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

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1 The Moment Generating Function As A Useful Tool in Understanding

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

2 The Moment Generating Function As A Useful Tool in Understanding

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.

3 Agenda

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

4 - First-Order Dissipation -

- First-Order Dissipation -

5 Model: First-Order Dissipation

Model: First-Order Dissipation

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

6 Example: First-Order Dissipation

Example: First-Order Dissipation

7 Some Processes that Follow First-Order Kinetics

Some Processes that Follow First-Order Kinetics

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

8 - The Moment Generating Function -

- The Moment Generating Function -

9 Definition: Moment Generating Function

Definition: Moment Generating Function

10 Example: Moment Generating Function

Example: Moment Generating Function

X ~ Gamma(?,?)

11 Relationship Between  First-Order Dissipation  and the Moment

Relationship Between First-Order Dissipation and the Moment

Generating Function

12 Random First-Order Dissipation

Random First-Order Dissipation

where r ~ PDF

Constant

13 Conceptual Model: Distribution of Dissipation Rates

Conceptual Model: Distribution of Dissipation Rates

r < 0

dCt1/dt = r1.Ct1

dCt2/dt = r2.Ct2

dCt3/dt = r3.Ct3

dCt4/dt = r4.Ct4

14 Transformation of r or t

Transformation of r or t

?= -t so substitute t = -? And treat rs as positive when necessary

r = -1.X

fr(r) = fX(-r)

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

r < 0

X = -r

Its easier to transform t, I.e., ? = -t

15 Typical Table of Distributions (Mood, Graybill & Boes

Typical Table of Distributions (Mood, Graybill & Boes

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

16 Some Possible Dissipation Rate Distributions

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.

17 Application to Dissipation Model: Uniform

Application to Dissipation Model: Uniform

No need to make ? = -t substitution

18 Application to Dissipation Model: Normal

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.

19 Application to Dissipation Model: Lognormal

Application to Dissipation Model: Lognormal

Note: Same as normal on the log scale.

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

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

Model)

Make ? = -t substitution

21 Distributed Loss Model

Distributed Loss Model

22 Key Paper: Gustafson & Holden (1990)

Key Paper: Gustafson & Holden (1990)

23 - Calculating the Variance -

- Calculating the Variance -

24 Example: Variance for the Gamma Case

Example: Variance for the Gamma Case

Make ? = -t substitution

25 - Random Initial Concentration -

- Random Initial Concentration -

26 Variable Initial Concentration: Product of Random Variables

Variable Initial Concentration: Product of Random Variables

Delta Method

Delta Method

27 - Other Non-Linear Models -

- Other Non-Linear Models -

28 Other Non-linear Models

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)..

29 First-order With Asymptote

First-order With Asymptote

30 Two Compartment Model

Two Compartment Model

31 Distributed Loss Model

Distributed Loss Model

32 Power Rate Model

Power Rate Model

33 - Half-lives -

- Half-lives -

34 Half-lives for Various Models (p = 0.5)

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.,

35 - References -

- References -

36 References

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]

37 Conclusions

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)

38 Questions

Questions

39 - Thank You

- Thank You

-

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