A computer program for estimating Gompertz curve using Gauss-Newton method of least squares is described in detail. It is based on the estimation technique proposed in Reddy (1985). The program is developed using Scilab (version 3.1.1), a freely available scientific software package that can be downloaded from http://www.scilab.org/. Data is to be fed into the program from an external disk file which should be in Microsoft Excel format. The output will contain sample size, tolerance limit, a list of initial as well as the final estimate of the parameters, standard errors, value of Gauss-Normal equations namely GN_1 GN_2 and GN_3, No. of iterations, variance(sigma^2), Durbin-Watson statistic, goodness of fit measures such as R^2, D value, covariance matrix and residuals. It also displays a graphical output of the estimated curve vis a vis the observed curve. It is an improved version of the program proposed in Dastidar (2005).

Surajit Ghosh Dastidar

English

Research Papers in Economics

JSS Journal of Statistical Software
April 2006, Volume 15, Issue 12. http://www.jstatsoft.org/
Gompertz: A Scilab Program for Estimating
Gompertz Curve Using Gauss-Newton Method of
Least Squares
Surajit Ghosh Dastidar
ICFAI University
Abstract
A computer program for estimating Gompertz curve using Gauss-Newton method of
least squares is described in detail. It is based on the estimation technique proposed in
Reddy (1985). The program is developed using Scilab (version 3.1.1), a freely available
scientiﬁc software package that can be downloaded from http://www.scilab.org/. Data
is to be fed into the program from an external disk ﬁle which should be in Microsoft Excel
format. The output will contain sample size, tolerance limit, a list of initial as well as
the ﬁnal estimate of the parameters, standard errors, value of Gauss-Normal equations
namely GN1 GN2 and GN3, No. of iterations, variance(σ2), Durbin-Watson statistic,
goodness of ﬁt measures such as R2, D value, covariance matrix and residuals. It also
displays a graphical output of the estimated curve vis a vis the observed curve. It is an
improved version of the program proposed in Dastidar (2005)
Keywords: Scilab, Gompertz, statistical software, growth modeling.
1. Introduction
Despite the diversity of available statistical software packages such as SAS (SAS Institute Inc
2005), SPSS (SPSS Inc 2003), PASS (NCSS Statistical Software 2005) there is a shortage of
stand-alone programs that allows the user to estimate the parameters of the Gompertz curve
using Gauss-Newton method of least squares. SPSS 12.0 (SPSS Inc 2003) provides SPSS
Regression Models as add-on enhancements to the full SPSS (SPSS Inc 2003) Base System
which has a general non-linear regression. SAS 9.1 (SAS Institute Inc 2005) however does
provide a programming interface to develop a program for Gompertz growth models using
modiﬁed Gauss-Newton method. PASS (NCSS Statistical Software 2005) provides users with a
modiﬁed Gompertz growth model but the method for estimating the parameters of Gompertz
curve cannot be speciﬁed by the user.2 Gompertz: A Scilab Program for Estimating Gompertz Curve
This paper documents a computer program for estimating the parameters of the Gompertz
curve using Gauss-Newton method of least squares. It is an improved version of the program
proposed in Dastidar (2005). The program is based on the estimation technique proposed in
Reddy (1985). In this program
• The sample size need not be a multiple of three.
• The user can specify the range of the data for calculating the initial estimates of A0,
B0 and C0 but the range should be a multiple of three.
In this program the critical input data to apply the computer code are:
1. Sample Size
2. Tolerance Limit
3. Input Data File
4. First Observation No.
5. Last Observation No.
It should be noted that (Last Observation No.-First Observation No.) should be a multiple
of three.
The output will contain the following:
1. Sample Size
2. Tolerance Limit
3. Initial estimate of the parameters
4. Final estimate of the parameters
5. Standard Errors
6. No. of iterations
7. Value of Gauss-Normal equations namely GN1, GN2 and GN3
8. Variance (σ2)
9. R2
10. D where D =
Pn
t=1(yt − ˆ yt)2/
Pn
t=1(yt − ¯ yt)2
11. Covariance Matrix
12. Residuals
13. Graphical display of the estimated Gompertz curve vis a vis the observed data.Journal of Statistical Software 3
The program is written in Scilab (version 3.1.1) (INRIA-ENPC 2005) and was run on a
desktop PC with a Pentium IV processor and 256 KB of RAM. Scilab is a scientiﬁc software
package to develop programs that requires numerical computations. It provides a powerful
open computing environment for engineering and scientiﬁc applications. It can be downloaded
freely from http://www.scilab.org/
2. A brief survey of well known statistical software
SPSS 12.0 (SPSS Inc 2003) provides general non-linear regression in the add-on enhancement
(SPSS Regression Models) to the full SPSS Base System. However, SAS 9.1 (SAS Institute
Inc 2005) does provide a programming interface to develop a program using NLIN procedure.
PASS (NCSS Statistical Software 2005) software uses the following modiﬁed Gompertz growth
model in its package. But the embedded procedure for calculating the estimated parameters
cannot be speciﬁed by the user.
Y = Aexp(−exp(−B(X − C))) (1)
This model is mentioned in Seber and Wild (1989)
3. Program description
The program proceeds as follows:
Before executing the program the following points needs to be checked.
1. The Scilab (version 3.1.1) software should be installed in the computer.
2. A new directory named Scilab should be created in C:\
3. The necessary input ﬁle should be created in Microsoft Excel format and the data should
be entered in the format as speciﬁed.(Figure 3) It should be noted that separate excel
ﬁles need to be created for separate variables.
4. All required input ﬁles should be saved in C:\Scilab directory.
5. The executable ﬁle ’Gompertz.sce’ also needs to be stored in C:\Scilab directory.
6. The program is executed using the following command in the Scilab command prompt.
exec(’C:\Scilab\Gompertz.sce’);
The critical user input to apply the computer program are sample size and tolerance limit.
After these two inputs are entered from the keyboard, the program asks for the input ﬁle
(stored in C:\Scilab directory). After the necessary data ﬁle has been chosen, it displays
the input data and then the program output. There is only one output format listing sample
size, tolerance limit, the names of the parameters, initial estimates, ﬁnal estimates, standard
errors, value of Gauss-Normal equations namely GN1, GN2 and GN3, No. of iterations,
variance (σ2), Durbin-Watson statistic (DW), goodness of ﬁt measures such as R2, D value
where D =
Pn
t=1((yt − ˆ yt)2/
Pn
t=1((yt − ¯ yt)2 ,covariance matrix and residuals. It also displays
a graphical output of the estimated curve vis a vis the observed curve.(Figure 1, Figure 2)4 Gompertz: A Scilab Program for Estimating Gompertz Curve
4. Notation and theory
Suppose we have observed time series data yt for t = 1,2,...,n
yt = abct
ut (2)
Where a, b and c are positive constants and logut’s are independent and identically distributed
normal random variates with mean 0 and variance σ2. Using logarithmic transformation we
have
logyt = loga + ct logb + logut or, (3)
Yt = A + CtB + Ut
where Yt = logyt, C = c, B = logb and Ut = logut.
From the least squares technique that is by taking partial derivatives of
Pn
t=1 Ut
2 with respect
to A, B and C and equating each one of them to zero we obtain
GN1 =
n X
t=1
(Yt − A − BCt) = 0
GN2 =
n X
t=1
(Yt − A − BCt)Ct = 0 (4)
GN3 =
n X
t=1
(Yt − A − BCt)BtCt−1 = 0
The estimates of A, B and C are obtained by solving the above system of Gauss-Normal
Equations 4. Also, it is easy to note that the least squares residual sum of squares is given by
[
n X
t=1
(Yt − ¯ Y )2] = [
n X
t=1
(Yt − ¯ Y )2(Ct − C)2]/[
n X
t=1
(Ct − ¯ C)2] (5)
where ¯ Y = (1/n)
Pn
t=1 Yt and ¯ C = (1/n)
Pn
t=1 Ct
Unlike the least squares estimators in the linear regression, the estimates of A, B and C may
not be unique.
Further, it may be noted here that it is not possible to obtain analytical expressions for C
which satisﬁes Equations 4 and also minimizes Equation 5 (Reddy 1977). However, given
such a value of C, the estimates of A and B are given by
ˆ A = ¯ Y − ˆ B ¯ C (6)
ˆ B =
n X
t=1
(Yt − ¯ Y )(Ct − C)/
n X
t=1
(Ct − ¯ C)2 (7)
Gauss-Newton method of least squares is applied in the algorithm to obtain the estimators
of A, B and C and their standard errors apart from the usual goodness of ﬁt measures such
as R2 and D where
D =
n X
t=1
(yt − ˆ yt)2/
n X
t=1
(yt − ¯ yt)2 (8)Journal of Statistical Software 5
where log ˆ yt = logˆ a+ ˆ ct.logˆ b. This Gauss-Newton method consists of taking linear expansion
of Yt = ft(A,B,C) = A + B.Ct around A0, B0 and C0 and retaining the ﬁrst degree terms
and then using ordinary least squares method to obtain A, B and C. The initial values of
A0, B0 and C0 are given as follows:
C0 = (D2/D1)1/r (9)
B0 = [(1 − C0)/C0].[D3
1/(D1 − D2)2] (10)
A0 = (1/3r)[(S1 + S2 + S3) − [(D2
1 + D1D2 + D2
2)/(D1 − D2)]] (11)
Where
S1 =
r X
t=1
Yt,
S2 =
2r X
t=r+1
Yt,
S3 =
3r X
t=2r+1
Yt,
D1 = S1 − S2,
D2 = S2 − S3.
And without the loss of generality it is assumed that the sample size for calculating the initial
estimates A0, B0 and C0 is a multiple of three(n=3r). The above analytical expressions for
A0, B0 and C0 are obtained by assuming the deterministic relation.
Yt = A + BCt (12)
For t = 1,2,...,n and solving for A, B and C using essentially three sets of equations. The
estimation procedure of A0, B0 and C0 is termed as ’three point’ method and this is a modiﬁed
version of the method given in Chakravarti and Laha (1967).
Using vector notation let
θ ´ = [θ1,θ2,θ3] = [A,B,C], (13)
Y 0 = (Y1,Y2,...,Yn), (14)
f0(θ) = (f1(θ1),f2(θ2),...,fn(θn)). (15)
Let F be an n × 3 matrix of partial derivatives where F = δfi(θi)/δ(θj); i = 1,2,...,n and
j = 1,2,3 and let F0 be the value of F evaluated at the initial values θ0 = (θ10,θ20,θ30). From
the Gauss-Newton method we get the estimate of
(θ − θ0) = (F0
0 F0)−1 F0
0 (Y − f(θ0)) (16)
Or the estimate of θ is given by
ˆ θ0 = θ0 + (F0
0 F0)−1 F0
0 (Y − f(θ0)) (17)
This procedure is repeated with the new values as the starting values and the procedure is
terminated if the successive values of (θ − θ0) or ((θ − θ0)/θ0) are very small and less than a
pre-speciﬁed limit.6 Gompertz: A Scilab Program for Estimating Gompertz Curve
Since the residuals Ui are independent and identically distributed normal variates with mean
0 and variance (σ2) and if ˆ θ is the ﬁnal estimate of θ, then
ˆ σ2 = (1/n)
n X
t=1
(Yi − fi(ˆ θ))2,
R2 = 1 −
 
n ˆ σ2/
n X
t=1
(Yi − ¯ Y )2
!
.
And ˆ θ is approximately normally distributed with mean θ and covariance matrix σ2 [F(θ)0F(θ)]−1.
The estimate of the covariance matrix is given by
ˆ σ2[F(ˆ θ)0F(ˆ θ)]−1 (18)
The estimates of GN1, GN2 and GN3 are obtained by substituting the ﬁnal estimates of A,
B and C in Equations 4.
To ﬁt Gompertz curve for the given data, using Gauss-Newton method of least squares the
following procedure is adopted in the computer program:
1. the estimates of A, B and C obtained from the ’three point’ method are used as the
initial values of θ0
0 = (θ10,θ20,θ30)
2. ﬁnal convergence of θ occurs at the rth iteration if
(θ
(r+1)
io ) − θr
io)/θr
io ≤ tolerance limit (user input) (19)
For all values of i, where θ
(r+1)
io denotes the values of i obtained at the (r+1) th iteration. The
tolerance limit can be speciﬁed by the user.
The advantage of Gauss-Newton method is that it is possible to conclude statistically that
1. y is increasing at an increasing rate, i.e., there is acceleration in y if logb > 0 and c > 1
2. y is increasing at a decreasing rate, i.e., there is deceleration in y if logb < 0 and c < 1
3. there is neither acceleration nor deceleration in y if the 95% conﬁdence interval of logb
contains 0 and the 95% conﬁdence interval of c contains 1.
5. Restriction
All input parameters are checked for nullity and a fault message is returned if there is an
illegal entry. Added to that the following are checked:
1. Sample size should be greater than ﬁve
2. Tolerance limit should be greater than 0
3. Input ﬁle name cannot be left blankJournal of Statistical Software 7
6. Sample input/output cases
The output of the computer program is in tabular form. It will contain a list of sample
size, tolerance limit, initial as well as the ﬁnal estimate of the parameters, standard errors,
value of Gauss-Normal equations namely GN1, GN2 and GN3, No. of iterations, variance,
Durbin-Watson statistic (DW), goodness of ﬁt measures such as R2, D, covariance matrix
and residuals. It also displays a graphical output of the estimated curve (Figure 1, Figure 2).
The program has been analyzed using , data on Tiwari’s estimates of Gross Domestic Prod-
uct (TGDP), Industrial Production (TIP), see Table 1. The sample size taken was 15 and
tolerance limit = 0.005. All the 15 observation were taken for calculating the initial estimates
A0, B0 and C0.
The input ﬁle will be an Excel format as shown in Figure 3. The output ﬁle will be in
text format. The name of the executable ﬁle is Gompertz.sce which contains the code for
estimating Gomepertz curve.
Figure 1: Plot of TGDP against year
The TGDP series is depicted in Figure 1, the output from Scilab is given below.
REPORT SHOWING RESULTS
----------------------
Sample Size = 15 Tolerance Limit = 0.0050008 Gompertz: A Scilab Program for Estimating Gompertz Curve
PARAMETER INITIAL ESTIMATES FINAL ESTIMATES STD. ERRORS DW
------- ----------------- ---------------- ------------ ------
A 3.583742 3.448600 0.571081 2.578518
B 0.961720 1.095294 0.560680
C 1.032668 1.029317 0.012232
No. of Iterations: = 3
GN1 = 0.0000000
GN2 = 0.0000000
GN3 = 0.0000001
Sigma_Hat_Square= 0.000232 R_Square= 0.992328 D=0.007672
COVARIANCE MATRIX
-------------------------
0.326134 -0.320167 0.006975
-0.320167 0.314363 -0.006851
0.006975 -0.006851 0.000150
SHOWING RESIDUALS
Y = 0.007635
Y = -0.003788
Y = -0.034316
Y = 0.019377
Y = 0.010319
Y = 0.007159
Y = 0.019571
Y = -0.023907
Y = 0.008351
Y = -0.016757
Y = 0.006385
Y = 0.007299
Y = -0.011531
Y = -0.006885
Y = 0.011088Journal of Statistical Software 9
Figure 2: Plot of TIP against year
Figure 3: Input Excel data ﬁle for TIP10 Gompertz: A Scilab Program for Estimating Gompertz Curve
The TIP series is depicted in Figure 1, the corresponding input Excel data ﬁle can be seen in
Figure 3 and the output from Scilab is given below.
REPORT SHOWING RESULTS
----------------------
Sample Size = 15 Tolerance Limit = 0.005000
PARAMETER INITIAL ESTIMATES FINAL ESTIMATES STD. ERRORS DW
------- ----------------- ---------------- ------------ ------
A 1.270229 1.726868 0.293523 1.250812
B 1.436541 0.999303 0.281053
C 1.035029 1.045774 0.009470
No. of Iterations: = 3
GN1 = -0.0000005
GN2 = -0.0000009
GN3 = -0.0000106
Sigma_Hat_Square= 0.000349 R_Square= 0.995587 D=0.004413
COVARIANCE MATRIX
-------------------------
0.086156 -0.082460 0.002769
-0.082460 0.078991 -0.002656
0.002769 -0.002656 0.000090
SHOWING RESIDUALS
Y = 0.020478
Y = -0.010949
Y = -0.013880
Y = -0.020121
Y = -0.011014
Y = 0.015730
Y = 0.046414
Y = 0.008207
Y = -0.013036
Y = -0.031044
Y = -0.002874
Y = 0.000186
Y = 0.006381
Y = 0.015183
Y = -0.009662Journal of Statistical Software 11
Year TGDP TIP
1951-52 100.01 16.59
1952-53 100.36 17.39
1953-54 109.67 18.21
1954-55 112.67 19.41
1955-56 116.56 21.11
1956-57 122.61 23.11
1957-58 122.10 23.68
1958-59 131.31 24.75
1959-60 133.50 26.03
1960-61 142.61 28.76
1961-62 149.18 31.09
1962-63 153.20 33.83
1963-64 161.28 37.04
1964-65 172.30 39.36
Table 1: Data on Tiwari’s estimates of Gross Domestic Product (TGDP), Industrial Produc-
tion (TIP), given in crores (10 million) of Indian Rupees in 1960/61 prices.
Acknowledgments
The author gratefully acknowledges Prof. V. N. Reddy of ICFAI Institute for Management
Teachers (IIMT), ICFAI University for his excellent guidance.
References
Chakravarti IM, Laha RG (1967). Handbook of Methods of Applied Statistics. John Wiley
and Sons, New York.
Dastidar SG (2005). “Gompertz (Version 1.00): A Computer Program for Estimating Gom-
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SAS Institute Inc (2005). SAS/STAT R 
 User’s Guide, Version 9.1.3. Cary, NC.
Seber GAF, Wild CJ (1989). Nonlinear Regression. John Wiley and Sons, New York.12 Gompertz: A Scilab Program for Estimating Gompertz Curve
SPSS Inc (2003). SPSS Regression Models 12.0. URL http://www.spss.com/.
Aﬃliation:
Surajit Ghosh Dastidar
ICFAI Institute for Management Teachers (IIMT)
3rd Floor, Astral Heights, 6-3-352/2 & 3
Road No.1, Banjara Hills
Hyderabad - 500 034, India
E-mail: sghoshdastidar@gmail.com
Journal of Statistical Software http://www.jstatsoft.org/
published by the American Statistical Association http://www.amstat.org/
Volume 15, Issue 12 Submitted: 2005-08-29
April 2006 Accepted: 2006-04-13

Gompertz (Version 1.00): A Computer Program for Estimating Gompertz Curve Using

Handbook of Methods of Applied Statistics.

Nonlinear Regression. John Wiley and Sons, New York.12 Gompertz: A Scilab Program for Estimating Gompertz Curve SPSS Inc

On Estimating Gompertz Curve.” Indian Institute of Management(IIM),

SAS/STAT R User’s Guide, Version 9.1.3.

Scilab (Version 3.1.1).

Statistical Fitting of Growth Curves with illustrations from

Gompertz: A Scilab Program for Estimating Gompertz Curve Using Gauss-Newton Method of Least Squares

Gompertz: A Scilab Program for Estimating Gompertz Curve Using Gauss-Newton Method of Least Squares

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