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 The Moment Generating Function As A Useful Tool in Understanding The Moment Generating Function As A Useful Tool in Understanding Agenda - First-Order Dissipation - Model: First-Order Dissipation Example: First-Order Dissipation Some Processes that Follow First-Order Kinetics - The Moment Generating Function - Definition: Moment Generating Function Example: Moment Generating Function Relationship Between – First-Order Dissipation – and the Moment Random First-Order Dissipation Conceptual Model: Distribution of Dissipation Rates Transformation of r or t Typical Table of Distributions (Mood, Graybill & Boes Some Possible Dissipation Rate Distributions Application to Dissipation Model: Uniform Application to Dissipation Model: Normal Application to Dissipation Model: Lognormal Application to Dissipation Model: Gamma (Gustafson and Holden (1990) Distributed Loss Model Key Paper: Gustafson & Holden (1990) - Calculating the Variance - Example: Variance for the Gamma Case - Random Initial Concentration - Variable Initial Concentration: Product of Random Variables - Other “Non-Linear” Models - Other “Non-linear” Models First-order With Asymptote Two Compartment Model Distributed Loss Model Power Rate Model - Half-lives - Half-lives for Various Models (p = 0.5) - References - References Conclusions Questions - Thank You

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## Apple 16 октября

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### 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

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### 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.

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### 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

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### Model: First-Order Dissipation

Rate of change: Model: Transformation to linearity: Constant half-life:

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### Some Processes that Follow First-Order Kinetics

Radio-active decay Population decline (i. e., “death” processes) Compounded interest/depreciation Chemical decomposition Etc…

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X ~ Gamma(?,?)

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### Relationship Between – First-Order Dissipation – and the Moment

Generating Function

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where r ~ PDF

Constant

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r < 0

dCt1/dt = r1.Ct1

dCt2/dt = r2.Ct2

dCt3/dt = r3.Ct3

dCt4/dt = r4.Ct4

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### Transformation of r or t

?= -t so substitute t = -? And treat r’s as positive when necessary

r = -1.X

fr(r) = fX(-r)

E(rn) = (-1)n.E(Xn)

r < 0

X = -r

It’s easier to transform t, I.e., ? = -t

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### Typical Table of Distributions (Mood, Graybill & Boes

1974. Intro. To the Theory of Stats., 3rd Ed. McGraw-Hill. 564 pp.)

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### Some Possible Dissipation Rate Distributions

Uniform r ~ U(min, max) Normal r ~ N(?r, ?2 r) Lognormal r ~ LN(?r= e?+? 2/2, ?2r = ?r2.(e? 2-1)) ?? = ln[?r /?(1+ ?r2/?2r)],; ? 2 = ln[1+ (?r2/?2r)] Gamma r ~ ?(?r= ?/?, ?2r = ?/?2) ?? = ?r2/?2r; ? = ?r/?2r (distribution used in Gustafson and Holden 1990) * Where ?r and ?2r are the expected value and variance of the untransformed rates, respectively.

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### Application to Dissipation Model: Uniform

No need to make ? = -t substitution

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### Application to Dissipation Model: Normal

No need to make ? = -t substitution

Note: Begins increasing at t = -?r/?r2, and becomes >C0 after t = -2.?r/?r2.

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### Application to Dissipation Model: Lognormal

Note: Same as normal on the log scale.

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### Application to Dissipation Model: Gamma (Gustafson and Holden (1990)

Model)

Make ? = -t substitution

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### Example: Variance for the Gamma Case

Make ? = -t substitution

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Delta Method

Delta Method

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### 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)………..…

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### 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.,

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### 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]

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### 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”)

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### - Thank You

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