1. Chapter 4a Stochastic Modelingμx σx x
f (x)
Prof. Lei He
Electrical Engineering Department
University of California, Los Angeles
URL: eda.ee.ucla.edu

2. Outline
Introduction to Stochastic Modeling
Monte Carlo Simulation
Example: SRAM cell Yield Estimation with MC&QMC

3. R.V. X can take value from its domain randomly
Review of Probability
R.V. X can take value from its domain randomly
Domain can be continuous/discrete, finite/infinite
PDF vs. CDF x dx
f (x) 1 x
F (x)

4. Review of Probability Mean and Variance Normal Distribution σx μx
f (x)

5. Multivariate Distribution
Similar definition can be extended for multivariate cases
Joint PDF (JPDF), Covariance
Becomes much more complicated
Correlation MATTERS!!

6. 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),

7. Correlation Coefficient
Normalized covariance:
Always lies between -1 and 1
Correlation of 1  x ~ y, -1 

8. 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.

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

10. The 1st PC z1 is a minimum distance fit to a line in x space
Geometric Interpretation of Principal Components
The 1st PC z1 is a minimum distance fit to a line in x space
The 2nd PC z2 is a minimum distance fit to a line in the plane
perpendicular to the 1st PC
PCs are a series of linear least squares fits to a sample,
each orthogonal to all the previous.

11. Outline
Introduction to Stochastic Modeling
Monte Carlo Simulation

12. 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

13. 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

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

15. Quasi Monte Carlo Simulation
Basic idea
Use deterministic samples instead of pure random samples
Select deterministic samples to cover the whole sample space evenly

16. Discrepancy measures how evenly the samples are in the sample place
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

17. Low Discrepancy Sequence
Sample sequence with low discrepancy
Low discrepancy array generation algorithms
Faure sequence
Neiderreiter sequence
Sobol sequence
Halton Sequence

18. 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 = => = 1/8
5 = => = 5/8
6 = => = 3/8
7 = => = 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

19. Advantage and Disadvantage of QMC Simulation
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

20. Comparison of MC and QMC
QMC converges faster than MC

21. 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.”

22. 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 below
Initial Value:
BL_B=1; Q_B=0; Q=1; BL=1;
Variation Source:
Vth (threshold voltage) of Mn1 and Mn2
Leff of Mn1 and Mn2
Device Model:
Use BSIM3 model for all MOSFETs

23. netlist Netlist for 6-T cell SRAM * SRAM netlist Vdd dd 0 5
Mn nmos
Mn nmos
Mn nmos
Mn nmos
Mp5 3 2 dd dd pmos
Mp6 2 3 dd dd pmos
all MOSFETs should use BSIM3 model

24. 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