# Error Estimation in model fitting

## Errors - Monte Carlo simulations¶

### In this project, I will show how to extract errors on best-fit parameters estimated from the curve fitting techniques introduced in curve_fitting. I will use the same example (the limb data) for illustration.¶

In [1]:
## importing the libraries
import numpy as np
import math as m
import scipy
import matplotlib.pyplot as plt
from scipy import special
from scipy.optimize import curve_fit

In [2]:
def Erfc(x,sigma):

y = special.erfc(x/(sigma*np.sqrt(2)))
return y

## Introducing the model to be used later for the fitting

def SL_fit(x,w1,w2,w3,s1,s2,s3):

f = 0.5*(w1*Erfc(x,s1)+w2*Erfc(x,s2)+w3*Erfc(x,s3)+ (1-w1-w2-w3))
return f


#### From now on I will use scipy.optimize.curve_fit for the fitting problem¶

In [3]:
## Jacobian matrix
def DF(x,w1,w2,w3,s1,s2,s3):
dfw1 = 0.5*(Erfc(x,s1)-1)
dfw2 = 0.5*(Erfc(x,s2)-1)
dfw3 =  0.5*(Erfc(x,s3)-1)
dfs1 = w1*x*np.exp(-x**2/(2*s1**2))/ (np.sqrt(2*np.pi)*s1**2)
dfs2 = w2*x*np.exp(-x**2/(2*s2**2))/ (np.sqrt(2*np.pi)*s2**2)
dfs3 = w3*x*np.exp(-x**2/(2*s3**2))/ (np.sqrt(2*np.pi)*s3**2)
return np.transpose(np.array([dfw1,dfw2,dfw3,dfs1,dfs2,dfs3]))

In [4]:
## Loading the data:
path = "/home/fatima/Desktop/project_3/"
x = file[:,0]
y = file[:,1]

In [5]:
ind = np.where(x>=0)
x = x[ind]
y = y[ind]
weights = np.sqrt(np.abs(y)) ## Poisson weighting

In [6]:
p0=[0.3, 0.3, 0.2, 1, 2, 3] ## initial guess best-fit parameters
popt, pcov = curve_fit(SL_fit,x,y,p0,method='lm',sigma=weights,jac=DF,ftol=1e-8,xtol=1e-8,maxfev=5000)
chi_sq_w = np.sum((1/weights**2)*(SL_fit(x,*popt)-y)**2)
red_chi_sq = chi_sq_w/(len(y)-len(popt))
print popt # to print the best-fit parameters

[ 0.52750103  0.28882569  0.10191755  0.25905336  0.76540583  2.83343005]


#### In the lines above, popt is the array of best-fit parameters, while pcov is the covariance matrix, the diagonal of which can give an estimate of the errors by typing the following:¶

In [7]:
perr = np.sqrt(np.diag(pcov))
print perr ## to print the error

[ 0.00349415  0.0028457   0.00119396  0.00135734  0.00715294  0.04738134]


#### One can also use another method to compute the errors on the best-fit parameters. The so called "Monte Carlo" method.¶

In [8]:
### The "true" solution is assumed to be that returned by curve_fit, popt
w10, w20, w30, s10, s20, s30 = popt
w40 = 1-w10-w20-w30

In [9]:
### we will make a copy for our data set so we don't alter the original data set
xr = x.copy()
yr = y.copy()
ery = weights.copy()

In [10]:
N = len(y)
trials = 1000 ## number of synthetic data sets
omegas1, omegas2, omegas3,omegas4,sigmas1,sigmas2, sigmas3 = [],[],[],[],[],[],[] #defining the arrays of best-fit parameters corresponding to the number of trials "trials"
chi_s = [] ## the array of dimension "trials" of chi-square values


#### Now generating synthetic data sets using numpy.random.randint.¶

In [11]:
from numpy.random import normal, randint

In [12]:
for i in range(trials):
indx = randint(0, N, N)
xr[:] = x[indx]
yr[:] = y[indx]
ery[:] = weights[indx]
ok = (xr != xr[0]).any()

if (not ok):
print "All elements are the same. Invalid sample."
print xr, yr
else:

## now preforming the fit using curve_fit on the synthetic data sets and saving results to popti
popti, pcovi = curve_fit(SL_fit,xr,yr,p0,method='lm',sigma=ery,ftol=1e-8,xtol=1e-8,maxfev=5000)
chi_sq_wi = np.sum((1/ery**2)*(SL_fit(xr,*popti)-yr)**2) ## chi_squared values
red_chi_sqi = chi_sq_wi/(len(yr)-len(popti)) ## reduced chi_squared values
om1, om2,om3,sig1,sig2,sig3 = popti
om4 = 1-om1-om2-om3
## saving best-fit parameters and reduced chi_sq values
omegas1.append(om1)
omegas2.append(om2)
omegas3.append(om3)
omegas4.append(om4)
sigmas1.append(sig1)
sigmas2.append(sig2)
sigmas3.append(sig3)
chi_s.append(red_chi_sqi)
print popti ##to print the output for each trial

[ 0.51780298  0.29471219  0.10472405  0.25423208  0.74833391  2.72665156]
[ 0.52129061  0.29353422  0.10319298  0.2550758   0.75857336  2.79692301]
[ 0.51306274  0.30034848  0.10439981  0.25379152  0.73951306  2.77894058]
[ 0.53365379  0.28457255  0.10031272  0.26452477  0.7753637   2.87604064]
[ 0.52833911  0.28897794  0.10118776  0.2593886   0.77123206  2.86117997]
[ 0.52448991  0.29194521  0.10179544  0.25803698  0.76412923  2.84068826]
[ 0.54145759  0.27780062  0.09965041  0.26467896  0.78719177  2.91283197]
[ 0.53095848  0.28561657  0.10171466  0.26005834  0.76725018  2.8396865 ]
[ 0.52902465  0.28694202  0.10166209  0.25976067  0.76918532  2.79408516]
[ 0.54044506  0.27784204  0.1002904   0.26247875  0.78540453  2.8818541 ]
[ 0.52980677  0.2848685   0.10290806  0.25969338  0.76599433  2.77238101]
[ 0.52936404  0.28811663  0.10103488  0.25763627  0.77464682  2.86450959]
[ 0.52031571  0.29408812  0.10352346  0.25716309  0.75133869  2.79816134]
[ 0.53425524  0.28351565  0.10073671  0.26173649  0.78007644  2.86371769]
[ 0.52515741  0.28956341  0.10315384  0.25922337  0.75885906  2.79361747]
[ 0.51899082  0.2945735   0.10360002  0.255167    0.75219202  2.73482613]
[ 0.5218894   0.29408037  0.10216375  0.25756991  0.75511623  2.82536094]
[ 0.52853154  0.28842998  0.10141798  0.26000835  0.76844709  2.85127265]
[ 0.52815104  0.28843112  0.10192589  0.25950146  0.76886055  2.85364331]
[ 0.53360273  0.28576337  0.09928717  0.26173305  0.7830851   2.89852059]
[ 0.52204151  0.29233396  0.10328523  0.25873909  0.75201407  2.77735066]
[ 0.51866219  0.29387625  0.1049722   0.25642266  0.74445909  2.742996  ]
[ 0.52906746  0.2877442   0.10146078  0.26094375  0.76587507  2.8384926 ]
[ 0.53584421  0.28274972  0.10026819  0.26255011  0.78053902  2.90331845]
[ 0.51887346  0.29665653  0.10248813  0.25653415  0.74956729  2.80152054]
[ 0.53108297  0.28665445  0.1009335   0.26043897  0.7729498   2.87108095]
[ 0.52685987  0.2888314   0.10249911  0.25913196  0.76147299  2.82142032]
[ 0.54017045  0.27871703  0.0998245   0.26518477  0.78541285  2.8887926 ]
[ 0.53022644  0.28672234  0.1013302   0.25859413  0.77036623  2.84287173]
[ 0.54733647  0.27251123  0.09905496  0.26910571  0.79753272  2.91732056]
[ 0.51937106  0.29502811  0.10347749  0.25479263  0.74988821  2.79294131]
[ 0.51739323  0.29523463  0.10479817  0.25401795  0.74502829  2.74293187]
[ 0.52915109  0.28817407  0.10125565  0.25950374  0.76757865  2.86817532]
[ 0.53306179  0.28411622  0.10097527  0.26196024  0.7730975   2.84264287]
[ 0.52885325  0.28706008  0.10220443  0.25949511  0.76326387  2.81412979]
[ 0.52797155  0.28743078  0.10246057  0.26054388  0.76328628  2.79907809]
[ 0.53983871  0.27951028  0.09955391  0.26420222  0.78808278  2.91201211]
[ 0.53148125  0.28619176  0.10106313  0.26118841  0.77183837  2.88317282]
[ 0.52984477  0.28650414  0.10211959  0.26002912  0.76585282  2.84931195]
[ 0.53157433  0.28550868  0.10152619  0.2579675   0.77326808  2.85639131]
[ 0.53105041  0.28181661  0.10421625  0.26224994  0.76027829  2.72299557]
[ 0.53673428  0.2797915   0.10155536  0.26386323  0.77657085  2.81816039]
[ 0.5298782   0.28518858  0.10280104  0.26022372  0.7645057   2.79015965]
[ 0.52490549  0.29082052  0.10235153  0.2576387   0.76165986  2.81743732]
[ 0.04817047  0.65777412  0.20194525  0.00311151  0.35519261  1.65868417]
[ 0.52435533  0.29291365  0.10109621  0.25848203  0.76372694  2.85510656]
[ 0.5238041   0.29090919  0.10326393  0.25788355  0.75801648  2.79553583]
[ 0.52026936  0.2956229   0.10244452  0.25686709  0.75328162  2.83559207]
[ 0.53170099  0.28562563  0.10115767  0.26266288  0.77361804  2.85646775]
[ 0.53765856  0.28197284  0.09922508  0.25992331  0.78876513  2.9126246 ]
[ 0.52945642  0.28755166  0.10127724  0.26022237  0.77074051  2.84176161]
[ 0.53165007  0.28406189  0.10216629  0.25915915  0.76982017  2.80149857]
[ 0.52935689  0.28558829  0.10284512  0.26035905  0.76299218  2.79076794]
[ 0.53501335  0.28235071  0.10100644  0.26339321  0.77668433  2.85707295]
[ 0.51791949  0.29536824  0.10439053  0.25507422  0.74717649  2.76472253]
[ 0.53342661  0.28463055  0.10044943  0.26281666  0.77641539  2.8742317 ]
[ 0.53697732  0.28309165  0.09878887  0.26038958  0.79096146  2.92168894]
[ 0.52396484  0.29256016  0.10189808  0.25795387  0.76216299  2.85272257]
[ 0.04820996  0.65811469  0.20143962  0.00260661  0.34733674  1.64119181]
[ 0.09869318  0.6311405   0.17871208  0.00300649  0.38939967  1.78158904]
[ 0.52756506  0.29006987  0.10125325  0.25909286  0.76793509  2.8882305 ]
[ 0.52663053  0.28895002  0.10236282  0.25828862  0.76393726  2.80627512]
[ 0.50332216  0.30500018  0.10824363  0.25136455  0.72119488  2.64959239]
[ 0.04824597  0.66668275  0.19274531  0.00225873  0.36123416  1.68070973]
[ 0.52725424  0.28712168  0.10329474  0.2599475   0.7582369   2.77504338]
[ 0.52557529  0.28886487  0.10346344  0.25924467  0.75706314  2.79048071]

/usr/local/lib/python2.7/dist-packages/scipy/optimize/minpack.py:779: OptimizeWarning: Covariance of the parameters could not be estimated
category=OptimizeWarning)

[ 0.28731485  0.5292576   0.10167048  0.76798473  0.26162477  2.83761925]
[ 0.5247599   0.29226996  0.10126959  0.2578713   0.76648919  2.84495703]
[ 0.52591027  0.28932518  0.10281447  0.26021156  0.75974466  2.81205229]
[ 0.52037307  0.29353716  0.10386839  0.25815326  0.74868591  2.77357637]
[ 0.52476081  0.29207564  0.10170637  0.25701154  0.76487136  2.86056447]
[ 0.53456731  0.28252339  0.10129951  0.26120784  0.77489021  2.84490542]
[ 0.09315831  0.61875699  0.19572435  0.00336271  0.37164124  1.67974338]
[ 0.53191365  0.28413132  0.1019134   0.26093346  0.77191662  2.81049658]
[ 0.53334859  0.2847749   0.10058468  0.2608581   0.7765642   2.8836232 ]
[ 0.5260746   0.28839298  0.1031883   0.25905232  0.76100372  2.77683836]
[ 0.52388349  0.29319221  0.10129585  0.25735639  0.76616122  2.84936941]
[ 0.53808165  0.28137813  0.09941461  0.26333898  0.78439046  2.90885228]
[ 0.52392466  0.29098829  0.10279606  0.25801993  0.75619837  2.79025982]
[ 0.52959063  0.28844162  0.10051073  0.26101499  0.76919164  2.87444553]
[ 0.53387061  0.28275679  0.10162025  0.26333165  0.77316462  2.83513094]
[ 0.53275046  0.28412264  0.10163749  0.26110506  0.77028075  2.85729407]
[ 0.52350126  0.29231492  0.10218213  0.25553595  0.76303664  2.81295625]
[ 0.53461894  0.28300599  0.10098527  0.26410606  0.77491479  2.8673768 ]
[ 0.52996507  0.28631572  0.10201854  0.25958218  0.76919512  2.83426272]
[ 0.53768163  0.28074159  0.10030041  0.26426115  0.78185016  2.88738778]
[ 0.52808802  0.28874585  0.10170162  0.2579407   0.76579437  2.85824368]
[ 0.52793391  0.28862634  0.10185228  0.25903145  0.76978775  2.84036451]
[ 0.51596141  0.29577514  0.10537201  0.25479675  0.74013017  2.71476991]
[ 0.52024823  0.29444328  0.10340306  0.25636758  0.75373919  2.80613817]
[ 0.52842688  0.28800853  0.1017072   0.25961012  0.76540611  2.82977111]
[ 0.52810476  0.28974054  0.10058083  0.25857623  0.77441036  2.85946852]
[ 0.52010941  0.29432289  0.10346299  0.25588554  0.75528002  2.78771055]
[ 0.52207546  0.2923195   0.10340722  0.25811799  0.75580232  2.77928213]
[ 0.52571316  0.29052209  0.1019723   0.25828595  0.761784    2.83417689]
[ 0.52127484  0.29385726  0.10256796  0.25630281  0.75834375  2.78969464]
[ 0.53063608  0.28664035  0.10103743  0.25967061  0.77031866  2.8485453 ]
[ 0.52668923  0.28917258  0.10216876  0.25837876  0.76377929  2.81549101]
[ 0.53831612  0.28130419  0.09916822  0.26253678  0.7872912   2.91003431]
[ 0.51267625  0.29963155  0.10513996  0.25401855  0.73391206  2.73338008]
[ 0.53799073  0.28081994  0.10010491  0.26309897  0.78380204  2.89996546]
[ 0.53217017  0.28555031  0.10080887  0.25928077  0.77337876  2.86908183]
[ 0.52843261  0.28740251  0.10214367  0.26006555  0.76479631  2.8073539 ]
[ 0.52458872  0.28934816  0.10359292  0.25896272  0.75538009  2.76774731]
[ 0.52742177  0.29044311  0.10064737  0.25995096  0.7703261   2.8660103 ]
[ 0.52715763  0.29119084  0.10039792  0.2565737   0.77101801  2.88789429]
[ 0.53669113  0.28029765  0.10112486  0.26214718  0.77540674  2.8368101 ]
[ 0.5221116   0.2931623   0.10273945  0.25675468  0.75797176  2.80767979]
[ 0.53219672  0.28639422  0.10008798  0.25754499  0.77868186  2.89287432]
[ 0.52620887  0.28964929  0.10261971  0.25882503  0.76060212  2.84168073]
[ 0.52456655  0.2911372   0.10241308  0.25815847  0.76015179  2.81974419]
[ 0.53174718  0.28714574  0.0999884   0.25947232  0.77577987  2.90240833]
[ 0.52862451  0.28869241  0.1013857   0.25982705  0.76806424  2.87532395]
[ 0.52298098  0.29428238  0.10128944  0.25670279  0.76356445  2.86615625]
[ 0.52252536  0.29350432  0.10235644  0.25779508  0.75898026  2.83105879]
[ 0.53918651  0.27806765  0.10107459  0.2622425   0.78094466  2.84962486]
[ 0.53251231  0.28400849  0.10171876  0.26166664  0.7713953   2.83613609]
[ 0.52545535  0.29050358  0.10195923  0.25842814  0.76193598  2.80870963]
[ 0.52870608  0.28668473  0.10253607  0.25829897  0.76730997  2.80270869]
[ 0.52405615  0.29053071  0.10307536  0.25767571  0.75754558  2.7741122 ]
[ 0.53377187  0.2828234   0.1019679   0.26149681  0.7717643   2.8540883 ]
[ 0.52907221  0.28598805  0.10285333  0.260045    0.76212763  2.80026312]
[ 0.53650945  0.28291743  0.09976867  0.26312847  0.78471718  2.9336462 ]
[ 0.53105713  0.28740937  0.10037984  0.26252089  0.77225666  2.90372595]
[ 0.52714945  0.28951862  0.10160736  0.25864276  0.76570338  2.84655343]
[ 0.52537376  0.2904728   0.10223838  0.25885511  0.75932895  2.81532376]
[ 0.53034644  0.28568507  0.10241338  0.25990602  0.76544748  2.84103101]
[ 0.5208124   0.29226539  0.10430927  0.2568336   0.75210129  2.74159952]
[ 0.51592218  0.29489078  0.10628553  0.25540555  0.73966975  2.69811318]
[ 0.53766858  0.28100067  0.10035566  0.26270389  0.78262331  2.91102692]
[ 0.53553954  0.28328807  0.10007881  0.26040352  0.78175166  2.91012435]
[ 0.53727298  0.28162813  0.09999701  0.26362228  0.78456668  2.90361671]
[ 0.52818202  0.28700043  0.10270494  0.25949758  0.76230592  2.79725846]
[ 0.51272247  0.30206853  0.10286594  0.25257957  0.74670418  2.7769073 ]
[ 0.53002241  0.28669679  0.10186423  0.26030046  0.76872889  2.86202332]
[ 0.53653313  0.28340424  0.09915201  0.2602396   0.79003565  2.92934369]
[ 0.52395671  0.29336302  0.10127089  0.25747311  0.7650367   2.86741626]
[ 0.51710074  0.29765233  0.10306706  0.25476752  0.75219921  2.79161253]
[ 0.52937893  0.28702532  0.10184153  0.26056903  0.76691895  2.83322453]
[ 0.52134643  0.29157213  0.1047956   0.25590068  0.75247561  2.75339579]
[ 0.53701264  0.28083976  0.10068385  0.26385377  0.78121344  2.87183887]
[ 0.52696793  0.28766139  0.10309847  0.26118941  0.7582619   2.78452005]
[ 0.5352989   0.28401087  0.09952072  0.26091262  0.78505689  2.90418045]
[ 0.53033826  0.28689381  0.10103001  0.26103038  0.77254703  2.84637251]
[ 0.52508878  0.29095602  0.10196504  0.25790374  0.76289577  2.81676817]
[ 0.52112771  0.29259163  0.10364721  0.25826317  0.75257338  2.75236528]
[ 0.5384658   0.27972493  0.10060054  0.26617779  0.77863729  2.88673251]
[ 0.53542212  0.2835966   0.09998805  0.26181245  0.78377076  2.90709414]
[ 0.52448732  0.29208547  0.10166187  0.25816044  0.76582781  2.8361584 ]
[ 0.53189987  0.28406127  0.10219495  0.26362587  0.77089331  2.8205457 ]
[ 0.5332084   0.28584719  0.0999367   0.26230471  0.77668727  2.91719336]
[ 0.5131005   0.29725261  0.10658518  0.25488925  0.73313815  2.6831872 ]
[ 0.53253475  0.28293427  0.10270887  0.25903791  0.76905454  2.81722249]
[ 0.53260362  0.28476647  0.1011027   0.26241984  0.77303943  2.86025695]
[ 0.52973693  0.28614852  0.10270323  0.25926804  0.76365111  2.85706936]
[ 0.52881528  0.28864531  0.10103104  0.26150719  0.76889126  2.86427845]
[ 0.53546474  0.28187503  0.10121126  0.26289931  0.77547356  2.86366535]
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3.67769618e-01   1.68965158e+00]
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3.86911167e-01   1.75289880e+00]
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3.77813283e-01   1.73411554e+00]
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3.55016718e-01   1.67696556e+00]
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[ 0.52743209  0.28920702  0.10162935  0.25880255  0.7684052   2.83473272]
[ 0.51946858  0.29638948  0.10230223  0.25368213  0.75726451  2.82024381]
[ 0.53537141  0.28350969  0.10007002  0.26300454  0.77894133  2.91216582]
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[ 0.52377007  0.29119375  0.10319841  0.25919138  0.75641308  2.81250339]
[ 0.53015267  0.28567887  0.1020989   0.26106416  0.7674984   2.80959837]
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[ 0.52281197  0.29157578  0.10340297  0.2571364   0.75701685  2.78406993]
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3.49224325e-01   1.67197417e+00]
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[ 0.52676134  0.28962629  0.1020867   0.25971637  0.76263121  2.849982  ]
[ 0.53096013  0.28599651  0.10132206  0.26102624  0.7733681   2.83904765]
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[ 0.52771203  0.28719687  0.10299977  0.25861102  0.76124833  2.79532915]
[  7.13864084e-02   6.43552443e-01   1.92243668e-01   9.90038755e-04
3.62271221e-01   1.66563598e+00]
[ 0.52841079  0.28881544  0.10116992  0.26066672  0.76957296  2.85255336]
[ 0.53748314  0.28281583  0.09890411  0.26384694  0.78806422  2.94016195]
[ 0.53089353  0.28424391  0.10266871  0.26077172  0.76497477  2.79377089]
[ 0.52468294  0.29323155  0.10080495  0.25844071  0.7644761   2.88374157]
[ 0.52856263  0.28666958  0.10266642  0.26124391  0.76189338  2.80060798]
[ 0.52187814  0.2926676   0.10338985  0.25749876  0.75390421  2.79575108]
[ 0.53252939  0.28597645  0.10038814  0.26282542  0.77588385  2.90147454]
[ 0.5236712   0.29368162  0.10124636  0.25723466  0.76587674  2.86500493]
[ 0.52529396  0.29074346  0.10214809  0.25883767  0.75988511  2.83021831]
[ 0.52675731  0.29060069  0.10132859  0.25975769  0.76651393  2.87892118]
[ 0.53325516  0.28638121  0.09934416  0.26117588  0.78309773  2.91472751]
[ 0.52988921  0.2887574   0.10028533  0.25948238  0.77706472  2.90311953]
[ 0.53795289  0.27963987  0.10109944  0.26416497  0.77907     2.88032623]
[ 0.51906123  0.29431499  0.10409717  0.25489278  0.74868064  2.75130043]
[ 0.53938137  0.27905109  0.10026632  0.26602141  0.78126996  2.88866107]
[ 0.53254     0.28582747  0.10029792  0.26027107  0.77663162  2.88362016]
[ 0.53242062  0.28467651  0.10130878  0.25990816  0.77414209  2.85301061]
[ 0.52672878  0.2898127   0.10169636  0.25845187  0.76750841  2.83622944]
[ 0.52692607  0.28736549  0.1035058   0.25961257  0.7600043   2.77858015]
[ 0.53338757  0.28473032  0.10056772  0.26040871  0.77881532  2.88164688]
[ 0.53455009  0.28315956  0.10090712  0.2618802   0.77516591  2.87521866]
[ 0.52401699  0.29190515  0.10250628  0.2580886   0.75989484  2.83981721]
[ 0.53616127  0.28270729  0.10006049  0.26381802  0.78136094  2.9122556 ]
[ 0.54207776  0.27813236  0.09856762  0.26356645  0.79211254  2.92114314]
[ 0.5258879   0.28963515  0.10255799  0.25922681  0.75963392  2.82025446]
[ 0.52983933  0.28835302  0.10044129  0.25937693  0.77086135  2.87822085]
[ 0.51856872  0.29657675  0.10310135  0.2541727   0.75207935  2.81897886]
[ 0.52592707  0.29050379  0.10194356  0.25875737  0.76343657  2.84094004]
[ 0.52858308  0.28851164  0.10118884  0.25931086  0.76619205  2.84450938]
[ 0.53097776  0.28561953  0.1017676   0.25954643  0.77320758  2.84213667]
[ 0.52124465  0.29370989  0.10290339  0.25732938  0.75631952  2.79946241]
[ 0.52458588  0.29162825  0.10149375  0.25622207  0.76406089  2.81208125]
[ 0.52578693  0.28989576  0.10253662  0.25910238  0.76178743  2.81893128]
[ 0.54169812  0.27808933  0.09946962  0.2637004   0.78960884  2.94172628]
[ 0.53094338  0.28803627  0.10002704  0.26163979  0.77390157  2.91589849]
[ 0.52833051  0.28835288  0.10162424  0.25842743  0.76902444  2.83721103]
[ 0.52796721  0.29001763  0.10097134  0.25913759  0.76917688  2.9024221 ]
[ 0.52763328  0.28898571  0.10170951  0.25887647  0.76697375  2.84096154]
[ 0.53048754  0.28724643  0.10089604  0.26098032  0.77225632  2.87122151]
[ 0.53258068  0.28317182  0.10232105  0.26200757  0.76709755  2.81519185]
[ 0.52673862  0.28950797  0.10210245  0.25768315  0.76052414  2.83918912]
[ 0.52436905  0.2913004   0.10257355  0.2591361   0.76126523  2.82585915]
[ 0.53142045  0.28625123  0.10084861  0.26068265  0.77464387  2.86613393]
[ 0.53097593  0.2874495   0.10025114  0.25984928  0.7767134   2.8859161 ]
[ 0.52425067  0.29471028  0.10030069  0.25802118  0.76675389  2.93328853]
[ 0.53047763  0.28586576  0.10199251  0.26104112  0.76701649  2.83060097]
[ 0.52351259  0.29151098  0.10303191  0.25564088  0.75701715  2.80374168]
[ 0.52923131  0.28841511  0.10083273  0.25731952  0.77331325  2.86363404]
[ 0.52855331  0.29133154  0.09921841  0.25643501  0.77753993  2.93603201]
[ 0.52497109  0.2910064   0.10210152  0.25526823  0.76105129  2.82413231]
[ 0.29282152  0.52397429  0.10150452  0.76390521  0.25686506  2.8406086 ]
[ 0.526921    0.28955753  0.10193746  0.26100734  0.76351418  2.84183106]
[ 0.54045158  0.27866262  0.09963642  0.26624963  0.78911339  2.90315597]
[ 0.52568445  0.29094665  0.10211989  0.2576108   0.76472948  2.86544976]
[ 0.5331346   0.28419945  0.1009209   0.26316389  0.77219107  2.84872529]
[ 0.52105294  0.29345024  0.10348245  0.25775431  0.75140411  2.79487489]
[ 0.53705216  0.28002126  0.10119789  0.26103897  0.77797736  2.84522076]
[ 0.53445606  0.28415627  0.09993353  0.26163676  0.77793596  2.88199432]
[ 0.51999391  0.29439693  0.10366078  0.25701904  0.74963335  2.79831749]
[ 0.52150788  0.29458529  0.10234917  0.25616219  0.75616095  2.84570527]
[ 0.52046679  0.29362527  0.10376618  0.25595748  0.75014522  2.78747685]
[ 0.52444354  0.29133581  0.10230412  0.25760216  0.76423008  2.81468826]
[ 0.53308721  0.28400382  0.10140704  0.26111285  0.77332075  2.84666406]


#### Now let's create arrays of the fluctuations of the best fit parameters popti around the true solution popt0¶

In [13]:
omegas1 = np.array(omegas1) - w10
omegas2 = np.array(omegas2) - w20
omegas3 = np.array(omegas3) - w30
omegas4 = np.array(omegas4) - w40
sigmas1 = np.array(sigmas1) - s10
sigmas2 = np.array(sigmas2) - s20
sigmas3 = np.array(sigmas3) - s30

In [14]:
## now computing the standard deviation of the best-fit parameters population
stdw1, stdw2,stdw3,stdw4,stds1,stds2,stds3 = omegas1.std(), omegas2.std(), omegas3.std(),omegas4.std(), sigmas1.std(),sigmas2.std(),sigmas3.std()
print "Bootstrap errors in w1,w2,w3,w4,s1,s2,s3 are:", stdw1, stdw2,stdw3,stdw4,stds1,stds2,stds3

Bootstrap errors in w1,w2,w3,w4,s1,s2,s3 are: 0.0772531005887 0.0610754089333 0.0149780096104 0.00644211083616 0.121609755555 0.0840172033095 0.192045426395

In [ ]: