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Standard Gompertz Model

General Examples Using the Gompertz Models

Discrete (Success/Failure) Data

Reliability Data

Parameter Estimation for the Standard Gompertz Model

This section of the on-line reference includes the following subsections:

Linear Regression (Least Squares)

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]:

MATH

or in matrix form where bold letters indicate matrices:

 MATH (2)

where:

MATH MATH

and:

MATH

The vector $\beta $ holds the values of the parameters. Now let $\widehat{\beta }$ be the estimates of these parameters, or the regression coefficients. The vector of estimated regression coefficients is denoted by:

MATH

Solving for $\beta $ in Eqn. (2) requires the analyst to left multiply both sides by the transpose of $\QTR{bf}{X}$, $\QTR{bf}{X}^{T}$:

MATH

Now the term MATH becomes a square and invertible matrix. Then taking it to the other side of the equation gives:

MATH (3)

Nonlinear Regression

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 $R=ab^{c^{T}}$, let:

MATH (4)

where:

MATH

and:

MATH

The Gauss-Newton method can be used to solve for the parameters $a$, $b$ and $c$ by performing a Taylor series expansion on $f(T_{i},\delta ).$ Then approximate the nonlinear model with linear terms and employ ordinary least squares to estimate the parameters. This procedure is performed in an iterative manner 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 $g_{1}^{(0)},$ $g_{2}^{(0)}$ and $g_{3}^{(0)},$ where $^{(0)}$ is the iteration number. The Taylor series expansion approximates the mean response, $f(T_{i},\delta )$, around the starting values, $g_{1}^{(0)},$ $g_{2}^{(0)}$ and $g_{3}^{(0)}.$ For the $i^{th}$ observation:

MATH

where:

MATH

Let:

MATH

So Eqn. (4) becomes:

MATH

or by shifting $f_{i}^{(0)}$ to the left of the equation:

MATH

In matrix form this is given by:

MATH (5)

where:

MATHMATH

and:

MATH

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 $\nu ^{(0)}$ is given by:

 MATH

The revised estimated regression coefficients in matrix form are:

MATH

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:

MATH

And with the coefficients at the end of the first iteration, $g^{(1)}$, $Q$ is:

MATH

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:

MATH (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.

Choice of Initial Values

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, $R_{i}$, is measured at the specified times, $T_{i}$.

MATH

where:

The Gompertz reliability equation is given by:

MATH (7)

and:

MATH (8)

Define:

MATH (9)

MATH (10)

MATH (11)

Then:

MATH

Without loss of generality, take $T_{a_{0}}=0$; then:

MATH

Solving for $c$ yields:

MATH (12)

Considering Eqns. (9) and (10), then:

MATH

or:

MATH

Reordering the equation yields:

MATH (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:

 MATH (14)

Reordering Eqn. (14) yields:

MATH (15)

For the special case where $I=1$, from Eqns. (12), (13) and (15), the parameters are:

MATH (16)

MATH (17)

MATH (18)

To estimate the values of the parameters $a,b$ and $c$, do the following:

  1. 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.
  2. Calculate the natural log $R$.
  3. 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.
  4. Add the values of the natural log $R$ in each group, obtaining the sums identified as $S_{1}$, $S_{2}$ and $S_{3}$, starting with the lowest values of the natural log $R$.
  5. Calculate $c$ from Eqn. (16):

MATH

  1. Calculate $a$ from Eqn. (17):

MATH

  1. Calculate $b$ from Eqn. (18):

MATH

  1. Write the Gompertz reliability growth equation.
  2. 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 or exceeded by $T$.

Table 7.1 - Design and development time versus demonstrated reliability
data for a device

Group
Number

Growth Time
$T$ (months)

Reliability
$\QTR{small}{R}$ (%)

$ln(R)$

 

0

58

4.060

1

1

66

4.190

 

 

 

$S_{1}$ = 8.250

 

2

72.5

4.284

2

3

78

4.357

 

 

 

$S_{2}$ = 8.641

 

4

82

4.407

3

5

85

4.443

 

 

 

$S_{3}$ = 8.850