Week 10, March 4-8: Resampling Techniques, Bootstrap and Blocking

Topics

1. Reminder on optimization methods and top-down approach to what we need to add
2. Resampling Techniques and statistics: Bootstrap and Blocking
3. Discussion of codes
4. Video of lecture
5. Handwritten notes

Teaching Material, videos and written material

• Overview video on the Bootstrap method
• Marius Johnson's Master thesis on the Blocking Method

# 2-electron VMC code for 2dim quantum dot with importance sampling
# Using gaussian rng for new positions and Metropolis- Hastings 
# Added energy minimization
from math import exp, sqrt
from random import random, seed, normalvariate
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
from scipy.optimize import minimize
import sys
import os

# Where to save data files
PROJECT_ROOT_DIR = "Results"
DATA_ID = "Results/EnergyMin"

if not os.path.exists(PROJECT_ROOT_DIR):
    os.mkdir(PROJECT_ROOT_DIR)

if not os.path.exists(DATA_ID):
    os.makedirs(DATA_ID)

def data_path(dat_id):
    return os.path.join(DATA_ID, dat_id)

outfile = open(data_path("Energies.dat"),'w')


# Trial wave function for the 2-electron quantum dot in two dims
def WaveFunction(r,alpha,beta):
    r1 = r[0,0]**2 + r[0,1]**2
    r2 = r[1,0]**2 + r[1,1]**2
    r12 = sqrt((r[0,0]-r[1,0])**2 + (r[0,1]-r[1,1])**2)
    deno = r12/(1+beta*r12)
    return exp(-0.5*alpha*(r1+r2)+deno)

# Local energy  for the 2-electron quantum dot in two dims, using analytical local energy
def LocalEnergy(r,alpha,beta):
    
    r1 = (r[0,0]**2 + r[0,1]**2)
    r2 = (r[1,0]**2 + r[1,1]**2)
    r12 = sqrt((r[0,0]-r[1,0])**2 + (r[0,1]-r[1,1])**2)
    deno = 1.0/(1+beta*r12)
    deno2 = deno*deno
    return 0.5*(1-alpha*alpha)*(r1 + r2) +2.0*alpha + 1.0/r12+deno2*(alpha*r12-deno2+2*beta*deno-1.0/r12)

# Derivate of wave function ansatz as function of variational parameters
def DerivativeWFansatz(r,alpha,beta):
    
    WfDer  = np.zeros((2), np.double)
    r1 = (r[0,0]**2 + r[0,1]**2)
    r2 = (r[1,0]**2 + r[1,1]**2)
    r12 = sqrt((r[0,0]-r[1,0])**2 + (r[0,1]-r[1,1])**2)
    deno = 1.0/(1+beta*r12)
    deno2 = deno*deno
    WfDer[0] = -0.5*(r1+r2)
    WfDer[1] = -r12*r12*deno2
    return  WfDer

# Setting up the quantum force for the two-electron quantum dot, recall that it is a vector
def QuantumForce(r,alpha,beta):

    qforce = np.zeros((NumberParticles,Dimension), np.double)
    r12 = sqrt((r[0,0]-r[1,0])**2 + (r[0,1]-r[1,1])**2)
    deno = 1.0/(1+beta*r12)
    qforce[0,:] = -2*r[0,:]*alpha*(r[0,:]-r[1,:])*deno*deno/r12
    qforce[1,:] = -2*r[1,:]*alpha*(r[1,:]-r[0,:])*deno*deno/r12
    return qforce
    

# Computing the derivative of the energy and the energy 
def EnergyDerivative(x0):

    
    # Parameters in the Fokker-Planck simulation of the quantum force
    D = 0.5
    TimeStep = 0.05
    # positions
    PositionOld = np.zeros((NumberParticles,Dimension), np.double)
    PositionNew = np.zeros((NumberParticles,Dimension), np.double)
    # Quantum force
    QuantumForceOld = np.zeros((NumberParticles,Dimension), np.double)
    QuantumForceNew = np.zeros((NumberParticles,Dimension), np.double)

    energy = 0.0
    DeltaE = 0.0
    alpha = x0[0]
    beta = x0[1]
    EnergyDer = 0.0
    DeltaPsi = 0.0
    DerivativePsiE = 0.0 
    #Initial position
    for i in range(NumberParticles):
        for j in range(Dimension):
            PositionOld[i,j] = normalvariate(0.0,1.0)*sqrt(TimeStep)
    wfold = WaveFunction(PositionOld,alpha,beta)
    QuantumForceOld = QuantumForce(PositionOld,alpha, beta)

    #Loop over MC MCcycles
    for MCcycle in range(NumberMCcycles):
        #Trial position moving one particle at the time
        for i in range(NumberParticles):
            for j in range(Dimension):
                PositionNew[i,j] = PositionOld[i,j]+normalvariate(0.0,1.0)*sqrt(TimeStep)+\
                                       QuantumForceOld[i,j]*TimeStep*D
            wfnew = WaveFunction(PositionNew,alpha,beta)
            QuantumForceNew = QuantumForce(PositionNew,alpha, beta)
            GreensFunction = 0.0
            for j in range(Dimension):
                GreensFunction += 0.5*(QuantumForceOld[i,j]+QuantumForceNew[i,j])*\
	                              (D*TimeStep*0.5*(QuantumForceOld[i,j]-QuantumForceNew[i,j])-\
                                      PositionNew[i,j]+PositionOld[i,j])
      
            GreensFunction = exp(GreensFunction)
            ProbabilityRatio = GreensFunction*wfnew**2/wfold**2
            #Metropolis-Hastings test to see whether we accept the move
            if random() <= ProbabilityRatio:
                for j in range(Dimension):
                    PositionOld[i,j] = PositionNew[i,j]
                    QuantumForceOld[i,j] = QuantumForceNew[i,j]
                wfold = wfnew
        DeltaE = LocalEnergy(PositionOld,alpha,beta)
        DerPsi = DerivativeWFansatz(PositionOld,alpha,beta)
        DeltaPsi += DerPsi
        energy += DeltaE
        DerivativePsiE += DerPsi*DeltaE
            
    # We calculate mean values
    energy /= NumberMCcycles
    DerivativePsiE /= NumberMCcycles
    DeltaPsi /= NumberMCcycles
    EnergyDer  = 2*(DerivativePsiE-DeltaPsi*energy)
    return EnergyDer


# Computing the expectation value of the local energy 
def Energy(x0):
    # Parameters in the Fokker-Planck simulation of the quantum force
    D = 0.5
    TimeStep = 0.05
    # positions
    PositionOld = np.zeros((NumberParticles,Dimension), np.double)
    PositionNew = np.zeros((NumberParticles,Dimension), np.double)
    # Quantum force
    QuantumForceOld = np.zeros((NumberParticles,Dimension), np.double)
    QuantumForceNew = np.zeros((NumberParticles,Dimension), np.double)

    energy = 0.0
    DeltaE = 0.0
    alpha = x0[0]
    beta = x0[1]
    #Initial position
    for i in range(NumberParticles):
        for j in range(Dimension):
            PositionOld[i,j] = normalvariate(0.0,1.0)*sqrt(TimeStep)
    wfold = WaveFunction(PositionOld,alpha,beta)
    QuantumForceOld = QuantumForce(PositionOld,alpha, beta)

    #Loop over MC MCcycles
    for MCcycle in range(NumberMCcycles):
        #Trial position moving one particle at the time
        for i in range(NumberParticles):
            for j in range(Dimension):
                PositionNew[i,j] = PositionOld[i,j]+normalvariate(0.0,1.0)*sqrt(TimeStep)+\
                                       QuantumForceOld[i,j]*TimeStep*D
            wfnew = WaveFunction(PositionNew,alpha,beta)
            QuantumForceNew = QuantumForce(PositionNew,alpha, beta)
            GreensFunction = 0.0
            for j in range(Dimension):
                GreensFunction += 0.5*(QuantumForceOld[i,j]+QuantumForceNew[i,j])*\
	                              (D*TimeStep*0.5*(QuantumForceOld[i,j]-QuantumForceNew[i,j])-\
                                      PositionNew[i,j]+PositionOld[i,j])
      
            GreensFunction = exp(GreensFunction)
            ProbabilityRatio = GreensFunction*wfnew**2/wfold**2
            #Metropolis-Hastings test to see whether we accept the move
            if random() <= ProbabilityRatio:
                for j in range(Dimension):
                    PositionOld[i,j] = PositionNew[i,j]
                    QuantumForceOld[i,j] = QuantumForceNew[i,j]
                wfold = wfnew
        DeltaE = LocalEnergy(PositionOld,alpha,beta)
        energy += DeltaE
        if Printout: 
           outfile.write('%f\n' %(energy/(MCcycle+1.0)))            
    # We calculate mean values
    energy /= NumberMCcycles
    return energy

#Here starts the main program with variable declarations
NumberParticles = 2
Dimension = 2
# seed for rng generator 
seed()
# Monte Carlo cycles for parameter optimization
Printout = False
NumberMCcycles= 10000
# guess for variational parameters
x0 = np.array([0.9,0.2])
# Using Broydens method to find optimal parameters
res = minimize(Energy, x0, method='BFGS', jac=EnergyDerivative, options={'gtol': 1e-4,'disp': True})
x0 = res.x
# Compute the energy again with the optimal parameters and increased number of Monte Cycles
NumberMCcycles= 2**19
Printout = True
FinalEnergy = Energy(x0)
EResult = np.array([FinalEnergy,FinalEnergy])
outfile.close()
#nice printout with Pandas
import pandas as pd
from pandas import DataFrame
data ={'Optimal Parameters':x0, 'Final Energy':EResult}
frame = pd.DataFrame(data)
print(frame)

# Common imports
import os

# Where to save the figures and data files
DATA_ID = "Results/EnergyMin"

def data_path(dat_id):
    return os.path.join(DATA_ID, dat_id)

infile = open(data_path("Energies.dat"),'r')

from numpy import std, mean, concatenate, arange, loadtxt, zeros, ceil
from numpy.random import randint
from time import time


def tsboot(data,statistic,R,l):
    t = zeros(R); n = len(data); k = int(ceil(float(n)/l));
    inds = arange(n); t0 = time()
    
    # time series bootstrap
    for i in range(R):
        # construct bootstrap sample from
        # k chunks of data. The chunksize is l
        _data = concatenate([data[j:j+l] for j in randint(0,n-l,k)])[0:n];
        t[i] = statistic(_data)

    # analysis
    print ("Runtime: %g sec" % (time()-t0)); print ("Bootstrap Statistics :")
    print ("%8g %14g %15g" % (statistic(data), \
                             mean(t) - statistic(data), \
                             std(t) ))
    return t
# Read in data
X = loadtxt(infile)
# statistic to be estimated. Takes two args.
# arg1: the data
def stat(data):
    return mean(data)
t = tsboot(X, stat, 2**12, 2**10)

# Common imports
import os

# Where to save the figures and data files
DATA_ID = "Results/EnergyMin"

def data_path(dat_id):
    return os.path.join(DATA_ID, dat_id)

infile = open(data_path("Energies.dat"),'r')

from numpy import log2, zeros, mean, var, sum, loadtxt, arange, array, cumsum, dot, transpose, diagonal, sqrt
from numpy.linalg import inv

def block(x):
    # preliminaries
    n = len(x)
    d = int(log2(n))
    s, gamma = zeros(d), zeros(d)
    mu = mean(x)

    # estimate the auto-covariance and variances 
    # for each blocking transformation
    for i in arange(0,d):
        n = len(x)
        # estimate autocovariance of x
        gamma[i] = (n)**(-1)*sum( (x[0:(n-1)]-mu)*(x[1:n]-mu) )
        # estimate variance of x
        s[i] = var(x)
        # perform blocking transformation
        x = 0.5*(x[0::2] + x[1::2])
   
    # generate the test observator M_k from the theorem
    M = (cumsum( ((gamma/s)**2*2**arange(1,d+1)[::-1])[::-1] )  )[::-1]

    # we need a list of magic numbers
    q =array([6.634897,9.210340, 11.344867, 13.276704, 15.086272, 16.811894, 18.475307, 20.090235, 21.665994, 23.209251, 24.724970, 26.216967, 27.688250, 29.141238, 30.577914, 31.999927, 33.408664, 34.805306, 36.190869, 37.566235, 38.932173, 40.289360, 41.638398, 42.979820, 44.314105, 45.641683, 46.962942, 48.278236, 49.587884, 50.892181])

    # use magic to determine when we should have stopped blocking
    for k in arange(0,d):
        if(M[k] < q[k]):
            break
    if (k >= d-1):
        print("Warning: Use more data")
    return mu, s[k]/2**(d-k)


x = loadtxt(infile)
(mean, var) = block(x) 
std = sqrt(var)
import pandas as pd
from pandas import DataFrame
data ={'Mean':[mean], 'STDev':[std]}
frame = pd.DataFrame(data,index=['Values'])
print(frame)

Resampling methods are an indispensable tool in modern statistics. They involve repeatedly drawing samples from a training set and refitting a model of interest on each sample in order to obtain additional information about the fitted model. For example, in order to estimate the variability of a linear regression fit, we can repeatedly draw different samples from the training data, fit a linear regression to each new sample, and then examine the extent to which the resulting fits differ. Such an approach may allow us to obtain information that would not be available from fitting the model only once using the original training sample.

Resampling approaches can be computationally expensive, because they involve fitting the same statistical method multiple times using different subsets of the training data. However, due to recent advances in computing power, the computational requirements of resampling methods generally are not prohibitive. In this chapter, we discuss two of the most commonly used resampling methods, cross-validation and the bootstrap. Both methods are important tools in the practical application of many statistical learning procedures. For example, cross-validation can be used to estimate the test error associated with a given statistical learning method in order to evaluate its performance, or to select the appropriate level of flexibility. The process of evaluating a model’s performance is known as model assessment, whereas the process of selecting the proper level of flexibility for a model is known as model selection. The bootstrap is widely used.

Statistical analysis

1. Our simulations can be treated as computer experiments. This is particularly the case for Monte Carlo methods
2. The results can be analysed with the same statistical tools as we would use analysing experimental data.
3. As in all experiments, we are looking for expectation values and an estimate of how accurate they are, i.e., possible sources for errors.

1. As in other experiments, many numerical experiments have two classes of errors:
   1. Statistical errors
   2. Systematical errors

Statistical errors can be estimated using standard tools from statistics.

Systematical errors are method specific and must be treated differently from case to case.

The probability distribution function (PDF) is a function \( p(x) \) on the domain which, in the discrete case, gives us the probability or relative frequency with which these values of \( X \) occur:

$$ p(x) = \mathrm{prob}(X=x) $$

In the continuous case, the PDF does not directly depict the actual probability. Instead we define the probability for the stochastic variable to assume any value on an infinitesimal interval around \( x \) to be \( p(x)dx \). The continuous function \( p(x) \) then gives us the density of the probability rather than the probability itself. The probability for a stochastic variable to assume any value on a non-infinitesimal interval \( [a,\,b] \) is then just the integral:

$$ \mathrm{prob}(a\leq X\leq b) = \int_a^b p(x)dx $$

Qualitatively speaking, a stochastic variable represents the values of numbers chosen as if by chance from some specified PDF so that the selection of a large set of these numbers reproduces this PDF.

A particularly useful class of special expectation values are the moments. The \( n \)-th moment of the PDF \( p \) is defined as follows:

$$ \langle x^n\rangle \equiv \int\! x^n p(x)\,dx $$

The zero-th moment \( \langle 1\rangle \) is just the normalization condition of \( p \). The first moment, \( \langle x\rangle \), is called the mean of \( p \) and often denoted by the letter \( \mu \):

$$ \langle x\rangle = \mu \equiv \int\! x p(x)\,dx $$

A special version of the moments is the set of central moments, the n-th central moment defined as:

$$ \langle (x-\langle x \rangle )^n\rangle \equiv \int\! (x-\langle x\rangle)^n p(x)\,dx $$

The zero-th and first central moments are both trivial, equal \( 1 \) and \( 0 \), respectively. But the second central moment, known as the variance of \( p \), is of particular interest. For the stochastic variable \( X \), the variance is denoted as \( \sigma^2_X \) or \( \mathrm{var}(X) \):

$$ \begin{align} \sigma^2_X\ \ =\ \ \mathrm{var}(X) & = \langle (x-\langle x\rangle)^2\rangle = \int\! (x-\langle x\rangle)^2 p(x)\,dx \label{_auto1}\\ & = \int\! \left(x^2 - 2 x \langle x\rangle^{2} + \langle x\rangle^2\right)p(x)\,dx \label{_auto2}\\ & = \langle x^2\rangle - 2 \langle x\rangle\langle x\rangle + \langle x\rangle^2 \label{_auto3}\\ & = \langle x^2\rangle - \langle x\rangle^2 \label{_auto4} \end{align} $$

The square root of the variance, \( \sigma =\sqrt{\langle (x-\langle x\rangle)^2\rangle} \) is called the standard deviation of \( p \). It is clearly just the RMS (root-mean-square) value of the deviation of the PDF from its mean value, interpreted qualitatively as the spread of \( p \) around its mean.

Another important quantity is the so called covariance, a variant of the above defined variance. Consider again the set \( \{X_i\} \) of \( n \) stochastic variables (not necessarily uncorrelated) with the multivariate PDF \( P(x_1,\dots,x_n) \). The covariance of two of the stochastic variables, \( X_i \) and \( X_j \), is defined as follows:

$$ \begin{align} \mathrm{cov}(X_i,\,X_j) &\equiv \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle \nonumber\\ &= \int\!\cdots\!\int\!(x_i-\langle x_i \rangle)(x_j-\langle x_j \rangle)\, P(x_1,\dots,x_n)\,dx_1\dots dx_n \label{eq:def_covariance} \end{align} $$

with

$$ \langle x_i\rangle = \int\!\cdots\!\int\!x_i\,P(x_1,\dots,x_n)\,dx_1\dots dx_n $$

If we consider the above covariance as a matrix \( C_{ij}=\mathrm{cov}(X_i,\,X_j) \), then the diagonal elements are just the familiar variances, \( C_{ii} = \mathrm{cov}(X_i,\,X_i) = \mathrm{var}(X_i) \). It turns out that all the off-diagonal elements are zero if the stochastic variables are uncorrelated. This is easy to show, keeping in mind the linearity of the expectation value. Consider the stochastic variables \( X_i \) and \( X_j \), (\( i\neq j \)):

$$ \begin{align} \mathrm{cov}(X_i,\,X_j) &= \langle(x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle \label{_auto5}\\ &=\langle x_i x_j - x_i\langle x_j\rangle - \langle x_i\rangle x_j + \langle x_i\rangle\langle x_j\rangle\rangle \label{_auto6}\\ &=\langle x_i x_j\rangle - \langle x_i\langle x_j\rangle\rangle - \langle \langle x_i\rangle x_j\rangle + \langle \langle x_i\rangle\langle x_j\rangle\rangle \label{_auto7}\\ &=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle - \langle x_i\rangle\langle x_j\rangle + \langle x_i\rangle\langle x_j\rangle \label{_auto8}\\ &=\langle x_i x_j\rangle - \langle x_i\rangle\langle x_j\rangle \label{_auto9} \end{align} $$

If \( X_i \) and \( X_j \) are independent, we get \( \langle x_i x_j\rangle =\langle x_i\rangle\langle x_j\rangle \), resulting in \( \mathrm{cov}(X_i, X_j) = 0\ \ (i\neq j) \).

Also useful for us is the covariance of linear combinations of stochastic variables. Let \( \{X_i\} \) and \( \{Y_i\} \) be two sets of stochastic variables. Let also \( \{a_i\} \) and \( \{b_i\} \) be two sets of scalars. Consider the linear combination:

$$ U = \sum_i a_i X_i \qquad V = \sum_j b_j Y_j $$

By the linearity of the expectation value

$$ \mathrm{cov}(U, V) = \sum_{i,j}a_i b_j \mathrm{cov}(X_i, Y_j) $$

Now, since the variance is just \( \mathrm{var}(X_i) = \mathrm{cov}(X_i, X_i) \), we get the variance of the linear combination \( U = \sum_i a_i X_i \):

$$ \begin{equation} \mathrm{var}(U) = \sum_{i,j}a_i a_j \mathrm{cov}(X_i, X_j) \label{eq:variance_linear_combination} \end{equation} $$

And in the special case when the stochastic variables are uncorrelated, the off-diagonal elements of the covariance are as we know zero, resulting in:

$$ \mathrm{var}(U) = \sum_i a_i^2 \mathrm{cov}(X_i, X_i) = \sum_i a_i^2 \mathrm{var}(X_i) $$ $$ \mathrm{var}(\sum_i a_i X_i) = \sum_i a_i^2 \mathrm{var}(X_i) $$

which will become very useful in our study of the error in the mean value of a set of measurements.

A stochastic process is a process that produces sequentially a chain of values:

$$ \{x_1, x_2,\dots\,x_k,\dots\}. $$

We will call these values our measurements and the entire set as our measured sample. The action of measuring all the elements of a sample we will call a stochastic experiment since, operationally, they are often associated with results of empirical observation of some physical or mathematical phenomena; precisely an experiment. We assume that these values are distributed according to some PDF \( p_X^{\phantom X}(x) \), where \( X \) is just the formal symbol for the stochastic variable whose PDF is \( p_X^{\phantom X}(x) \). Instead of trying to determine the full distribution \( p \) we are often only interested in finding the few lowest moments, like the mean \( \mu_X^{\phantom X} \) and the variance \( \sigma_X^{\phantom X} \).

In practical situations a sample is always of finite size. Let that size be \( n \). The expectation value of a sample, the sample mean, is then defined as follows:

$$ \bar{x}_n \equiv \frac{1}{n}\sum_{k=1}^n x_k $$

The sample variance is:

$$ \mathrm{var}(x) \equiv \frac{1}{n}\sum_{k=1}^n (x_k - \bar{x}_n)^2 $$

its square root being the standard deviation of the sample. The sample covariance is:

$$ \mathrm{cov}(x)\equiv\frac{1}{n}\sum_{kl}(x_k - \bar{x}_n)(x_l - \bar{x}_n) $$

Note that the sample variance is the sample covariance without the cross terms. In a similar manner as the covariance in Eq. \eqref{eq:def_covariance} is a measure of the correlation between two stochastic variables, the above defined sample covariance is a measure of the sequential correlation between succeeding measurements of a sample.

These quantities, being known experimental values, differ significantly from and must not be confused with the similarly named quantities for stochastic variables, mean \( \mu_X \), variance \( \mathrm{var}(X) \) and covariance \( \mathrm{cov}(X,Y) \).

The law of large numbers states that as the size of our sample grows to infinity, the sample mean approaches the true mean \( \mu_X^{\phantom X} \) of the chosen PDF:

$$ \lim_{n\to\infty}\bar{x}_n = \mu_X^{\phantom X} $$

The sample mean \( \bar{x}_n \) works therefore as an estimate of the true mean \( \mu_X^{\phantom X} \).

What we need to find out is how good an approximation \( \bar{x}_n \) is to \( \mu_X^{\phantom X} \). In any stochastic measurement, an estimated mean is of no use to us without a measure of its error. A quantity that tells us how well we can reproduce it in another experiment. We are therefore interested in the PDF of the sample mean itself. Its standard deviation will be a measure of the spread of sample means, and we will simply call it the error of the sample mean, or just sample error, and denote it by \( \mathrm{err}_X^{\phantom X} \). In practice, we will only be able to produce an estimate of the sample error since the exact value would require the knowledge of the true PDFs behind, which we usually do not have.

Let us first take a look at what happens to the sample error as the size of the sample grows. In a sample, each of the measurements \( x_i \) can be associated with its own stochastic variable \( X_i \). The stochastic variable \( \overline X_n \) for the sample mean \( \bar{x}_n \) is then just a linear combination, already familiar to us:

$$ \overline X_n = \frac{1}{n}\sum_{i=1}^n X_i $$

All the coefficients are just equal \( 1/n \). The PDF of \( \overline X_n \), denoted by \( p_{\overline X_n}(x) \) is the desired PDF of the sample means.

The probability density of obtaining a sample mean \( \bar x_n \) is the product of probabilities of obtaining arbitrary values \( x_1, x_2,\dots,x_n \) with the constraint that the mean of the set \( \{x_i\} \) is \( \bar x_n \):

$$ p_{\overline X_n}(x) = \int p_X^{\phantom X}(x_1)\cdots \int p_X^{\phantom X}(x_n)\ \delta\!\left(x - \frac{x_1+x_2+\dots+x_n}{n}\right)dx_n \cdots dx_1 $$

And in particular we are interested in its variance \( \mathrm{var}(\overline X_n) \).

It is generally not possible to express \( p_{\overline X_n}(x) \) in a closed form given an arbitrary PDF \( p_X^{\phantom X} \) and a number \( n \). But for the limit \( n\to\infty \) it is possible to make an approximation. The very important result is called the central limit theorem. It tells us that as \( n \) goes to infinity, \( p_{\overline X_n}(x) \) approaches a Gaussian distribution whose mean and variance equal the true mean and variance, \( \mu_{X}^{\phantom X} \) and \( \sigma_{X}^{2} \), respectively:

$$ \begin{equation} \lim_{n\to\infty} p_{\overline X_n}(x) = \left(\frac{n}{2\pi\mathrm{var}(X)}\right)^{1/2} e^{-\frac{n(x-\bar x_n)^2}{2\mathrm{var}(X)}} \label{eq:central_limit_gaussian} \end{equation} $$

The desired variance \( \mathrm{var}(\overline X_n) \), i.e. the sample error squared \( \mathrm{err}_X^2 \), is given by:

$$ \begin{equation} \mathrm{err}_X^2 = \mathrm{var}(\overline X_n) = \frac{1}{n^2} \sum_{ij} \mathrm{cov}(X_i, X_j) \label{eq:error_exact} \end{equation} $$

We see now that in order to calculate the exact error of the sample with the above expression, we would need the true means \( \mu_{X_i}^{\phantom X} \) of the stochastic variables \( X_i \). To calculate these requires that we know the true multivariate PDF of all the \( X_i \). But this PDF is unknown to us, we have only got the measurements of one sample. The best we can do is to let the sample itself be an estimate of the PDF of each of the \( X_i \), estimating all properties of \( X_i \) through the measurements of the sample.

Our estimate of \( \mu_{X_i}^{\phantom X} \) is then the sample mean \( \bar x \) itself, in accordance with the the central limit theorem:

$$ \mu_{X_i}^{\phantom X} = \langle x_i\rangle \approx \frac{1}{n}\sum_{k=1}^n x_k = \bar x $$

Using \( \bar x \) in place of \( \mu_{X_i}^{\phantom X} \) we can give an estimate of the covariance in Eq. \eqref{eq:error_exact}

$$ \mathrm{cov}(X_i, X_j) = \langle (x_i-\langle x_i\rangle)(x_j-\langle x_j\rangle)\rangle \approx\langle (x_i - \bar x)(x_j - \bar{x})\rangle, $$

resulting in

$$ \frac{1}{n} \sum_{l}^n \left(\frac{1}{n}\sum_{k}^n (x_k -\bar x_n)(x_l - \bar x_n)\right)=\frac{1}{n}\frac{1}{n} \sum_{kl} (x_k -\bar x_n)(x_l - \bar x_n)=\frac{1}{n}\mathrm{cov}(x) $$

By the same procedure we can use the sample variance as an estimate of the variance of any of the stochastic variables \( X_i \)

$$ \mathrm{var}(X_i)=\langle x_i - \langle x_i\rangle\rangle \approx \langle x_i - \bar x_n\rangle\nonumber, $$

which is approximated as

$$ \begin{equation} \mathrm{var}(X_i)\approx \frac{1}{n}\sum_{k=1}^n (x_k - \bar x_n)=\mathrm{var}(x) \label{eq:var_estimate_i_think} \end{equation} $$

Now we can calculate an estimate of the error \( \mathrm{err}_X^{\phantom X} \) of the sample mean \( \bar x_n \):

$$ \begin{align} \mathrm{err}_X^2 &=\frac{1}{n^2}\sum_{ij} \mathrm{cov}(X_i, X_j) \nonumber \\ &\approx&\frac{1}{n^2}\sum_{ij}\frac{1}{n}\mathrm{cov}(x) =\frac{1}{n^2}n^2\frac{1}{n}\mathrm{cov}(x)\nonumber\\ &=\frac{1}{n}\mathrm{cov}(x) \label{eq:error_estimate} \end{align} $$

which is nothing but the sample covariance divided by the number of measurements in the sample.

In the special case that the measurements of the sample are uncorrelated (equivalently the stochastic variables \( X_i \) are uncorrelated) we have that the off-diagonal elements of the covariance are zero. This gives the following estimate of the sample error:

$$ \mathrm{err}_X^2=\frac{1}{n^2}\sum_{ij} \mathrm{cov}(X_i, X_j) = \frac{1}{n^2} \sum_i \mathrm{var}(X_i), $$

resulting in

$$ \begin{equation} \mathrm{err}_X^2\approx \frac{1}{n^2} \sum_i \mathrm{var}(x)= \frac{1}{n}\mathrm{var}(x) \label{eq:error_estimate_uncorrel} \end{equation} $$

where in the second step we have used Eq. \eqref{eq:var_estimate_i_think}. The error of the sample is then just its standard deviation divided by the square root of the number of measurements the sample contains. This is a very useful formula which is easy to compute. It acts as a first approximation to the error, but in numerical experiments, we cannot overlook the always present correlations.

For computational purposes one usually splits up the estimate of \( \mathrm{err}_X^2 \), given by Eq. \eqref{eq:error_estimate}, into two parts

$$ \mathrm{err}_X^2 = \frac{1}{n}\mathrm{var}(x) + \frac{1}{n}(\mathrm{cov}(x)-\mathrm{var}(x)), $$

which equals

$$ \begin{equation} \frac{1}{n^2}\sum_{k=1}^n (x_k - \bar x_n)^2 +\frac{2}{n^2}\sum_{k < l} (x_k - \bar x_n)(x_l - \bar x_n) \label{eq:error_estimate_split_up} \end{equation} $$

The first term is the same as the error in the uncorrelated case, Eq. \eqref{eq:error_estimate_uncorrel}. This means that the second term accounts for the error correction due to correlation between the measurements. For uncorrelated measurements this second term is zero.

Computationally the uncorrelated first term is much easier to treat efficiently than the second.

$$ \mathrm{var}(x) = \frac{1}{n}\sum_{k=1}^n (x_k - \bar x_n)^2 = \left(\frac{1}{n}\sum_{k=1}^n x_k^2\right) - \bar x_n^2 $$

We just accumulate separately the values \( x^2 \) and \( x \) for every measurement \( x \) we receive. The correlation term, though, has to be calculated at the end of the experiment since we need all the measurements to calculate the cross terms. Therefore, all measurements have to be stored throughout the experiment.

Let us analyze the problem by splitting up the correlation term into partial sums of the form:

$$ f_d = \frac{1}{n-d}\sum_{k=1}^{n-d}(x_k - \bar x_n)(x_{k+d} - \bar x_n) $$

The correlation term of the error can now be rewritten in terms of \( f_d \)

$$ \frac{2}{n}\sum_{k < l} (x_k - \bar x_n)(x_l - \bar x_n) = 2\sum_{d=1}^{n-1} f_d $$

The value of \( f_d \) reflects the correlation between measurements separated by the distance \( d \) in the sample samples. Notice that for \( d=0 \), \( f \) is just the sample variance, \( \mathrm{var}(x) \). If we divide \( f_d \) by \( \mathrm{var}(x) \), we arrive at the so called autocorrelation function

$$ \kappa_d = \frac{f_d}{\mathrm{var}(x)} $$

which gives us a useful measure of pairwise correlations starting always at \( 1 \) for \( d=0 \).

The sample error (see eq. \eqref{eq:error_estimate_split_up}) can now be written in terms of the autocorrelation function:

$$ \begin{align} \mathrm{err}_X^2 &= \frac{1}{n}\mathrm{var}(x)+\frac{2}{n}\cdot\mathrm{var}(x)\sum_{d=1}^{n-1} \frac{f_d}{\mathrm{var}(x)}\nonumber\\ &=& \left(1+2\sum_{d=1}^{n-1}\kappa_d\right)\frac{1}{n}\mathrm{var}(x)\nonumber\\ &=\frac{\tau}{n}\cdot\mathrm{var}(x) \label{eq:error_estimate_corr_time} \end{align} $$

and we see that \( \mathrm{err}_X \) can be expressed in terms the uncorrelated sample variance times a correction factor \( \tau \) which accounts for the correlation between measurements. We call this correction factor the autocorrelation time:

$$ \begin{equation} \tau = 1+2\sum_{d=1}^{n-1}\kappa_d \label{eq:autocorrelation_time} \end{equation} $$

For a correlation free experiment, \( \tau \) equals 1. From the point of view of eq. \eqref{eq:error_estimate_corr_time} we can interpret a sequential correlation as an effective reduction of the number of measurements by a factor \( \tau \). The effective number of measurements becomes:

$$ n_\mathrm{eff} = \frac{n}{\tau} $$

To neglect the autocorrelation time \( \tau \) will always cause our simple uncorrelated estimate of \( \mathrm{err}_X^2\approx \mathrm{var}(x)/n \) to be less than the true sample error. The estimate of the error will be too good. On the other hand, the calculation of the full autocorrelation time poses an efficiency problem if the set of measurements is very large.

The so-called time-displacement autocorrelation \( \phi(t) \) for a quantity \( \mathbf{M} \) is given by

$$ \phi(t) = \int dt' \left[\mathbf{M}(t')-\langle \mathbf{M} \rangle\right]\left[\mathbf{M}(t'+t)-\langle \mathbf{M} \rangle\right], $$

which can be rewritten as

$$ \phi(t) = \int dt' \left[\mathbf{M}(t')\mathbf{M}(t'+t)-\langle \mathbf{M} \rangle^2\right], $$

where \( \langle \mathbf{M} \rangle \) is the average value and \( \mathbf{M}(t) \) its instantaneous value. We can discretize this function as follows, where we used our set of computed values \( \mathbf{M}(t) \) for a set of discretized times (our Monte Carlo cycles corresponding to moving all electrons?)

$$ \phi(t) = \frac{1}{t_{\mathrm{max}}-t}\sum_{t'=0}^{t_{\mathrm{max}}-t}\mathbf{M}(t')\mathbf{M}(t'+t) -\frac{1}{t_{\mathrm{max}}-t}\sum_{t'=0}^{t_{\mathrm{max}}-t}\mathbf{M}(t')\times \frac{1}{t_{\mathrm{max}}-t}\sum_{t'=0}^{t_{\mathrm{max}}-t}\mathbf{M}(t'+t). \label{eq:phitf} $$

One should be careful with times close to \( t_{\mathrm{max}} \), the upper limit of the sums becomes small and we end up integrating over a rather small time interval. This means that the statistical error in \( \phi(t) \) due to the random nature of the fluctuations in \( \mathbf{M}(t) \) can become large.

One should therefore choose \( t \ll t_{\mathrm{max}} \).

Note that the variable \( \mathbf{M} \) can be any expectation values of interest.

The time-correlation function gives a measure of the correlation between the various values of the variable at a time \( t' \) and a time \( t'+t \). If we multiply the values of \( \mathbf{M} \) at these two different times, we will get a positive contribution if they are fluctuating in the same direction, or a negative value if they fluctuate in the opposite direction. If we then integrate over time, or use the discretized version of, the time correlation function \( \phi(t) \) should take a non-zero value if the fluctuations are correlated, else it should gradually go to zero. For times a long way apart the different values of \( \mathbf{M} \) are most likely uncorrelated and \( \phi(t) \) should be zero.

We can derive the correlation time by observing that our Metropolis algorithm is based on a random walk in the space of all possible spin configurations. Our probability distribution function \( \mathbf{\hat{w}}(t) \) after a given number of time steps \( t \) could be written as

$$ \mathbf{\hat{w}}(t) = \mathbf{\hat{W}^t\hat{w}}(0), $$

with \( \mathbf{\hat{w}}(0) \) the distribution at \( t=0 \) and \( \mathbf{\hat{W}} \) representing the transition probability matrix. We can always expand \( \mathbf{\hat{w}}(0) \) in terms of the right eigenvectors of \( \mathbf{\hat{v}} \) of \( \mathbf{\hat{W}} \) as

$$ \mathbf{\hat{w}}(0) = \sum_i\alpha_i\mathbf{\hat{v}}_i, $$

resulting in

$$ \mathbf{\hat{w}}(t) = \mathbf{\hat{W}}^t\mathbf{\hat{w}}(0)=\mathbf{\hat{W}}^t\sum_i\alpha_i\mathbf{\hat{v}}_i= \sum_i\lambda_i^t\alpha_i\mathbf{\hat{v}}_i, $$

with \( \lambda_i \) the \( i^{\mathrm{th}} \) eigenvalue corresponding to the eigenvector \( \mathbf{\hat{v}}_i \).

If we assume that \( \lambda_0 \) is the largest eigenvector we see that in the limit \( t\rightarrow \infty \), \( \mathbf{\hat{w}}(t) \) becomes proportional to the corresponding eigenvector \( \mathbf{\hat{v}}_0 \). This is our steady state or final distribution.

We can relate this property to an observable like the mean energy. With the probabilty \( \mathbf{\hat{w}}(t) \) (which in our case is the squared trial wave function) we can write the expectation values as

$$ \langle \mathbf{M}(t) \rangle = \sum_{\mu} \mathbf{\hat{w}}(t)_{\mu}\mathbf{M}_{\mu}, $$

or as the scalar of a vector product

$$ \langle \mathbf{M}(t) \rangle = \mathbf{\hat{w}}(t)\mathbf{m}, $$

with \( \mathbf{m} \) being the vector whose elements are the values of \( \mathbf{M}_{\mu} \) in its various microstates \( \mu \).

We rewrite this relation as

$$ \langle \mathbf{M}(t) \rangle = \mathbf{\hat{w}}(t)\mathbf{m}=\sum_i\lambda_i^t\alpha_i\mathbf{\hat{v}}_i\mathbf{m}_i. $$

If we define \( m_i=\mathbf{\hat{v}}_i\mathbf{m}_i \) as the expectation value of \( \mathbf{M} \) in the \( i^{\mathrm{th}} \) eigenstate we can rewrite the last equation as

$$ \langle \mathbf{M}(t) \rangle = \sum_i\lambda_i^t\alpha_im_i. $$

Since we have that in the limit \( t\rightarrow \infty \) the mean value is dominated by the the largest eigenvalue \( \lambda_0 \), we can rewrite the last equation as

$$ \langle \mathbf{M}(t) \rangle = \langle \mathbf{M}(\infty) \rangle+\sum_{i\ne 0}\lambda_i^t\alpha_im_i. $$

We define the quantity

$$ \tau_i=-\frac{1}{log\lambda_i}, $$

and rewrite the last expectation value as

$$ \langle \mathbf{M}(t) \rangle = \langle \mathbf{M}(\infty) \rangle+\sum_{i\ne 0}\alpha_im_ie^{-t/\tau_i}. \label{eq:finalmeanm} $$

The quantities \( \tau_i \) are the correlation times for the system. They control also the auto-correlation function discussed above. The longest correlation time is obviously given by the second largest eigenvalue \( \tau_1 \), which normally defines the correlation time discussed above. For large times, this is the only correlation time that survives. If higher eigenvalues of the transition matrix are well separated from \( \lambda_1 \) and we simulate long enough, \( \tau_1 \) may well define the correlation time. In other cases we may not be able to extract a reliable result for \( \tau_1 \). Coming back to the time correlation function \( \phi(t) \) we can present a more general definition in terms of the mean magnetizations $ \langle \mathbf{M}(t) \rangle$. Recalling that the mean value is equal to $ \langle \mathbf{M}(\infty) \rangle$ we arrive at the expectation values

$$ \phi(t) =\langle \mathbf{M}(0)-\mathbf{M}(\infty)\rangle \langle \mathbf{M}(t)-\mathbf{M}(\infty)\rangle, $$

resulting in

$$ \phi(t) =\sum_{i,j\ne 0}m_i\alpha_im_j\alpha_je^{-t/\tau_i}, $$

which is appropriate for all times.

If the correlation function decays exponentially

$$ \phi (t) \sim \exp{(-t/\tau)}$$

then the exponential correlation time can be computed as the average

$$ \tau_{\mathrm{exp}} = -\langle \frac{t}{log|\frac{\phi(t)}{\phi(0)}|} \rangle. $$

If the decay is exponential, then

$$ \int_0^{\infty} dt \phi(t) = \int_0^{\infty} dt \phi(0)\exp{(-t/\tau)} = \tau \phi(0),$$

which suggests another measure of correlation

$$ \tau_{\mathrm{int}} = \sum_k \frac{\phi(k)}{\phi(0)}, $$

called the integrated correlation time.

from numpy import *
from numpy.random import randint, randn
from time import time

def jackknife(data, stat):
    n = len(data);t = zeros(n); inds = arange(n); t0 = time()
    ## 'jackknifing' by leaving out an observation for each i                                                                                                                      
    for i in range(n):
        t[i] = stat(delete(data,i) )

    # analysis                                                                                                                                                                     
    print("Runtime: %g sec" % (time()-t0)); print("Jackknife Statistics :")
    print("original           bias      std. error")
    print("%8g %14g %15g" % (stat(data),(n-1)*mean(t)/n, (n*var(t))**.5))

    return t


# Returns mean of data samples                                                                                                                                                     
def stat(data):
    return mean(data)


mu, sigma = 100, 15
datapoints = 10000
x = mu + sigma*random.randn(datapoints)
# jackknife returns the data sample                                                                                                                                                
t = jackknife(x, stat)

Bootstrapping is a nonparametric approach to statistical inference that substitutes computation for more traditional distributional assumptions and asymptotic results. Bootstrapping offers a number of advantages:

1. The bootstrap is quite general, although there are some cases in which it fails.
2. Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small.
3. It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically.
4. It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples).

%matplotlib inline

from numpy import *
from numpy.random import randint, randn
from time import time
from scipy.stats import norm
import matplotlib.pyplot as plt

# Returns mean of bootstrap samples                                                                                                                                                
def stat(data):
    return mean(data)

# Bootstrap algorithm                                                                                                                                                              
def bootstrap(data, statistic, R):
    t = zeros(R); n = len(data); inds = arange(n); t0 = time()

    # non-parametric bootstrap                                                                                                                                                     
    for i in range(R):
        t[i] = statistic(data[randint(0,n,n)])

    # analysis                                                                                                                                                                     
    print("Runtime: %g sec" % (time()-t0)); print("Bootstrap Statistics :")
    print("original           bias      std. error")
    print("%8g %8g %14g %15g" % (statistic(data), std(data),\
                             mean(t), \
                             std(t)))
    return t


mu, sigma = 100, 15
datapoints = 10000
x = mu + sigma*random.randn(datapoints)
# bootstrap returns the data sample                                                                                                          t = bootstrap(x, stat, datapoints)
# the histogram of the bootstrapped  data  
t = bootstrap(x, stat, datapoints)
# the histogram of the bootstrapped  data                                            
n, binsboot, patches = plt.hist(t, bins=50, density='true',histtype='bar', color='red', alpha=0.75)

# add a 'best fit' line                                                                                                                                                          
y = norm.pdf( binsboot, mean(t), std(t))
lt = plt.plot(binsboot, y, 'r--', linewidth=1)
plt.xlabel('Smarts')
plt.ylabel('Probability')
plt.axis([99.5, 100.6, 0, 3.0])
plt.grid(True)

plt.show()