.get_xlim()[0], ax3.get_xlim()[1])
    ax3.set_xlabel(units)

Figure 1

In [6]:
n_data = 3
n_graphs = 4
f = figure(figsize=(8,11))

data = words
data_inst = 1
units = 'Word Frequency'
plot_basics(data, data_inst, f, units)

data_inst = 2
#data = city
#units = 'City Population'
data = worm
units = 'Neuron Connections'
plot_basics(data, data_inst, f, units)

data = blackouts
data_inst = 3
units = 'Population Affected\nby Blackouts'
plot_basics(data, data_inst, f, units)

f.subplots_adjust(left=None, bottom=None, right=None, top=None, wspace=.3, hspace=.2)
f.savefig('FigWorkflow.eps', bbox_inches='tight')
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
In [95]:
blackouts = blackouts/10**3

Introduction

In [8]:
data = blackouts
####
import powerlaw
fit = powerlaw.Fit(data)
fit.power_law.alpha
fit.power_law.sigma
fit.distribution_compare('power_law', 'exponential')
Calculating best minimal value for power law fit
Out[8]:
(12.754562675882063, 0.1522925560442657)

Basic Methods

Visualization

PDF Linear vs Logarithmic Bins

In [9]:
data = words
####
figPDF = powerlaw.plot_pdf(data, color='b')
powerlaw.plot_pdf(data, linear_bins=True, color='r', ax=figPDF)
####
figPDF.set_ylabel(r"$p(X)$")
figPDF.set_xlabel(r"Word Frequency")
savefig('FigPDF.eps', bbox_inches='tight')

Figure 2

In [10]:
data = words
fit = powerlaw.Fit(data, discrete=True)
####
figCCDF = fit.plot_pdf(color='b', linewidth=2)
fit.power_law.plot_pdf(color='b', linestyle='--', ax=figCCDF)
fit.plot_ccdf(color='r', linewidth=2, ax=figCCDF)
fit.power_law.plot_ccdf(color='r', linestyle='--', ax=figCCDF)
####
figCCDF.set_ylabel(r"$p(X)$,  $p(X\geq x)$")
figCCDF.set_xlabel(r"Word Frequency")
savefig('FigCCDF.eps', bbox_inches='tight')
Calculating best minimal value for power law fit
In [11]:
data = blackouts
fit = powerlaw.Fit(data)
###
x, y = fit.cdf()
bin_edges, probability = fit.pdf()
y = fit.lognormal.cdf(data=[300,350])
y = fit.lognormal.pdf()
Calculating best minimal value for power law fit

Identifying the Scaling Range

In [12]:
data = blackouts
####
import powerlaw
fit = powerlaw.Fit(data)
fit.xmin
fit.fixed_xmin
fit.alpha
fit.D
fit = powerlaw.Fit(data, xmin=1.0)
fit.xmin
fit.fixed_xmin
fit.alpha
fit.D
Calculating best minimal value for power law fit
Out[12]:
0.37601504850371759
In [13]:
data = blackouts
####
fit = powerlaw.Fit(data, xmin=(250.0, 300.0))
fit.fixed_xmin
fit.given_xmin
fit.xmin
Calculating best minimal value for power law fit
Out[13]:
272.0
In [14]:
data = blackouts
fit = powerlaw.Fit(data)
####
fit = powerlaw.Fit(data, xmax=10000.0)
fit.xmax
fit.fixed_xmax
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Out[14]:
True

Figure 3

In [9]:
data = words
#FigCCDFmax = powerlaw.plot_ccdf(data, linewidth=3)
fit = powerlaw.Fit(data, discrete=True, xmax=None)
FigCCDFmax = fit.plot_ccdf(color='b', label=r"Empirical, no $x_{max}$")
fit.power_law.plot_ccdf(color='b', linestyle='--', ax=FigCCDFmax, label=r"Fit, no $x_{max}$")
fit = powerlaw.Fit(data, discrete=True, xmax=1000)
fit.plot_ccdf(color='r', label=r"Empirical, $x_{max}=1000$")
fit.power_law.plot_ccdf(color='r', linestyle='--', ax=FigCCDFmax, label=r"Fit, $x_{max}=1000$")
#x, y = powerlaw.ccdf(data, xmax=max(data))
#fig1.plot(x,y)
####
FigCCDFmax.set_ylabel(r"$p(X\geq x)$")
FigCCDFmax.set_xlabel(r"Word Frequency")
handles, labels = FigCCDFmax.get_legend_handles_labels()
leg = FigCCDFmax.legend(handles, labels, loc=3)
leg.draw_frame(False)
savefig('FigCCDFmax.eps', bbox_inches='tight')
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit

Continuous vs. Discrete Data

In [16]:
data = blackouts
fit = powerlaw.Fit(data)
####
fit = powerlaw.Fit(data, xmin=230.0)
fit.discrete
fit = powerlaw.Fit(data, xmin=230.0, discrete=True)
fit.discrete 
Calculating best minimal value for power law fit
Out[16]:
True

Comparing Candidate Distributions

In [17]:
data = blackouts
fit = powerlaw.Fit(data)
####
fit.power_law
fit.power_law.alpha
fit.power_law.parameter1
fit.power_law.parameter1_name
fit.lognormal.mu
fit.lognormal.parameter1_name
fit.lognormal.parameter2_name
fit.lognormal.parameter3_name == None
Calculating best minimal value for power law fit
Out[17]:
True
In [18]:
data = blackouts
####
fit = powerlaw.Fit(data)
R, p = fit.distribution_compare('power_law', 'exponential', normalized_ratio=True)
print R, p
Calculating best minimal value for power law fit
1.43148048496 0.152292556044

Generative Mechanisms

In [5]:
data = worm
fit = powerlaw.Fit(data, discrete=True)
####
fit.distribution_compare('power_law', 'exponential')
fit.distribution_compare('power_law', 'truncated_power_law')
Calculating best minimal value for power law fit
Assuming nested distributions
Out[5]:
(-0.081336372762826459, 0.68670761175575712)
In [57]:
data = worm
fit = powerlaw.Fit(data, discrete=True)
####
fit.distribution_compare('power_law', 'exponential')
fit.distribution_compare('power_law', 'truncated_power_law')
Calculating best minimal value for power law fit
Assuming nested distributions
Out[57]:
(-422.43094963299131, 0.0)
In [6]:
data = words
fit = powerlaw.Fit(data, discrete=True)
####
print fit.distribution_compare('power_law', 'exponential', normalized_ratio=True)
print fit.distribution_compare('power_law', 'truncated_power_law')
Calculating best minimal value for power law fit
(9.1359147187770002, 6.4856142413795328e-20)
Assuming nested distributions
(-0.91712308337346782, 0.17562683168707593)
In [68]:
print fit.distribution_compare('power_law', 'truncated_power_law')
Assuming nested distributions
(-13026.994760437048, 0.0)

Figure 4

In [20]:
data = words
fit = powerlaw.Fit(data, discrete=True)
####
fit.distribution_compare('power_law', 'lognormal')
fig = fit.plot_ccdf(linewidth=3, label='Empirical Data')
fit.power_law.plot_ccdf(ax=fig, color='r', linestyle='--', label='Power law fit')
fit.lognormal.plot_ccdf(ax=fig, color='g', linestyle='--', label='Lognormal fit')
####
fig.set_ylabel(r"$p(X\geq x)$")
fig.set_xlabel(r"Word Frequency")
handles, labels = fig.get_legend_handles_labels()
fig.legend(handles, labels, loc=3)
savefig('FigLognormal.eps', bbox_inches='tight')
Calculating best minimal value for power law fit

Creating Simulated Data

In [127]:
empirical_data = blackouts
####
fit = powerlaw.Fit(empirical_data)
simulated_data = fit.power_law.generate_random(10000)

theoretical_distribution = powerlaw.Power_Law(xmin=5.0, parameters=[2.5])
simulated_data = theoretical_distribution.generate_random(10000)
Calculating best minimal value for power law fit
In [42]:
theoretical_distribution = powerlaw.Power_Law(xmin=5.0, parameters=[2.5])
simulated_data = theoretical_distribution.generate_random(10000)
####
fit = powerlaw.Fit(simulated_data)
fit.power_law.xmin, fit.power_law.alpha
Calculating best minimal value for power law fit
Out[42]:
(5.5253208595576631, 2.5298369451264975)
In [47]:
powerlaw.plot_pdf(simulated_data,linewidth=3)
fit.power_law.plot_pdf(simulated_data,linestyle='--',color='r')
Out[47]:
<matplotlib.axes.AxesSubplot at 0xd829dd0>

Advanced Considerations

Discrete Distribution Calculation and Estimation

In [22]:
data = blackouts
####
fit = powerlaw.Fit(data, discrete=True, estimate_discrete=True)
fit.power_law.alpha
fit.power_law.estimate_discrete
fit = powerlaw.Fit(data, discrete=True, estimate_discrete=False)
fit.power_law.alpha
fit.power_law.estimate_discrete
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Out[22]:
2.2691417084814285
In [23]:
data = blackouts
####
fit = powerlaw.Fit(data, discrete=True, xmin=230.0, xmax=9000, discrete_approximation='xmax')
fit.lognormal.mu
fit = powerlaw.Fit(data, discrete_approximation=100000, xmin=230.0, discrete=True)
fit.lognormal.mu
fit = powerlaw.Fit(data, discrete_approximation='round', xmin=230.0, discrete=True)
fit.lognormal.mu
Out[23]:
0.39905257607693184
In [125]:
theoretical_distribution = powerlaw.Power_Law(xmin=5.0, parameters=[2.5], discrete=True)
simulated_data = theoretical_distribution.generate_random(10000, estimate_discrete=True)
In [126]:
empirical_data = blackouts
####
theoretical_distributionibution = powerlaw.Power_Law(xmin=5.0, parameters=[2.5], discrete=True, estimate_discrete=False)
simulated_data = theoretical_distribution.generate_random(10000)

fit = powerlaw.Fit(empirical_data, discrete=True, estimate_discrete=True)
simulated_data = fit.power_law.generate_random(10000)
Calculating best minimal value for power law fit

Nested Distributions

In [7]:
data = blackouts
fit = powerlaw.Fit(data)
####
fit.distribution_compare('power_law', 'truncated_power_law')
fit.distribution_compare('exponential', 'stretched_exponential')
Calculating best minimal value for power law fit
Assuming nested distributions
Assuming nested distributions
Out[7]:
(-13.018342321459194, 3.3499122220614908e-07)

Restricted Parameter Range

In [24]:
data = blackouts
####
fit = powerlaw.Fit(data)
fit.power_law.alpha, fit.power_law.sigma, fit.xmin

fit = powerlaw.Fit(data, sigma_threshold=.1)
fit.power_law.alpha, fit.power_law.sigma, fit.xmin
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Out[24]:
(1.8833765811180314, 0.094168259953067143, 124.0)
In [123]:
parameter_range = {'alpha': [2.3, None], 'sigma': [None, .2]}
fit = powerlaw.Fit(data, parameter_range=parameter_range)
fit.power_law.alpha, fit.power_law.sigma, fit.xmin
Calculating best minimal value for power law fit
Out[123]:
(2.3000107113942141, 0.17069976919260452, 234.0)
In [124]:
parameter_range = lambda(self): self.sigma/self.alpha < .05
fit = powerlaw.Fit(data, parameter_range=parameter_range)
fit.power_law.alpha, fit.power_law.sigma, fit.xmin
Calculating best minimal value for power law fit
Out[124]:
(1.8833765811180314, 0.094168259953067143, 124.0)

Multiple Possible Fits

Figure 5

In [25]:
data = blackouts
fit = powerlaw.Fit(data, sigma_threshold=.1)
print fit.xmin, fit.D, fit.alpha
fit = powerlaw.Fit(data)
print fit.xmin, fit.D, fit.alpha
####
from matplotlib.pylab import plot
plot(fit.xmins, fit.Ds, label=r'$D$')
plot(fit.xmins, fit.sigmas, label=r'$\sigma$', linestyle='--')
plot(fit.xmins, fit.sigmas/fit.alphas, label=r'$\sigma /\alpha$', linestyle='--')
####
ylim(0, .4)
legend(loc=4)
xlabel(r'$x_{min}$')
ylabel(r'$D,\sigma,\alpha$')
savefig('FigD.eps', bbox_inches='tight')
Calculating best minimal value for power law fit
50.0 0.0998297854528 1.78313986533
Calculating best minimal value for power law fit
230.0 0.0606737962944 2.27263721983

No Possible Fits

In [26]:
data = blackouts
####
fit = powerlaw.Fit(data, sigma_threshold=.001)
fit.power_law.alpha, fit.power_law.sigma, fit.xmin, fit.noise_flag
Calculating best minimal value for power law fit
No valid fits found.
No valid fits found.
Out[26]:
(10.500000000422041, 5.1423189016918585, False)
In [ ]:
fit.lognormal.mu, fit.lognormal.sigma
range_dict = {'mu': [10.5, None]}
fit.lognormal.parameter_range(range_dict)
fit.lognormal.mu, fit.lognormal.sigma, fit.lognormal.noise_flag

initial_parameters = (12, .7)
fit.lognormal.parameter_range(range_dict, initial_parameters)
fit.lognormal.mu, fit.lognormal.sigma, fit.lognormal.noise_flag

Supporting Information

In [38]:
from numpy import logspace
from scipy.stats import variation
import pandas as pd

def validate(xmin, alpha, discrete='continuous', n_data=10000, n_trials=1):
    
    if n_trials>1:
        return array([validate(xmin, alpha, discrete=discrete, n_data=n_data, n_trials=1) for trial in arange(n_trials)]).T
        
    if discrete=='continuous':
        discrete = False
        estimate_discrete = False
    elif discrete == 'discrete':
        discrete = True
        estimate_discrete = False
    elif discrete == 'discrete_estimate':
        discrete = True
        estimate_discrete = True

    theoretical_distribution = powerlaw.Power_Law(xmin=xmin, parameters=[alpha], discrete=discrete)
    simulated_data = theoretical_distribution.generate_random(n_data, estimate_discrete=estimate_discrete)
    fit = powerlaw.Fit(simulated_data, discrete=discrete, estimate_discrete=estimate_discrete)
    return fit.xmin, fit.alpha
In [180]:
%%prun
xmin=1
alpha=1.5
n_data=10000
discrete='continuous'
validate(xmin, alpha, discrete=discrete, n_data=n_data, n_trials=10)
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit

In [39]:
theoretical_xmins = unique(floor(logspace(0.0,2.0,num=20)))
theoretical_alphas = array([1.5,2.0,2.5,3.0,3.5])
#theoretical_xmins = [2]
#theoretical_alphas = [2.5]
distribution_types = ['continuous','discrete']
n_trials = 10
n_data = 10000

ind = [(d,a,x) for d in distribution_types for a in theoretical_alphas for x in theoretical_xmins]
ind = pd.MultiIndex.from_tuples(ind, names=['type', 'alpha','xmin'])
df = pd.DataFrame(columns=['alpha_mean', 'alpha_sd', 'xmin_mean', 'xmin_sd'], index=ind)
In [182]:
i = 0
for dt, alpha, xmin in ind:
    i += 1
    print(i)
    data = validate(xmin, alpha, discrete=dt, n_data=n_data, n_trials=n_trials)
    df.ix[dt,alpha,xmin] = (mean(data[1]), std(data[1]), mean(data[0]), std(data[0]))

filename = 'powerlaw_validation_%itrials_%idata.h5'%(int(n_trials),int(n_data))
df.to_hdf(filename,'df')
1
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
13
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
14
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
15
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
16
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
17
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
18
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
19
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
20
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
21
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
22
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
23
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
24
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
25
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
26
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
27
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
28
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
29
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
30
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
31
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
32
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
33
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
34
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
35
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
36
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
37
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
38
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
39
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
40
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
41
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
42
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
43
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
44
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
45
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
46
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
47
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
48
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
49
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
50
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
51
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
52
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
53
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
54
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
55
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
56
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
57
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
58
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
59
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
60
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
61
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
62
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
63
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
64
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
65
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
66
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
67
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
68
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
69
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
70
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
71
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
72
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
73
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
74
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
75
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
76
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
77
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
78
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
79
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
80
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
81
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
82
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
83
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
84
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
85
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
86
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
87
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
88
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
89
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
90
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
91
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
92
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
93
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
94
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
95
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
96
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
97
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
98
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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99
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
100
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
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101
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
102
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
103
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
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104
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
105
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
106
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
107
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
108
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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109
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
110
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
111
Calculating best minimal value for power law fit
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112
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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113
Calculating best minimal value for power law fit
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114
Calculating best minimal value for power law fit
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115
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
116
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
117
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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118
Calculating best minimal value for power law fit
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119
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
120
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
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121
Calculating best minimal value for power law fit
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122
Calculating best minimal value for power law fit
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123
Calculating best minimal value for power law fit
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124
Calculating best minimal value for power law fit
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125
Calculating best minimal value for power law fit
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126
Calculating best minimal value for power law fit
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127
Calculating best minimal value for power law fit
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128
Calculating best minimal value for power law fit
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129
Calculating best minimal value for power law fit
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130
Calculating best minimal value for power law fit
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131
Calculating best minimal value for power law fit
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132
Calculating best minimal value for power law fit
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133
Calculating best minimal value for power law fit
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134
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
135
Calculating best minimal value for power law fit
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136
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
137
Calculating best minimal value for power law fit
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138
Calculating best minimal value for power law fit
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139
Calculating best minimal value for power law fit
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140
Calculating best minimal value for power law fit
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141
Calculating best minimal value for power law fit
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142
Calculating best minimal value for power law fit
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143
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
144
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
145
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
146
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
147
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
148
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
149
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
150
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
151
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
152
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
153
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
154
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
155
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
156
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
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Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
165
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
166
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
167
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
168
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
169
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
170
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
Calculating best minimal value for power law fit
In [40]:
#filename = 'powerlaw_validation_%itrials_%idata.h5'%(int(n_trials),int(n_data))
#df = pd.read_hdf(filename,'df')
In [41]:
subplot(2,2,1)
for a in theoretical_alphas:
    y_vals = df.ix['continuous', a]['alpha_mean']
    error = df.ix['continuous', a]['alpha_sd']

    plot(theoretical_xmins, y_vals, label=a)
    fill_between(theoretical_xmins, y_vals-error, y_vals+error, alpha=.1)

xscale('log')
#xlabel(r"$x_{min}$")
ylabel(r"Fitted $\alpha$")
yticks(theoretical_alphas)
setp(gca().get_xticklabels(), visible=False)
title("Continuous")

#########
subplot(2,2,2)
for a in theoretical_alphas:
    y_vals = df.ix['discrete', a]['alpha_mean']
    error = df.ix['discrete', a]['alpha_sd']

    plot(theoretical_xmins, y_vals, label=a)
    fill_between(theoretical_xmins, y_vals-error, y_vals+error, alpha=.1)

xscale('log')
#xlabel(r"$x_{min}$")
#ylabel(r"Fitted $\alpha$")
setp(gca().get_xticklabels(), visible=False)
setp(gca().get_yticklabels(), visible=False)
title("Discrete")

########
subplot(2,2,3)
for a in theoretical_alphas:
    y_vals = df.ix['continuous', a]['xmin_mean']
    error = df.ix['continuous', a]['xmin_sd']
    up = y_vals+error
    down = y_vals-error
    ind = down<theoretical_xmins
    down[ind] = theoretical_xmins[ind]
    
    plot(theoretical_xmins, y_vals, label=a)
    fill_between(theoretical_xmins, down, up, alpha=.1)

xlim(xmin=1)
ylim(ymin=1)
plot(xlim(),xlim(),linestyle='--', color='k')
xscale('log')
yscale('log')
xlabel(r"$x_{min}$ of Data")
ylabel(r"Fitted $x_{min}$")


########
legend_refs = []
########
subplot(2,2,4,sharey=gca())
for a in theoretical_alphas:
    y_vals = df.ix['discrete', a]['xmin_mean']
    error = df.ix['discrete', a]['xmin_sd']
    up = y_vals+error
    down = y_vals-error
    ind = down<theoretical_xmins
    down[ind] = theoretical_xmins[ind]

    line = plot(theoretical_xmins, y_vals, label=a)
    legend_refs += line
    fill_between(theoretical_xmins, down, up, alpha=.1)

xlim(xmin=1)
ylim(ymin=1)
plot(xlim(),xlim(),linestyle='--', color='k')
xscale('log')
yscale('log')
xlabel(r"$x_{min}$ of Data")
#ylabel(r"Fitted $x_{min}$")
setp(gca().get_yticklabels(), visible=False)


#######
#figlegend(legend_refs[::-1], theoretical_alphas[::-1],'center right', title=r'$\alpha$ of Data')
subplots_adjust(wspace=.15, hspace=.1)
legend( legend_refs[::-1], theoretical_alphas[::-1], loc = 'center right', bbox_to_anchor = (.1,0,1,1),
            bbox_transform = plt.gcf().transFigure, title=r'$\alpha$ of Data' )
savefig('Fig_powerlaw_validation_%itrials_%idata.pdf'%(int(n_trials),int(n_data)), bbox_inches='tight')