3.3 Bootstrap Confidence interval
3.3.3 The Normal Method
When the distribution of is asymptotically normal, we can get a confidence interval from the familiar normal distribution. The estimation is the estimated from the original data and the estimate, , is from each bootstrap sample. The 1 - α/2 and α/2 quantile confidence interval is
(18)
The Bb is the bootstrap bias estimation, and sb is the bootstrap SE from (16) and (17). The flowchart is shown in the Figure 6.
Figure 6. Flowchart of Bias-Corrected and Accelerated confidence interval ˆ ]
ˆ ,
[ Bbz(1/2)sb Bbz(/2)sb
ˆ
ˆ
ˆi
15
3.4 Botstrap Confidence Intervals for the Correlation Coefficient
Pearson’s correlation coefficient, rxy, is often used to measure the association between two sampled (data size n) variables, x and y. For binormally distributed data, Fisher’s (1921) transformation yields z being asymptotically distributed with variance 1/(n-3).
However, most cases are nonnormal data distributions and make normal-theory-based intervals questionable. The bootstrap is a good candidate to solve these problems. There are no distributionary assumptions because it uses the data to simulate the distribution. We can use it to find a confidence interval for the correlation coefficient of two path delays.
Figure 7. Flow of silicon data bootstrap
From Figure 7, we measurer two path delays and use bootstrap re-sampling. We can get an approximation of the population distribution. If the path delay is approximated normal, we can use Bootstrap normal confidence interval to estimate spatial path delay correlations. If the path delay is skewed, we can use Bootstrap Bias-Corrected and Accelerated confidence interval to estimate delay correlations. If we know nothing about path delay, we can use Bootstrap percentile confidence interval to estimate delay correlations.
1 ) log(1 5 . 0
xy xy
r z r
Chapter 4
Experimental Results
4.1 Experiment Setup
We design a novel test chip to run HSPICE to simulate path delay spatial correlations. Many papers propose the Ring Oscillator (RO) to find spatial process variation [14]. To get an efficient experiment result, we use the multiplexor with a 5 by 5 inverter trend in an inverter array. Each inverter trend has 900 inverters and the inverter trend array is 1640 um by 1640 um. The layout is shown in Figure 8. The inverter trend array can reduce process variations including photo impact, etch impact, CMP impact and micro loading impact. We simulate the 5 by 5 inverter trend delay with HSPICE simulations.
Figure 8. Test Chip layout with multiplexor
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4.2 Spatial Path Correlations with Multivariate Normal Gate Length Variations
To detail understanding the correlation of CD variation and delays, we first simulate the single inverter delay with gate length CD variations. Suppose the gate length variation is normal and its histogram is shown in Figure 9. Its delay after Hspice simulation is shown in Figure 10 and is high correlated to CD variationsFigure 9. Gate length histogram
68.5
L gate vs Delay
Figure 10. A single inverter delay simulation with gate CD variations
For the 5 by 5 inverter trend, the gate CD variation is multivariate normal distribution with correlation matrix
To validate the accuracy of path base spatial correlation, we run Monte Carlo to generate 100000 patterns and set it as a golden set. The path delay distribution of the golden set is shown in Figure 11.
It shows inverter trend 1 and 2 has higher correlation than inverter trend of 1 and 3 due to the spatial distance. Close path distance has higher correlations.
Figure 11. Path delay correlation distributions of inverter trend 1, 2 and 3
We select 100 samples from the golden set and run 2000 bootstrap re-sampling for the 5 by 5 inverter trend. The delay spatial correlation of this is in Figure 12.
Figure 12. The Histograms of correlation coefficient of Bootstrap of 5 by 5 inverter trend
19
The histogram is approximated as normal distribution. Inverter trend 1 and 6 (in Figure 8) are most close and have the highest correlation distribution. The correlation of inverter trend 1 and 5 is lower due to the distance of them is far. The histogram variation of them is wide.
Confidence interval coverage rate is the probability that the confidence interval includes the true parameter from the population. If the coverage rate is the same as the stated size of the confidence interval, the intervals are accurate. We simulate 5 kinds of bootstrap confidence interval: Percentile method, Bias-Corrected and Accelerated method, Normal method, Normal method with outlier filter and Fisher Z. The normal with outlier filter is normal method excluding outlier with 3 standard variations.
The coverage rates are listed in Figure 13 with population size 100000, sample size 3000 and confidence α=95%
Coverage Rate for different Bootstrap times
0.94 0.942 0.944 0.946 0.948 0.95 0.952 0.954 0.956
1000 2000 3000
Bootstrap time
Coverage Rate BCA_Percentile
Normal
Normal_Z_Filter Percentile Fisher_Z
Figure 13. Coverage rate for BCA, Percentile, Normal, Normal with filter and Fisher Z
The coverage rate is around 95% under confidence α=95%. The Bootstrap normal confidence interval is better than other bootstrap ones due to normal distributions of gate length variations. The Fisher Z confidence interval has the same performance with Bootstrap methods. This is also because normal gate length variations. From Figure 12, we investigate the coverage rate is better as bootstrap times increase.
The correlations of path delays to delta x direction distance and delta y direction distance are listed in the Figure 14 and 15. The delta x distance is two inverter trend distance in the x direction and the delta y distance is two inverter trend distance in the y direction. The path delay correlation decrease as the path distance increase. When the trends are closer, the bound is smaller. This is because long distances have low correlations and cause more variations from them.
Delta X distance vs Path correlation (Sample=3000)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 1 2 3 4 5
Delta X distance (normalized)
Path Correlation B=1000 Upper Bound
B=1000 Lower Bound B=100 Upper Bound B=100 Lower Bound Real
Figure 14. Confidence interval bound for delta x distance
Delta Y distance vs Path correlation (Sample=3000)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 1 2 3 4 5
Delta Y distance (normalized)
Path Correlation B=1000 Upper Bound
B=1000 Lower Bound B=100 Upper Bound B=100 Lower Bound Real
Figure 15. Confidence interval bound for delta Y distance
21
4.3 Correlations with Multivariate Chi Square Gate Length Variations
Most path delays are not normal distributed. We demonstrate a gate length variation with chi square distribution. The Figure 16 shows the histogram of a single gate length variation.
Figure 16. The histogram of a single gate length variation with chi square distribution
For the inverter trend array, multi gate length variations are multivariate chi square distribution with the same correlation matrix (19) and ρ=0.98. We use Bootstrap methodology to resample the correlation coefficient of the path delay. The result is shown in Figure17. The histogram shows the correlation coefficient distribution is non-normal.
Figure 17. The histogram of the correlation coefficient of Bootstrap of 1 and 2 inverter trend
We simulate 5 kinds of Bootstrap confidence interval under the 95% confidence. The coverage rate is shown in Figure 18. The Bootstrap confidence intervals have 94.5% coverage rate but Fisher Z is lower about 93%. This is because the path delay is non-normal distribution and it is not suitable for Fisher Z confidence interval.
Figure 18. Coverage rate for BCA, Percentile, normal, normal with filter and Fisher Z
The correlations of path delays to delta x direction distances and delta y direction distances are listed in the Figure 19 and 20.
Delta X distance vs Path correlation (Sample=3000)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
0 1 2 3 4 5
Delta X distance (normalized)
Path Correlation B=1000 Upper Bound
B=1000 Lower Bound B=100 Upper Bound B=100 Lower Bound Real
Figure 19. Confidence interval bound for delta x distance
23
Delta Y distance vs Path correlation (Sample=3000)
0 0.2 0.4 0.6 0.8 1 1.2
0 1 2 3 4 5
Delta Y distance (normalized)
Path Correlation B=1000 Upper Bound
B=1000 Lower Bound B=100 Upper Bound B=100 Lower Bound Real
Figure 20. Confidence interval bound for delta Y distance
The gap of the upper bound and the lower bound is big than normal case. This is because the data are skewed seriously.
Chapter 5
Conclusion
In this thesis, we have applied bootstrap re-sampling algorithm and path-based learning methodology to estimate the spatial correlation of the path delay on the test chip. From this way, we can get a confidence interval for spatial correlation with a good coverage rate. It is also applied to non-normal correlation distribution. The plot of the path distance and the path delay correlation give us a predation way and give designer guidance for timing analysis. For the future work, we can select an initial correlation matrix which fall into Bootstap confidence intervals and put the initial correlation matrix to SSTA models. With the Bayesian theory or other learning methodologies, we use new silicon data to adjust the initial correlation matrix. After repeating this, we can get a converged correlation matrix which approximates to the true value.
25
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