(21S) Gompertz Curve
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02-09-2023, 05:11 AM
Post: #1
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(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 Code: 01: 61, 41, A; LBL A 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. |
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