**5. Conclusion**

**Table 3** presents the results for our framework in comparison to hSpice using these 10 test paths. **Figure 6** gives the histogram comparison of one of the paths between hSpice and GTSSTA.Results

Experimental results show our framework consumes only ~2% more runtime than quadratic

The quadratic delay model in Eq. (7) has a fixed number of operations, that is, 120 multiplications and 66 additions for a one-input gate and 224 multiplications and 120 additions for a two-input gate. The number of operations using MARSP models is not fixed, and it depends on which subspace the data sample falls into. Basically, calculating a MARSP model will have comparisons first and based on the comparison results, different equations (linear, quadratic etc.) are used for calculations. In average, the number of operations for the MARSP model is

3 c880 N1 to N878 57 2000 4148 189 185 −0.31 0.53 −14.10 25.4

avg. — — 81 2000 6608 273 268 2.72 1.25 −11.86 25.8

**Table 3.** Experimental results on 10 test paths for MARSP models and quadratic models (errors compared to golden

**Running Time (s) Path-delay** 

**SPICE Our Quad. Mean** 

60 2000 4211 198 192 3.21 1.21 −10.99 24.9

43 2000 2417 140 138 0.25 0.54 −13.95 26.6

42 2000 2412 143 144 3.54 1.49 −10.68 26.7

72 2000 5878 258 250 4.12 1.92 −10.76 26.9

54 2000 4001 184 180 2.90 1.16 −15.81 25.1

77 2000 6719 276 270 2.75 1.10 −5.94 27.5

71 2000 6454 255 251 2.53 1.55 −11.72 26.3

221 2000 19,898 704 698 4.29 1.50 −14.43 23.1

116 2000 9948 380 378 3.87 1.48 −10.23 25.8

**error per sample (OurSSTA)**

> **SD (%)**

**(%)**

**Path-delay error per sample (quadratic)**

**Mean (%) SD (%)**

in **Table 3** also show that quadratic model has limited accuracy for the 10 test paths.

**4.2. Runtime analysis**

58 Topics in Splines and Applications

**Path Circuit name**

1 c432 N102 to

2 c499 N85 to

4 c1355 G11 to

5 c1908 N19 to

6 c2670 N227 to

7 c3540 N33 to

8 c5315 N335 to

9 c6288 N290 to

10 c7552 N18 to

SPICE results).

delay model but achieves much better accuracy.

close to that of the quadratic delay model.

**Num. of stages** **Num. of samples**

**Primary input to primary output**

N421

N724

G1352

N2890

N3851

N5360

N8128

N6287

N11334

This chapter talks about the technique called multivariate adaptive regression splines (MARSP). MARSP is a nonparametric regression without taking any pre-assumed form. Instead, it adaptively constructs the model according to the provided data. MARSP has been widely used in high-dimension problems and particularly popular in data mining.

This chapter also gives an application of MARSP in semiconductor field, more specifically, in standard cell characterization. The objective of standard cell characterization is to create a set of high-quality models of a standard cell library that accurately and efficiently model cell behavior. In this work, the MARSP method is employed to characterize the gate delay as a function of many parameters including process-voltage-temperature parameters. Due to its ability of capturing essential nonlinearities and interactions, MARSP method helps to achieve significant accuracy improvement.

Some future work that is worth investigating includes extending the aging-aware MARSPbased timing analyzer to 3D integrated circuits (IC) to study the reliability of 3D ICs which tend to have reliability challenges due to the stronger heat issues. 3D ICs requires more sophisticated thermal models [27–29] and more complicated power-grid analysis [30]. As mentioned earlier, the methodology in this chapter is general to support other thermal and IR-drop models
