**1. Introduction**

Multivariate adaptive regression splines (MARSP) was first proposed by Friedman [1] for solving regression-type problems. MARSP is widely used to predict the values of an outcome variable from a set of predictor variables. There are many methods for model fitting, and MARSP is one of them. Other modeling techniques include linear regression (e.g., general

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linear model), nonlinear regression (e.g., generalized linear/nonlinear models), and regression trees (e.g., classification and regression trees), and so on. It is also worth noting that a neural network, which is very popular nowadays in an era of artificial intelligence and machine learning, is also a modeling technique.

for best delay, power, area, and so on. The standard cell library provides a way for designers to

Multivariate Adaptive Regression Splines in Standard Cell Characterization for Nanometer…

http://dx.doi.org/10.5772/intechopen.74854

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The objective of standard cell characterization is to create a set of high-quality models of a standard cell library that could accurately and efficiently model cell behavior. Cell behavior may come from a variety of parameters in different aspects like capacitance, power, timing, current, waveform, and so on. Among them, one of the most important models in standard cell characterization is timing delay models. In this chapter, for simplicity, we focus on the characterization of the pin-to-pin propagation delay of standard cells. Propagation delay is the time required for the input to be propagated to the output. In other words, it is defined as the time it takes for the effect change in input to be visible at the output. Propagation delay is important because it has a direct effect on the speed at which a digital device, such as a computer, can operate. This is true of memory chips as well as microprocessors. As mentioned earlier that a gate and a cell are often interchangeable in this chapter, cell delay and gate delay

**3.** The process parameters (threshold voltages, channel lengths) of the transistors that the cell

**5.** The temperature (Although temperature is not a factor with significant impact, it is still an

Among the different factors above, the process parameters are included because of the emergingly pronounced effect called process variations, which is introduced in details in Section 2.1.1. In the later sections, we use process-voltage-temperature (PVT) parameters to denote

As mentioned above, one of the most important tasks in standard cell characterization is to find a model which can accurately capture the relationship between the cell propagation delay and the parameters that have impact on cell delay (as shown in the paragraph above). Here, the cell propagation delay is the response variable, and the impacting parameters (input transition time, output loads, VDD, and the process parameters) are the explanatory parameters. We have not talked about the number of explanatory parameters yet. But as mentioned in Section 1, MARSP is suitable for the high-dimension problem while capturing essential nonlinearities and interactions. In the following subsections, we introduce the high-dimension parameter space when characterizing the delay models of standard cells, especially when the

place cells in rows, and it enables the automated layout generations for digital ASICs.

Cell propagation delays change with many factors, including the following:

**1.** The transition time of the input causing transition at the output.

**2.** The output load being felt by the logic cell/gate.

are interchangeable as well.

is consisted of.

impacting factor).

**4.** The power supply voltage (VDD).

process parameters, VDD, and temperature.

process variations and aging effect are included [3–7].

**2.1. Introduction (problem formation)**

MARSP is a nonparametric regression procedure that makes no assumptions about the underlying functional relationships between dependent and independent variables. The form of MARSP and its coefficients are entirely derived from the regression data. The modeling strategy is called "divide and conquer," by which the input space is partitioned into a number of regions, with each region having its own regression equation. This makes MARSP particularly efficient for high-dimension problems, where other techniques most likely have accuracy issues.

As the name suggests, MARSP uses splines as its main component. Splines are piecewise curves from polynomial functions. When different splines are smoothly connected, it can result in a flexible model which can handle both linear and nonlinear situations. The connection points between different pieces are called knots, which connect the end of one region of data and the beginning of another.

The MARSP technique has been particularly popular in data mining because it does not require or assume any particular type or any class of relationship (e.g., logistic, linear, etc.) between the outcome variable of interest and the predictor variables. Instead, MARSP derives useful models (i.e., models that yield accurate predictions) even in situations where the relationship between the predictor variable and the predictor variables is difficult to approximate with parametric models. If you are interested in more information about MARSP and how it compares to other methods for nonlinear regression (or regression trees), please refer to Chapter 9 of [2].
