Comparison of IC-CAP's Analysis Methods

IC-CAP has several statistical analysis options for different situations: direct Monte Carlo, generated Monte Carlo, corner modeling, parametric boundary modeling, and Non-Parametric Boundary modeling.

The difference between direct and generated Monte Carlo is that direct uses the original extracted data and skips fitting a multidimensional Gaussian PDF.

The strength of direct Monte Carlo is that it can be used with the smallest amounts of data, it provides a yield estimate, and it is non-parametric. While a Monte Carlo yield estimate is noisy, the level of noise goes essentially as S-0.5, where S is the number of samples. However, one gets some feel for robustness even if the estimate is corrupted. The disadvantage, which it shares with Non-Parametric Boundary analysis, is that it does not provide factor (independent variable) reduction or allow for discarding the raw data set. For large enough data sets and given sufficient time and computing resources, direct Monte Carlo gives increasingly accurate yield estimates.

Generated Monte Carlo shares the primary strength of direct Monte Carlo —a yield estimate. Additionally, generated Monte Carlo allows for factor reduction via factor analysis or principal component analysis. Additionally, the method requires the calculation of a Gaussian PDF covariance matrix and mean vector from which the Monte Carlo samples are generated. Generated Monte Carlo requires more raw data than direct Monte Carlo because a sufficiently accurate covariance matrix and mean vector are required. Given that the data really is Gaussian and given enough data to calculate the covariance matrix, then the raw data can be discarded. As with direct Monte Carlo the compute time and resources can be prohibitive.

Corner and parametric boundary analyses both allow for factor reduction and discarding the raw data. The previous caveats regarding data volume/covariance accuracy and appropriateness of the Gaussian PDF also apply. Given the exponential increase of corner models as the number of factors (independent variables) increases, parametric boundary analysis is more desirable. In fact, corner modeling can lead to such high numbers of generated model points that parametric modeling is almost always more desirable. Additionally, parametric boundary modeling has the potential for providing a lower bound on the circuit yield. Corner modeling could be made to do the same thing, but the relationship between placement of generated points and the enclosure fraction is not direct. Corner analysis is included as an option for users of traditional analysis methods. We strongly recommend parametric or Non-Parametric Boundary analysis instead.

Parametric boundary analysis is a shortcut compared to a full-blown Monte Carlo analysis. Computing resources are saved by judiciously picking a much smaller set of component parameter points to simulate. These points are likely to be the extremes that cause the overall design not to meet its performance specifications. As mentioned earlier the enclosure percentage corresponding to a particular set of boundary points can, under the proper circumstances, place a lower bound on the yield of the design.

Non-parametric boundary analysis is also a shortcut to a full-blown Monte Carlo analysis. Non-parametric shares all of the strengths of parametric analysis discussed except factor reduction is lost and the raw data must be retained. Further advantages of Non-Parametric Boundary analysis are that it works on any distribution of data since the method does not require the fitting of a PDF. This quality alleviates the effort involved and the inherent loss of accuracy arising from fitting any function to real data. Non-parametric analysis also avoids the problem of returning component parameter values that cannot physically exist; the method picks actual instances (or samples) from the raw data set provided.

All of the parametric methods have the potential for returning non-physical component values. If, for various reasons, the fitted PDF is not appropriate for the actual data, then this problem of unrealistic returned values becomes quite serious. A simple example of this is shown at the end of this section. The non-parametric method even gives better results when the amount of raw data is small and the dimension of the data is moderate to large. This circumstance arises from the limited accuracy of the covariance matrix in high dimensional problems with limited numbers of samples. Non-parametric boundary analysis has no dimensional limit. Furthermore, the computational effort of Non-Parametric Boundary analysis does not suffer from the curse of dimensionality. (The curse of dimensionality is the phenomena that an algorithm, which operates on a problem of varying dimension, exhibits an exponential increase in computational effort as a function of problem dimension.)