bruce h. stanley, oct. 16, 2003 del. chapter of asa meeting: mgf and 1 st -order dissipation slide 1...

Bruce H. Stanley, Oct. 16, 2003Del. Chapter of ASA Meeting: MGF and 1st-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. StanleyDuPont Crop Protection

Stine-Haskell Research Center

Newark, Delaware


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.


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


- First-Order Dissipation -


Model: First-Order Dissipation

tt Cr

dt

dC

trt eCC 0

r

t 21ln

21

trCCt 0lnln

Rate of change:

Model:

Transformation to linearity:

Constant half-life:


First-Order Dissipation

0.1

1

10

100

1000

0 50 100 150 200 250 300

Time (days)

Co

nce

ntr

atio

n (

pp

m)

Example: First-Order Dissipation

trCCt 0lnln


Some Processes that Follow First-Order Kinetics

• Radio-active decay

• Population decline (i. e., “death” processes)

• Compounded interest/depreciation

• Chemical decomposition

• Etc…


- The Moment Generating Function -


Definition: Moment Generating Function

teEtm r

n

t

n

n

Edt

tmdr

0


Example: Moment Generating Function

t

dxexeeEtm xtxtX 1

0

2

2

2

2

00

2

222

0

1

0

0

)1()var(

tt

X

tt

X

n

t

n

n

dt

tdm

dt

tmdXEX

tdt

tdm

Edt

tmdr

X ~ Gamma(,)


Relationship Between – First-Order Dissipation –

and the Moment Generating Function


Random First-Order Dissipation

tt eCC r

0

tt eECCE r0

where r ~ PDF

Constant


Conceptual Model:Distribution of Dissipation Rates

dCt4/dt = r4

.Ct4

dCt3/dt = r3

.Ct3

dCt2/dt = r2

.Ct2

dCt1/dt = r1

.Ct1

0

0.020.04

0.060.08

0.10.12

0.140.16

0.180.2

-10 -8 -6 -4 -2 0

r

f(r)

r < 0


Transformation of r or t?

00.020.040.060.080.10.120.140.160.180.2

-10 -8 -6 -4 -2 0

r

f(r)

00.020.040.060.080.1

0.120.140.160.180.2

0 2 4 6 8 10

x = -r

f(x)

r < 0 X = -r

r = -1.X

fr(r) = fX(-r)

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

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

= -tso substitute

t = -And treat r’s as positive

when necessary


Typical Table of Distributions(Mood, Graybill & Boes. 1974. Intro. To the Theory of Stats., 3rd Ed. McGraw-Hill. 564 pp.)


Some Possible Dissipation Rate Distributions

• Uniform r ~ U(min, max)

• Normal r ~ N(r, 2 r)

• Lognormal r ~ LN(r= e+ 2/2, 2r = r

2.(e 2-1))

= ln[r /(1+ r2/2

r)],;

2 = ln[1+ (r2/2

r)]

• Gamma r ~ (r= /, 2r = /2)

= r2/2

r; = r/2r

(distribution used in Gustafson and Holden

1990)* Where r and 2

r are the expected value and variance of the untransformed rates, respectively.


Application to Dissipation Model: Uniform

tt eECCE r0

t

eeCCE

tt

tminmax

0

minmax

rr

rr

No need to make = -t substitution


Application to Dissipation Model: Normal

tt eECCE r0

20

22 tt

t

rr

eCCE

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

2.

No need to make = -t substitution


Application to Dissipation Model: Lognormal

Note: Same as normal on the log scale.


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

tt eECCE r0

2

2

2

2

2

0

200

1r

r

r

r

t

C

tC

tCCE

r

r

rr

rt

Make = -t substitution


Distributed Loss Model

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

100.00

0 2 4 6 8 10

t

C(t

)


Key Paper: Gustafson & Holden (1990)


- Calculating the Variance -


Example: Variance for the Gamma Case

2

2

2

2 2

2220

2

20

2220

22

2

2

r

r

r

r

ttC

ttC

eEeECCECECVar

rr

r

rr

r

ttttt

rr

Make = -t substitution


- Random Initial Concentration -


Variable Initial Concentration:Product of Random Variables

0

220

2

000

200

2200

2

0

000

220

0

000

2

2

,

,2

,

CVareEeVarCE

eEeCECECE

eEeCECEeE

eEeCECEeCCov

eCCoveECECVareEeVarCECVar

eECE

eCCoveECEeCECE

tt

tt

ttt

ttt

ttttt

t

tttt

rr

rr

rrr

rrr

rrrr

r

rrr

Delta Method

Delta Method


- 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)………..…

Other “Non-linear” Models ii tr

t eCC 0

t

ii dfactornDegradatiox0

_

ii xr

t eCC 0

ii tr

t eCCCC 0

tktkt eeC i 2

21

t

CCt

10

1

0

0

1 trC

CCt


First-order With Asymptote

10.00

100.00

0 2 4 6 8 10

t

C(t

)


Two Compartment Model

10.00

100.00

0 2 4 6 8 10

t

C(t

)


Distributed Loss Model

10.00

100.00

0 2 4 6 8 10

t

C(t

)


Power Rate Model

10.00

100.00

0 2 4 6 8 10

t

C(t

)


- Half-lives -


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.,

r

pt p

ln

i

ijjj

i r

trp

tp

ln

t

dfactornDegradatiox0

_

r

CCCCp

t p

0

0ln

slow

pfast r

pt

r

p lnln

1

2

2r

r

ptr

rp

11

0

prC

t p


- References -


ReferencesDuffy, 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]


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”)


Questions?


- Thank You! -

bruce h. stanley, oct. 16, 2003 del. chapter of asa meeting: mgf and 1 st -order dissipation slide 1...

Documents

order dissipation slide

order dissipation slide

function slide

function slide

order fo dissipation

delaware slide

variance of dissipation

chapter of asa meeting