The Moment Generating Function As A Useful Tool

The Moment Generating Function As A Useful Tool in Understanding Random Effects on First-Order Environmental Dissipation Processes Dr. Bruce H. Stanley Du. Pont Crop Protection Stine-Haskell Research Center Newark, Delaware Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 1 Bruce H. Stanley, Oct. 16, 2003

The Moment Generating Function As A Useful Tool in Understanding Random Effects on First. Order Environmental Dissipation Processes Abstract Many physical and, thus, environmental processes follow firstorder kinetics, where the rate of change of a substance is proportional to its concentration. The rate of change may be affected by a variety of factors, such as temperature or light intensity, that follow a probability distribution. The moment generating function provides a quick method to estimate the mean and variance of the process through time. This allows valuable insights for environmental risk assessment or process optimization. Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 2 Bruce H. Stanley, Oct. 16, 2003

Agenda • First-order (FO) dissipation • The moment generating function (MGF) • Relationship between FO dissipation and MGF • Calculating the variance of dissipation • Other “curvilinear” models • Half-lives of the models • References • Conclusions Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 3 Bruce H. Stanley, Oct. 16, 2003

- First-Order Dissipation - Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 4 Bruce H. Stanley, Oct. 16, 2003

Model: First-Order Dissipation Rate of change: Model: Transformation to linearity: Constant half-life: Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 5 Bruce H. Stanley, Oct. 16, 2003

Example: First-Order Dissipation Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 6 Bruce H. Stanley, Oct. 16, 2003

Some Processes that Follow First-Order Kinetics • Radio-active decay • Population decline (i. e. , “death” processes) • Compounded interest/depreciation • Chemical decomposition • Etc… Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 7 Bruce H. Stanley, Oct. 16, 2003

- The Moment Generating Function - Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 8 Bruce H. Stanley, Oct. 16, 2003

Definition: Moment Generating Function Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 9 Bruce H. Stanley, Oct. 16, 2003

Example: Moment Generating Function X ~ Gamma( , ) Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 10 Bruce H. Stanley, Oct. 16, 2003

Relationship Between – First-Order Dissipation – and the Moment Generating Function Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 11 Bruce H. Stanley, Oct. 16, 2003

Random First-Order Dissipation where r ~ PDF Constant Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 12 Bruce H. Stanley, Oct. 16, 2003

Conceptual Model: Distribution of Dissipation Rates d. Ct 1/dt = r 1. Ct 1 d. Ct 2/dt = r 2. Ct 2 d. Ct 3/dt = r 3. Ct 3 d. Ct 4/dt = r 4. Ct 4 r<0 Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 13 Bruce H. Stanley, Oct. 16, 2003

Transformation of r or t? r<0 X = -r It’s easier to transform t, I. e. , = -t so substitute t = - And treat r’s as positive when necessary r = -1. X fr(r) = f. X(-r) E(rn) = (-1)n. E(Xn) Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 14 Bruce H. Stanley, Oct. 16, 2003

Typical Table of Distributions (Mood, Graybill & Boes. 1974. Intro. To the Theory of Stats. , 3 rd Ed. Mc. Graw-Hill. 564 pp. ) Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 15 Bruce H. Stanley, Oct. 16, 2003

Some Possible Dissipation Rate Distributions • Uniform r ~ U(min, max) • Normal r ~ N( r, 2 r) • Lognormal r ~ LN( r= 2/2 + e , 2 r= r 2. (e 2 -1)) = ln[ r / (1+ r 2/ 2 r)], ; 2 = ln[1+ ( r 2/ 2 r)] • Gamma r ~ ( r= / , 2 r = / 2) = r 2/ 2 r; = r/ 2 r (distribution used in Gustafson and Holden 1990) * Where r and 2 r are the expected value and variance of the untransformed rates, respectively. Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 16 Bruce H. Stanley, Oct. 16, 2003

Application to Dissipation Model: Uniform No need to make = -t substitution Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 17 Bruce H. Stanley, Oct. 16, 2003

Application to Dissipation Model: Normal No need to make = -t substitution Note: Begins increasing at t = - r/ r 2, and becomes >C 0 after t = -2. r/ r 2. Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 18 Bruce H. Stanley, Oct. 16, 2003

Application to Dissipation Model: Lognormal Note: Same as normal on the log scale. Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 19 Bruce H. Stanley, Oct. 16, 2003

Application to Dissipation Model: Gamma (Gustafson and Holden (1990) Model) Make = -t substitution Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 20 Bruce H. Stanley, Oct. 16, 2003

Distributed Loss Model Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 21 Bruce H. Stanley, Oct. 16, 2003

Key Paper: Gustafson & Holden (1990) Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 22 Bruce H. Stanley, Oct. 16, 2003

- Calculating the Variance - Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 23 Bruce H. Stanley, Oct. 16, 2003

Example: Variance for the Gamma Case Make = -t substitution Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 24 Bruce H. Stanley, Oct. 16, 2003

- Random Initial Concentration - Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 25 Bruce H. Stanley, Oct. 16, 2003

Variable Initial Concentration: Product of Random Variables Delta Method Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 26 Bruce H. Stanley, Oct. 16, 2003

- Other “Non-Linear” Models - Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 27 Bruce H. Stanley, Oct. 16, 2003

Other “Non-linear” Models • Bi- (or multi-) first-order model ………. . . • Non-linear functions of time, …………. . …… e. g. , t = degree days or cum. rainfall (Nigg et al. 1977) • First-order with asymptote (Pree et al. 1976). . • Two-compartment first-order………………. . • Distributed loss rate…………. …… (Gustafson and Holden 1990) • Power-rate model (Hamaker 1972)………. . … Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 28 Bruce H. Stanley, Oct. 16, 2003

First-order With Asymptote Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 29 Bruce H. Stanley, Oct. 16, 2003

Two Compartment Model Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 30 Bruce H. Stanley, Oct. 16, 2003

Distributed Loss Model Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 31 Bruce H. Stanley, Oct. 16, 2003

Power Rate Model Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 32 Bruce H. Stanley, Oct. 16, 2003

- Half-lives - Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 33 Bruce H. Stanley, Oct. 16, 2003

Half-lives for Various Models (p = 0. 5) • First-order*……………. • Multi-first-order*………………… • First-order with asymptote ……… • Two-compartment first-order …… • Distributed loss rate ……………. . • Power-rate model ………………. * Can substitute cumulative environmental factor for time, i. e. , Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 34 Bruce H. Stanley, Oct. 16, 2003

- References - Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 35 Bruce H. Stanley, Oct. 16, 2003

References Duffy, M. J. , M. K. Hanafey, D. M. Linn, M. H. Russell and C. J. Peter. 1987. Predicting sulfonylurea herbicide behavior under field conditions Proc. Brit. Crop Prot. Conf. – Weeds. 2: 541 -547. [Application of 2 -compartment first-order model] Gustafson, D. I. And L. R. Holden. 1990. Nonlinear pesticide dissipation in Soil: a new model based upon spatial variability. Environ. Sci. Technol. 24 (7): 1032 -1038. [Distributed rate model] Hamaker, J. W. 1972. Decomposition: quantitative aspects. Pp. 253 -340 In C. A. I. Goring and J. W. Hamaker (eds. ) Organic Chemicals in the Soil Environment, Vol 1. Marcel Dekker, Inc. , NY. [Power rate model] Nigg, H. N. , J. C. Allen, R. F. Brooks, G. J. Edwards, N. P. Thompson, R. W. King and A. H. Blagg. 1977. Dislodgeable residues of ethion in Florida citrus and relationships to weather variables. Arch. Environ. Contam. Toxicol. 6: 257 -267. [First-order model with cumulative environmental variables] Pree, D. J. , K. P. Butler, E. R. Kimball and D. K. R. Stewart. 1976. Persistence of foliar residues of dimethoate and azinphosmethyl and their toxicity to the apple maggot. J. Econ. Entomol. 69: 473 -478. [First-order model with non-zero asymptote] Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 36 Bruce H. Stanley, Oct. 16, 2003

Conclusions • Moment-generating function is a quick way to predict the effects of variability on dissipation • Variability in dissipation rates can lead to “nonlinear” (on log scale) dissipation curves • Half-lives are not constant when variability is present • A number of realistic mechanisms can lead to a curvilinear dissipation curve (i. e. , model is not “diagnostic”) Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 37 Bruce H. Stanley, Oct. 16, 2003

Questions? Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 38 Bruce H. Stanley, Oct. 16, 2003

- Thank You! Del. Chapter of ASA Meeting: MGF and 1 st-Order Dissipation Slide 39 Bruce H. Stanley, Oct. 16, 2003