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

Bruce H. Stanley, Oct. 16, 2003 Del. Chapter of ASA Meeting: MGF and 1 st -Order Dissipation Slide 2 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 first- order 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.

Bruce H. Stanley, Oct. 16, 2003 Del. Chapter of ASA Meeting: MGF and 1 st -Order Dissipation Slide 3 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

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

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

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

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

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

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

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

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

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

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

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

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

Bruce H. Stanley, Oct. 16, 2003 Del. Chapter of ASA Meeting: MGF and 1 st -Order Dissipation Slide 16 Some Possible Dissipation Rate Distributions Uniformr ~ U(min, max) Normalr ~ N(  r,  2 r ) Lognormalr ~ LN(  r = e  +  2 /2,  2 r =  r 2. (e  2 -1))  = ln[  r /  (1+  r 2 /  2 r )],;  2 = ln[1+ (  r 2 /  2 r )] Gammar ~  (  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.

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

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

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

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

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

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

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

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

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

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

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

Bruce H. Stanley, Oct. 16, 2003 Del. Chapter of ASA Meeting: MGF and 1 st -Order Dissipation Slide 28 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)………..… Other “Non-linear” Models

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

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

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

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

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

Bruce H. Stanley, Oct. 16, 2003 Del. Chapter of ASA Meeting: MGF and 1 st -Order Dissipation Slide 34 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.,

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

Bruce H. Stanley, Oct. 16, 2003 Del. Chapter of ASA Meeting: MGF and 1 st -Order Dissipation Slide 36 References Duffy, M. J., M. K. Hanafey, D. M. Linn, M. H. Russell and C. J. Peter Predicting sulfonylurea herbicide behavior under field conditions Proc. Brit. Crop Prot. Conf. – Weeds. 2: [Application of 2-compartment first-order model] Gustafson, D. I. And L. R. Holden Nonlinear pesticide dissipation in Soil: a new model based upon spatial variability. Environ. Sci. Technol. 24 (7): [Distributed rate model] Hamaker, J. W Decomposition: quantitative aspects. Pp 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 Dislodgeable residues of ethion in Florida citrus and relationships to weather variables. Arch. Environ. Contam. Toxicol. 6: [First-order model with cumulative environmental variables] Pree, D. J., K. P. Butler, E. R. Kimball and D. K. R. Stewart Persistence of foliar residues of dimethoate and azinphosmethyl and their toxicity to the apple maggot. J. Econ. Entomol. 69: [First-order model with non-zero asymptote]

Bruce H. Stanley, Oct. 16, 2003 Del. Chapter of ASA Meeting: MGF and 1 st -Order Dissipation Slide 37 Conclusions Moment-generating function is a quick way to predict the effects of variability on dissipation Variability in dissipation rates can lead to “non- linear” (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”)

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

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