AllanTools 2016.11 - now with Confidence Intervals!

at2016-11_wtih_ci

I've added confidence interval estimation to allantools, based on a 2004 paper by Greenhall & Riley: "Uncertainty of stability variances based on finite differences"

So far not much is automated so you have to run everything manually. After a normal call to allantools.adev() to calculate the ADEV we loop through each (tau, adev) pair and first call allantools.edf_greenhall() to get the number of equivalent-degrees-of-freedom (EDF), and then evaluate a confidence interval with allantools.confidence_interval(). A knowledge or estimate of the noise type "alpha" is required for edf_greenhall() - here we just assume alpha=0.

This example is on github at: https://github.com/aewallin/allantools/blob/master/examples/ci_demo.py

import numpy
import matplotlib.pyplot as plt
 
import allantools as at
 
# this demonstrates how to calculate confidence intervals for ADEV
# using the algorithms from Greenhall2004
data_file = '../tests/phasedat/PHASE.DAT'
 
def read_datafile(filename):
    p = []
    with open(filename) as f:
        for line in f:
            if not line.startswith("#"):  # skip comments
                p.append(float(line))
    return p
 
# read input data from file
phase = read_datafile(data_file)
# normal ADEV computation, giving naive 1/sqrt(N) errors
(taus,devs,errs,ns) = at.adev(phase, taus='octave')
 
# Confidence-intervals for each (tau,adev) pair separately.
cis=[]
for (t,dev) in zip(taus,devs):
    # Greenhalls EDF (Equivalent Degrees of Freedom)
    # alpha     +2,...,-4   noise type, either estimated or known
    # d         1 first-difference variance, 2 allan variance, 3 hadamard variance
    #           we require: alpha+2*d >1     (is this ever false?)
    # m         tau/tau0 averaging factor
    # N         number of phase observations
    edf = at.edf_greenhall( alpha=0, d=2, m=t, N=len(phase), overlapping = False, modified=False )
    # with the known EDF we get CIs 
    # for 1-sigma confidence we set
    # ci = scipy.special.erf(1/math.sqrt(2)) = 0.68268949213708585
    (lo,hi) = at.confidence_intervals( dev=dev, ci=0.68268949213708585, edf=edf )
    cis.append( (lo,hi) )
 
# now we are ready to print and plot the results
print "Tau\tmin Dev\t\tDev\t\tMax Dev"
for (tau,dev,ci) in zip(taus,devs,cis):
    print "%d\t%f\t%f\t%f" % (tau, ci[0], dev, ci[1] )
""" output is
Tau	min Dev		Dev		Max Dev
1	0.285114	0.292232	0.299910
2	0.197831	0.205102	0.213237
4	0.141970	0.149427	0.158198
8	0.102541	0.110135	0.119711
16	0.056510	0.062381	0.070569
32	0.049153	0.056233	0.067632
64	0.027109	0.032550	0.043536
128	0.026481	0.033855	0.055737
256	0.007838	0.010799	0.031075
"""
plt.figure(figsize=(12,8))
plt.gca().set_yscale('log')
plt.gca().set_xscale('log')
err_lo = [ d-ci[0] for (d,ci) in zip(devs,cis)]
err_hi = [ ci[1]-d for (d,ci) in zip(devs,cis)]
 
plt.errorbar(taus, devs, yerr=[ err_lo, err_hi ] ,fmt='o')
plt.grid()
plt.xlabel('Tau (s)')
plt.ylabel('ADEV')
plt.title('AllanTools 2016.11 - now with Confidence Intervals!')
# just to check plot the intervals as dots also
plt.plot(taus, [ci[0] for ci in cis],'r.')
plt.plot(taus, [ci[1] for ci in cis],'g.')
 
plt.show()

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