HP 21S: Gompertz Curve
Introduction
This program fits sequential data to the Gompertz Curve:
y = c * a ^ (b ^ x)
where:
b = ( (S3 - S1) / (S2 - S1) ) ^ 1/m
a = e^( (S2 - S1) / b * (b - 1) / (b^m - 1)^2 )
c = e^( 1 / m * (S1 * S3 - S2^2) / (S1 + S3 - 2 * S2) )
where:
m = number of data points / 3
S1 = Σ ln x_i of the first tier
S2 = Σ ln x_i of the second tier
S3 = Σ ln x_i of the third tier
HP 21S Program Code: Gompertz Curve
Labels:
A: initialize the program
B: enter data
C: calculate b, a, c
Registers:
R1 = natural log of the sum of the first-third of data points
R2 = natural log of the sum of the second-third of data points
R3 = natural log of the sum of the third-third of data points
R4 = n = number of data points ÷ 3
R5 = a
R6 = b
R7 = c
Program Code:
step #: key code; key
01: 61, 41, A; LBL A
02: 51, 75; CLRG
03: 61, 26; RTN
04: 61, 41, b; LBL B
05: 13; LN
06: 21, 75, 1; STO+ 1
07: 26; R/S
08: 13; LN
09: 21, 75, 2; STO+ 2
10: 26; R/S
11: 13; LN
12: 21, 75, 3; STO+ 3
13: 1; 1
14: 21, 75, 4; STO+ 4
15: 22, 4; RCL 4
16: 61, 26; RTN
17: 61, 41, C; LBL C
18: 33; (
19: 22, 3; RCL 3
20: 65; ×
21: 22, 2; RCL 2
22: 34; )
23: 45; ÷
24: 33; (
25: 22, 2; RCL 2
26: 65; -
27: 22, 1; RCL 1
28: 34; )
29: 74; =
30: 14; y^x
31: 22, 4; RCL 4
32: 15; 1/x
33: 74; =
34: 21, 6; STO 6
35: 26; R/S
36: 33; (
37: 22, 2; RCL 2
38: 65; -
39: 22, 1; RCL 1
40: 34; )
41: 55; ×
42: 33; (
43: 22, 6; RCL 6
44: 65; -
45: 1; 1
46: 34; )
47: 45; ÷
48: 22, 6; RCL 6
49: 45; ÷
50: 33; (
51: 22, 6; RCL 6
52: 14; y^x
53: 22, 4; RCL 4
54: 65; -
55: 1; 1
56: 34; )
57: 51, 11; x^2
58: 74; =
59: 12; e^x
60: 21, 5; STO 5
61: 26; R/S
62: 33; (
63: 22, 1; RCL 1
64: 55; ×
65: 22, 3; RCL 3
66: 65; -
67: 22, 2; RCL 2
68: 51, 11; x^2
69: 34; )
70: 45; ÷
71: 33; (
72: 22, 4; RCL 4
73: 55; ×
74: 33; (
75: 22, 1; RCL 1
76: 75; +
77: 22, 3; RCL 3
78: 65; -
79: 2; 2
80: 55; ×
81: 22, 2; RCL 2
82: 34; )
83: 34; )
84: 74; =
85: 12; e^x
86: 21, 7; STO 7
87: 61, 26; RTN
Example
Fit the following data into a Gompertz curve:
(1, 16)
(2, 18)
(3, 19)
(4, 25)
(5, 28)
(6, 29)
(7, 32)
(8, 36)
(9, 37)
First, order the data points. Second, divide the data into three equal parts. For this example, we have 9 data points. From our example:
16, 18, 19, 25, 28, 29, 32, 36, 37
is divided into:
Group I: 16, 18, 19
Group II: 25, 28, 29
Group III: 32, 36, 37
Initialize the program: XEQ A
Enter the data:
16 XEQ B, 25 R/S, 32 R/S
18 XEQ B, 28 R/S, 36 R/S
19 XEQ B, 29 R/S, 37 R/S
Calculate the parameters: XEQ C
b: 0.82711002676, R/S
a: 2.33690515E-1, R/S
c: 48.2137954914
The Gompertz Curve is
y = 48.2137954914 * 2.33690515E-1 ^ (0.82711002676 ^ x)
Predict the 10th point:
Key strokes: RCL 6, y^x ,10, =, RCL 5, y^x, LAST, ×, RCL 7
Result: 38.776391586
Source
HP-37E & HP-38E/38C: Marketing and Forecasting Applications Hewlett Packard. May 1979.
Until next time,
Eddie
All original content copyright, © 2011-2023. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
