9. Convergence and 9. Convergence and Monte Carlo ErrorsMonte Carlo Errors

9. Convergence and 9. Convergence and Monte Carlo ErrorsMonte Carlo Errors

Measuring Convergence to

Equilibrium• Variation distance

1 2 1 2

1 2

|| || max| ( ) ( )|

1 | ( ) ( ) |

2

A

i

P P P A P A

P i P i

where P1 and P2 are two probability distributions, A is a set of states, i is a single state.

Eigenvalue Problem• Consider the matrix S defined by

[S]ij = pi½ W(i->j) pj

-½

then S is real and symmetric and eigenvalues of S satisfy|n| ≤ 1

• One of the eigenvalue must be 0=1 with eigenvector pj

½.

Spectrum Decomposition

• Then we haveUTSU = Λ, or S = U Λ UT

where Λ is a diagonal matrix with diagonal elements k and U is orthonormal matrix, U UT = I.

• W can be expressed in U, P, and Λ asW = P-½UΛUTP½

Evolution in terms of eigen-states

• Pn= P0Wn

= P0 P-½UΛUTP½ P-½UΛUTP½…

= P0 P-½UΛnUTP½

• In component form, this means

Pn(j) = ∑iP0(i) pi

-½pj½∑

kk

n uikujk

Discussion• In the limit n goes to ∞,

Pn(j) ≈ ∑iP0(i) pi-½pj

½ ui0uj0 = pj

• The leading correction to the limit isPn(j) ≈ pj + a 1

n = pj + a e-n/

Exponential Correlation Time

• We define by the next largest eigenvalue

= - 1/log 1

This number characterizes the theoretical rate of convergence in a Markov chain.

Measuring Error

• Let Qt be some quantity of interest at time step t, then sample average is

QN = (1/N) ∑t Qt

• We treat QN as a random variable. By central limit theorem, QN is normal distributed with a mean <QN>=<Q> and variance σN

2 = <QN2>-<QN>2.

<…> standards for average over the exact distribution.

Confidence Interval

• The chance that the actual mean <Q> is in the interval[ QN – σN, QN + σN ]is about 68 percents.

• σN cannot be computed (exactly) in a single MC run of length N.

Estimating Variance

22

1, 1

int

1

var( ) ( ) 1

var( )

N

N t s t st s

N

t N

Q Q Q QN

tQf t

N N

QN

The calculation of var(Q) = <Q2>-<Q>2 and int can be done in a single run of length N.

Error Formula• The above derivation gives the famous

error estimate in Monte Carlo as:

where var(Q) = <Q2>-<Q>2 can be estimated by sample variance of Qt.

intvar( ) 1Error N

QN N

Time-Dependent Correlation function and integrated correlation

time

• We define

and

22( ) s s t s s t

s s

Q Q Q Qf t

Q Q

int0, 1, 2,... 1

( ) 1 2 ( )t t

f t f t

Circular Buffer for Calculating f(t)

Qt, Current time t

Qt-1 Previous time t-1

Earliest time t-(M-1)

We store the values of Qs of the previous M-1 times and the current value Qt

Qs

An Example of f(t)

Time-dependent correlation function for 3D Ising at Tc on a 163 lattice; Swendsen-Wang dynamics.

From J S Wang, Physica A 164 (1990) 240.

Efficient Method for Computing int

We compute int by the formula

int = N σN2/var(Q)

For small value N and then extrapolating N to ∞.

From J S Wang, O Kozan and R H Swendsen, Phys Rev E 66 (2002) 057101.

Exponential and integrated correlation

times

where 1 < 1 is the second largest eigenvalue of W matrix. This result says that exponential correlation time (=-1/log1) is related to the largest integrated correlation time.

1int

1

1sup ( )

1 Q

Q

Critical Slowing Down

Tc T

The correlation time becomes large near Tc. For a finite system (Tc) Lz, with dynamical critical exponent z ≈ 2 for local moves

Relaxation towards Equilibrium

Time t

Magnetization m

T < Tc

T = Tc

T > Tc

Schematic curves of relaxation of the total magnetization as a function of time. At Tc relaxation is slow, described by power law:

m t -β/(zν)

Jackknife Method• Let n be the number of independent

samples• Let c be some estimate using all n

samples• Let ci be the same estimate but using n-

1 samples, with i-th sample removed• Then Jackknife error estimate is

2

1

( )n

J ii

c c