Chapter 4a

Stochastic Modeling

Prof. Lei HeElectrical Engineering Department

University of California, Los Angeles

URL: eda.ee.ucla.eduEmail: lhe@ee.ucla.eduμx

σx

x

f (x)

Outline

Introduction to Stochastic Modeling

Monte Carlo Simulation

Example: SRAM cell Yield Estimation with MC&QMC

Review of Probability

R.V. X can take value from its domain randomly Domain can be continuous/discrete, finite/infinite

PDF vs. CDF

x

dxxfxXPxF

dxxfdxxXxP

)()()(: (c.d.f.)Function on Distributi Cumulative

)()(: (p.d.f.)Function Density y Probabilit

x

dx

f (x)

1

0x

F (x)

Review of Probability

Mean and Variance

Normal Distribution

dxxfxxE

dxxfxxE

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x

x

x

xExpxf

μx

σx

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f (x)

Multivariate Distribution

Similar definition can be extended for multivariate cases

Joint PDF (JPDF), Covariance

Becomes much more complicated

Correlation MATTERS!!

)()(),(:tIndependen

0or 0),cov( :edUncorrelat

: Coeffcientn Correlatio

}{}{}{),cov(: Covariance

),(),()},({ :Mean

),(),( :JPDF

yfxfyxf

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yExExyEyx

dxdyyxfyxyxE

dxdyyxfdyyYydxxXxP

YX

xy

yx

xyxy

xy

Independent Random Variables

Two events A and B are independent P(A ∩ B) = P(A)P(B).

Two random variables X and Y are independent events A={X<=a} and B={Y<=b} are independent.

For two independent variables X and Y, we have E[X Y] = E[X] E[Y] var(X + Y) = var(X) + var(Y),

Correlation Coefficient

Normalized covariance:

Always lies between -1 and 1Correlation of 1 x ~ y, -1

Correlation( , )x y xyxy

x y

2

xy

~1

Principal Component Analysis

PAC helps to compress and classify data It reduces the dimensionality of a data set (sample) by

finding a new set of variables The new set has a smaller number of variables The new set nonetheless retains most of the original

information.

By information we mean the variation present in the sample, given by the correlations between the original variables. The new variables, called principal components (PCs), are uncorrelated, and are ordered by the fraction of the total information each retains from high to low.

Geometric Interpretation of Principal Components

A sample of n observations in the 2-D space x = (x1, x2)

Goal: to account for the variation in a sample

in as few variables as possible, to some accuracy

• The 1st PC z1 is a minimum distance fit to a line in x space

PCs are a series of linear least squares fits to a sample,

each orthogonal to all the previous.

• The 2nd PC z2 is a minimum distance fit to a line in the plane

perpendicular to the 1st PC

Geometric Interpretation of Principal Components

Outline

Introduction to Stochastic Modeling

Monte Carlo Simulation

Monte Carlo Simulation

Problem Formulation Given a set of random variables X=(X1, X2, … Xn)T and a

function of X, Y=f(X), estimate the distribution of the Y

Method Generate N samples of X=(X1, X2, … Xn)T

For each sample of X, calculate the correspondent sample of Y=f(X)

Obtain the distribution of Y from the samples of Y

Advantage and Disadvantage of MC simulation

Pro: Accurate

– – Error→0 when N→∞

Flexible– Works for any arbitrary distribution of X– Works for any arbitrary function of f

Simple– Easy to implement

Usually used as golden case in statistical analysis Con:

Not efficient– Need large N to obtain high accuracy– Need to run large number of iterations

Not suitable for statistical optimization

NError /1

Example

Given X1 and X2 are independent standard Gaussian RVs, estimate the distribution of max(X1, X2)

Quasi Monte Carlo Simulation

Basic idea Use deterministic samples instead of pure random samples Select deterministic samples to cover the whole sample

space evenly

Discrepancy

Definition

N is total number of samples, A(B, P) is the number of points in bounding box B, λs(B) is the volume of B

Discrepancy measures how evenly the samples are in the sample place

Low Discrepancy Sequence

Sample sequence with low discrepancy Low discrepancy array generation algorithms

Faure sequence Neiderreiter sequence Sobol sequence Halton Sequence

Example: Halton Sequence

Basic idea Choose a prime number as base (let's say 2) Write natural number sequence 1, 2, 3, ... in base Reverse the digits, including the decimal sign Convert back to base 10:

– 1 = 1.0 => 0.1 = 1/2 – 2 = 10.0 => 0.01 = 1/4 – 3 = 11.0 => 0.11 = 3/4 – 4 = 100.0 => 0.001 = 1/8 – 5 = 101.0 => 0.101 = 5/8 – 6 = 110.0 => 0.011 = 3/8 – 7 = 111.0 => 0.111 = 7/8

High dimensional array Use different base for different dimension Example 2-d array, X-base 2, y-base 3

– 1 => x=1/2 y=1/3– 2 => x=1/4 y=2/3– 3 => x=3/4 y=1/9– 4 => x=1/8 y=4/9– 5 => x=5/8 y=7/9– 6 => x=3/8 y=2/9– 7 => x=7/8 7=5/9

Advantage and Disadvantage of QMC Simulation

Advantage Efficient

– Use fewer sample than random Monte Carlo simulation

Disadvantage Only works in low dimension cases Very slow when number of random variations become

large Not very common in statistical analysis

Comparison of MC and QMC

QMC converges faster than MC

Reference

L. I. Smith. “A Tutorial on Principal Components Analysis”. Cornell University, USA, 2002.

Singhee, A., Rutenbar, R. “From Finance to Flip Flops: A study of Fast Quasi-Monte Carlo Methods from Computational Finance Applied to Statistical Circuit Analysis.”

Homework 5

Yield Estimation using Monte Carlo Method Consider “access time failure” : the time that voltage

difference between BL_B and BL becomes larger than certain value.

The schematic are shown as belowInitial Value:

BL_B=1; Q_B=0; Q=1; BL=1;

Variation Source:

1. Vth (threshold voltage) of Mn1 and Mn2

2. Leff of Mn1 and Mn2

Device Model:

Use BSIM3 model for all MOSFETs

netlist

Netlist for 6-T cell SRAM

* SRAM netlist

Vdd dd 0 5Mn1 3 2 0 0 nmosMn2 3 5 4 4 nmosMn3 2 3 0 0 nmosMn4 2 5 1 1 nmosMp5 3 2 dd dd pmosMp6 2 3 dd dd pmos

• all MOSFETs should use BSIM3 model

Detailed Steps

Performance Constraint: The voltage difference between BL_B and BL should be larger

than ∆v at the time-step tthresh.

Use Monte-Carlo and Quasi-Monte Carlo to calculate the yield Y, which is the percentage of circuits with satisfied performance.

Steps:– (1) Use MC and QMC to generate random sequences for

two variable parameters with Matlab code. – (2) Perform transient simulations with these sequences,

and compare the variable performance with constraint.– (3) Calculate the yield rate with definition.

•Nominal Values, Performance Constraint and Matlab code will be provided soon

•Due Feb 20th