Jackknife

To generate jackknife error bars, please use the module

import latqcdtools.statistics.jackknife

The central method of this file is the jackknife method. A call to

jackknife(func, data, numb_blocks = 20, conf_axis = 1, return_sample = False, args = (), nproc=1)

returns a jackknife average, error, and (if desired) the covariance, in that order. Here data is a time series of raw data you are interested in, and func is some function of that data. numb_blocks is the number of jackknife blocks, args are the arguments to pass to func. The conf_axis is needed when one wants to pass a 2D array as data, which can happen, for instance, if func depends on multiple observables. An example will be given below.

This jackknife is quite general. Something easy one might do with the jackknife is calculate the Polyakov loop susceptibility from Polyakov loops. A rule of thumb to use when using the jackknife code is to “write your function the way it appears using math symbols”. For example with the susceptibility one calculates \(\chi_P=N_\sigma^3\left(\langle P^2\rangle - \langle P\rangle\right)\), so one correspondingly defines

Ns=32
def susc(P):
  return Ns**3*( np.mean(P**2) - np.mean(P)**2 )

where \(P\) is the time series of measurements of \(P\), and then calls the jackknife by

Pmean, Perr = jackknife(susc, P)

Sometimes your function may depend on multiple observables; for instance one may be interested in the susceptibility of \(|P|\), for which one needs the real and imaginary parts of \(P\). One can accomplish this using 32 jackknife blocks with, for instance,

def suscA(data):
  ReP = data[0]
  ImP = data[1]
  return Ns**3*( np.mean(ReP**2+ImP**2) - np.mean(ReP)**2 )

AbsPmean, AbsPerr = jackknife(suscA, [ReP, ImP], 32, 1)

Note that these Polyakov loop examples are meant to be instructional. A complete set of functions measuring Polyakov loop observables is given in polyakovTools.py, described in part in the part of the documentation for physics modules.

WARNING: Although the jackknife method is very general, there are at least a couple kinds of functions that may fail when passed to jackknife:

Don’t pass a lambda function. This is because the jackknife can be parallelized using concurrent.futures, which is not able to pickle lambda functions. We discourage the use of lambda functions in general.
Don’t pass a function that manipulates data indices. You may not be manipulating the indices you think, because the jackknife first splits your input data into bins, and then the indices of the binned data will get manipulated.