SYSTEM ANALYSIS USING RESPONSE SURFACE MODELING - Advanced Signal Integrity for High-Speed Digital Design - page 611

where SS model is the sum of squares of the model, SS error the sum of squares of
the residual, and SS total the total sum of squares.
Since the response of the system is determined by the combined effect of
the model and the error in estimation, we know that the sum of squares for the
system (SS total ) is made up of the model sum of squares and the error sum of
squares. These quantities are expressed as

i = 1 y i

2

y T y SS total = − (14-17)

n

i = 1 y i

2

b T x T y

SS model = SS total − SS error = − (14-18)

n

Equation (14-16) suggests that a perfect model fit will have SS error = 0, result-

ing in R 2 1. Although a near-unity value of R 2 is a good sign, it does not

guarantee that the model is good. This is because adding a term to the model
will always increase R 2 , even if the term is not statistically significant (we dis-
cuss significance testing in Section 14.5). An additional correlation metric that
is frequently employed is the adjusted coefficient

=

SS error / dF error
SS total / dF total

1
n − k −

n −

R adj − R 2 )

= = 1 −

1 ( 1

(14-19)

where

dF total = n − 1 (14-20)

We can use (14-15) and (14-20) to calculate the number of degrees of freedom
of the model as

dF model = k − 1 (14-21)

The inclusion of the degrees of freedom of the system and the error means that
the adjusted R 2 statistic can decrease if an insignificant term is added to the
model. Thus, dramatically different values of R 2 and R adj indicate that the model
may contain unnecessary terms.
For signaling systems, we recommend using R 2 0 . 95 and R adj 0 . 90 as fit

criteria. Systems that contain significant nonlinearity in response due to physical
effects such as reflections may have difficulty in meeting these criteria, requiring
either adjustment to the criteria or adjustments to the model.
A final check that we often use is to check that the root-mean-square error
(RMSE):

≥ ≥

SS error
dF error

=

RMSE (14-22)

is less than one-tenth of the difference between the minimum and maximum
responses observed.