This section of the on-line reference includes the following subsections:
The method of least squares requires that a straight line be fitted to a set of data points. If the regression is on Y, then the sum of the squares of the vertical deviations from the points to the line is minimized. If the regression is on X, the line is fitted to a set of data points such that the sum of the squares of the horizontal deviations from the points to the line is minimized. To illustrate the method, this section presents a regression on Y. Consider the linear model [2]:
or in matrix form where bold letters indicate matrices:
(2)
where:
and:
The vector 
holds the values of the parameters.
Now let 
be the estimates of these parameters,
or the regression coefficients. The vector of estimated regression coefficients
is denoted by:
Solving for 
in Eqn. 2 requires that you
left multiply both sides by the transpose of X,
XT:
Now the term (XT X) becomes a square and invertible matrix. Then taking it to the other side of the equation gives:
(3)
Nonlinear regression is similar to linear regression, except that a
curve is fitted to the data set instead of a straight line. Just as in
the linear scenario, the sum of the squares of the horizontal and vertical
distances between the line and the points are to be minimized. In the
case of the nonlinear Gompertz model 
,
let
(4)
where:
and:
The Gauss-Newton method can be used to solve for the parameters a,
b
and c
by performing a Taylor series expansion on 
Then approximate the nonlinear model with linear terms and
employ ordinary least squares to estimate the parameters. This procedure
is performed in iterative nature and it generally leads to a solution
of the nonlinear problem.
This procedure starts by using initial estimates of the parameters a,
b
and c,
denoted as 
and 
where (0)
is the iteration number. The Taylor series expansion approximates the
mean response, 
, around the starting values,
and 
For the ith observation:
where:
Let:
So Eqn. 4 becomes:
or by shifting 
to the left of the equation:
In matrix form this is given by:
(5)
where:
and:
Note that Eqn. 5 is in the form of the general linear regression model of Eqn. 2. According to Eqn. 3, the estimate of the parameters v(0) is given by:
The revised estimated regression coefficients in matrix form are:
The least squares criterion measure, Q, should be checked to examine whether the revised regression coefficients will lead to a reasonable result. According to the Least Squares Principle, the solution to the values of the parameters are those values that minimize Q. With the starting coefficients, g(0), Q is:
And with the coefficients at the end of the first iteration, g(1), Q is:
For the Gauss-Newton method to work properly and to satisfy the Least Squares Principle, the relationship Q(k+1) < Q(k) has to hold for all k, meaning that g(k+1) gives a better estimate than g(k). The problem is not yet completely solved. Now g(1) are the starting values, producing a new set of values g(2). The process is continued until the following relationship has been satisfied:
(6)
When using the Gauss-Newton method or some other estimation procedure, it is advisable to try several sets of starting values to make sure that the solution gives relatively consistent results.
The choice of the starting values is not an easy task. A poor choice may result in a lengthy computation with many iterations. It may also lead to divergence, or to a convergence due to a local minimum. Therefore, good initial values will result in fast computations with few iterations and if multiple minima exist, it will lead to a solution that is a minimum.
Various methods were developed for obtaining valid initial values for the regression parameters. The following procedure is described by Virene [1] in estimating the Gompertz parameters. This procedure is rather simple. It will be used to get the starting values for the Gauss-Newton method, or for any other method that requires initial values. Some analysts are using this method to calculate the parameters if the data set is divisible into three groups of equal size. However, if the data set is not equally divisible, it can still, provide good initial estimates.
Consider the case where m observations are available in the form shown next. Each reliability value, Ri, is measured at the specified times, Ti.
where:
m
= 3n, n is equal
to the number of items in each equally sized group
Ti - Ti-1 = const
i = 0, 1, ..., m - 1
The Gompertz reliability equation is given by:
(7)
and:
(8)
Define:
(9)
(10)
(11)
Then:
Without loss of generality, take Ta0 = 0; then:
Solving for c yields:
(12)
Considering Eqns. 9 and 10, then:
or:
Reordering the equation yields:
(13)
If the reliability values are in percent then a needs to be divided by 100 to return the estimate in decimal format. Consider Eqns. 9 and 10 again, where:
(14)
Reordering Eqn. 14 yields:
(15)
For the special case where I = 1, from Eqns. 12, 13 and 15, the parameters are:
(16)
(17)
(18)
To estimate the values of the parameters a, b and c, do the following:
Arrange the currently available data in terms of T and R as in Table 7.1. The T values should be chosen at equal intervals and increasing in value by 1, such as one month, one hour, etc.
Table 7.1 - Design and development time versus demonstrated reliability data for a device
|
Calculate the natural log R.
Divide the column of values for log R into three groups of equal size, each containing n items. There should always be three groups. Each group should always have the same number, n, of items, measurements or values.
Add the values of the natural log R in each group, obtaining the sums identified as S1, S2 and S3, starting with the lowest values of the natural log R.
Calculate c from Eqn. 16:
Calculate b from Eqn. 17:
Calculate c from Eqn. 18:
Write the Gompertz reliability growth equation.
Substitute the value of T, the time at which the reliability goal is to be achieved, to see if the reliability is indeed to be attained if not even exceeded by T.
See Also:
Gompertz Models (Standard and Modified)
Go to weibull.com
Go
to ReliaSoft.com
©1992-2005. ReliaSoft Corporation. ALL RIGHTS RESERVED.