HP 21S: Gompertz Curve

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.