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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
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 1
Bruce H. Stanley, Oct. 16, 2003
The Moment Generating Function As A Useful
Tool in Understanding Random Effects on FirstOrder 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 1st-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 1st-Order Dissipation
Slide 3
Bruce H. Stanley, Oct. 16, 2003
- First-Order Dissipation -
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 4
Bruce H. Stanley, Oct. 16, 2003
Model: First-Order Dissipation
Rate of change:
dC t
dt
Model:
Transformation to
linearity:
Constant half-life:
пЂЅ r пѓ— Ct
Ct пЂЅ C0 пѓ— e
ln пЂЁC t пЂ© пЂЅ ln пЂЁC 0 пЂ© пЂ« r пѓ— t
t1 пЂЅ
2
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
r пѓ—t
Slide 5
пЂЁ 2пЂ©
ln 1
r
Bruce H. Stanley, Oct. 16, 2003
Example: First-Order Dissipation
F irst-O rd e r D issip a tio n
C o n cen t rat io n ( p p m )
1000
100
ln пЂЁC t пЂ© пЂЅ ln пЂЁC 0 пЂ© пЂ« r пѓ— t
10
1
0.1
0
50
100
150
200
250
300
T im e ( d ays )
Del. Chapter of ASA Meeting: MGF and 1st-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 1st-Order Dissipation
Slide 7
Bruce H. Stanley, Oct. 16, 2003
- The Moment Generating Function -
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 8
Bruce H. Stanley, Oct. 16, 2003
Definition: Moment Generating Function
пЃ› пЃќ
m пЂЁt пЂ© пЂЅ E e
d m пЂЁt пЂ©
n
dt
пЃ› пЃќ
пЂЅ E r
n
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
r пѓ—t
n
tп‚® 0
Slide 9
Bruce H. Stanley, Oct. 16, 2003
Example: Moment Generating Function
X ~ Gamma(пЃЎ,пЃў)
пЃ› пЃќпЂЅ пѓІ
m пЂЁt пЂ© пЂЅ E e
X пѓ—t
п‚Ґ
e
x пѓ—t
пѓ—
0
пЃЎ
пЃў
пЃ‡ пЂЁпЃЎ пЂ©
пѓ—x
d m пЂЁt пЂ©
n
dt
пЃ­X пЂЅ
пЃ› пЃќ пЂ­ пЂЁпЃ­
var( X ) пЂЅ E X
2
2
пЂ©
X
dm пЂЁt пЂ©
dt
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
пѓ¦ пЃў пѓ¶
пѓ·пѓ·
пѓ— dx пЂЅ пѓ§пѓ§
пѓЁ пЃў пЂ­tпѓё
n
tп‚® 0
пЂЅ
tп‚® 0
d m пЂЁt пЂ©
dt
пѓ—e
пЂ­ пЃў пѓ—x
пЃ› пЃќ
пЂЅ E r
n
пЃЎ пѓ¦
пЃў пѓ¶
пѓ·пѓ·
пѓ— пѓ§пѓ§
пЃў пѓЁпЃў пЂ­tпѓё
пЃЎ пЂ«1
пЂЅ
tп‚® 0
2
tп‚® 0
пЃЎ
пЃў
пѓ¦ dm пЂЁt пЂ©
пѓ¶
пЃЎ пѓ— (пЃЎ пЂ« 1) пѓ¦ пЃЎ пѓ¶
пЃЎ
пѓ· пЂЅ
пѓ§
пѓ·
пЂ­ пѓ§пѓ§
пЂ­
пЂЅ
2
2
пѓ§пЃў пѓ·
пѓ·
dt
пЃў
пЃў
пѓЁ пѓё
tп‚® 0 пѓё
пѓЁ
2
2
пЂЅ
пЃЎ пЂ­1
пЃЎ
Slide 10
2
Bruce H. Stanley, Oct. 16, 2003
Relationship Between
–
First-Order Dissipation
–
and the
Moment Generating Function
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 11
Bruce H. Stanley, Oct. 16, 2003
Random First-Order Dissipation
Ct пЂЅ C0 пѓ— e
r пѓ—t
where r ~ PDF
Constant
пЃ› пЃќ
E пЃ›C t пЃќ пЂЅ C 0 пѓ— E e
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 12
r пѓ—t
Bruce H. Stanley, Oct. 16, 2003
Conceptual Model:
Distribution of Dissipation Rates
dCt1/dt = r1.Ct1
dCt2/dt = r2.Ct2
dCt3/dt = r3.Ct3
f(r)
dCt4/dt = r4.Ct4
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
r<0
-10
-8
-6
-4
-2
0
r
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 13
Bruce H. Stanley, Oct. 16, 2003
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
f(x)
f(r)
Transformation of r or t?
r<0
-10
-8
-6
-4
-2
0
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
X = -r
0
r
2
4
6
8
10
x = -r
It’s easier to transform t, I.e.,  = -t
пЃґ= -t
so substitute
t = -пЃґ
And treat r’s as positive
when necessary
r = -1.X
fr(r) = fX(-r)
n
E(rn) = (-1) .E(Xn)
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 14
Bruce H. Stanley, Oct. 16, 2003
Typical Table of Distributions
(Mood, Graybill & Boes. 1974. Intro. To the Theory of Stats., 3rd Ed. McGraw-Hill. 564 pp.)
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 15
Bruce H. Stanley, Oct. 16, 2003
Some Possible Dissipation Rate
Distributions
• Uniform
r ~ U(min, max)
• Normal
r ~ N(пЃ­r, пЃі2 r)
• Lognormal
r ~ LN(пЃ­r=
2/2
пЃ­
+
пЃі
e
,
пЃі2
r=
2
2.
пЃі
пЃ­r (e -1))
пѓћпЃ­ = ln[пЃ­r /пѓ–(1+ пЃіr2/пЃ­2r)],;
пЃі 2 = ln[1+ (пЃіr2/пЃ­2r)]
• Gamma
r ~ пЃ‡(пЃ­r= пЃЎ/пЃў, пЃі2r = пЃЎ/пЃў2)
пѓћпЃЎ = пЃ­r2/пЃі2r; пЃў = пЃ­r/пЃі2r
(distribution used in Gustafson and Holden 1990)
*
Where пЃ­r and пЃі2r are the expected value and variance of the untransformed rates, respectively.
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 16
Bruce H. Stanley, Oct. 16, 2003
Application to Dissipation Model:
Uniform
No need to make пЃґ = -t substitution
пЃ› пЃќ
E пЃ›C t пЃќ пЂЅ C 0 пѓ— E e
r max пѓ—t
r пѓ—t
r min пѓ—t
пѓ© e
пѓ№
пЂ­e
E пЃ›C t пЃќ пЂЅ C 0 пѓ— пѓЄ
пѓє
пЂЁ
пЂ©
r
пЂ­
r
пѓ—
t
max
min
пѓ«
пѓ»
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 17
Bruce H. Stanley, Oct. 16, 2003
Application to Dissipation Model:
Normal
No need to make пЃґ = -t substitution
пЃ› пЃќ
E пЃ›C t пЃќ пЂЅ C 0 пѓ— E e
E пЃ›C t пЃќ пЂЅ C 0 пѓ— e
пЃ­ r пѓ—t пЂ«
r пѓ—t
пЃі
2
r
пѓ—t
2
2
Note: Begins increasing at t = -пЃ­r/пЃіr2, and becomes >C0 after t = -2.пЃ­r/пЃіr2.
Del. Chapter of ASA Meeting: MGF and 1st-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 1st-Order Dissipation
Slide 19
Bruce H. Stanley, Oct. 16, 2003
Application to Dissipation Model: Gamma
(Gustafson and Holden (1990) Model)
пЃ› пЃќ
E пЃ›C t пЃќ пЂЅ C 0 пѓ— E e
r пѓ—t
Make пЃґ = -t substitution
пЃ­r
2
пѓ¦
E пЃ›C t пЃќ пЂЅ C 0 пѓ— пѓ§пѓ§
пѓЁ
пЂЅ
пЃЎ
пѓ¦
пѓ¶пЃі r
пЃў пѓ¶
пЂ­ пЃ­r
пѓ·
пѓ·пѓ· пЂЅ C 0 пѓ— пѓ§
2
пѓ§
пѓ·
пЃў пЂ­tпѓё
пЂ­
пЃ­
пЂ­
пЃі
пѓ—
t
r
r
пѓЁ
пѓё
2
C0
пЃ­r
2
2
пѓ¦
пѓ¶пЃі r
пЃіr
пѓ§1 пЂ«
пѓ— t пѓ·пѓ·
пѓ§
пЂ­ пЃ­r пѓё
пѓЁ
2
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 20
Bruce H. Stanley, Oct. 16, 2003
Distributed Loss Model
100.00
90.00
80.00
C(t)
70.00
60.00
50.00
40.00
30.00
20.00
10.00
0
2
4
6
8
10
t
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 21
Bruce H. Stanley, Oct. 16, 2003
Key Paper: Gustafson & Holden (1990)
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 22
Bruce H. Stanley, Oct. 16, 2003
- Calculating the Variance -
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 23
Bruce H. Stanley, Oct. 16, 2003
Example: Variance for the Gamma Case
пЃ› пЃќ пЂ­ пЂЁE пЃ›C пЃќпЂ©
Var пЃ›C t пЃќ пЂЅ E C
2
2
t
t
пЃ›
пЃ›
пЂЅC пѓ— E e
2
0
2 пѓ—r пѓ—t
пЃќ пЂ­ пЂЁE пЃ›e пЃќпЂ© пЃќ
r пѓ—t
2
Make пЃґ = -t substitution
пѓ©пѓ¦
пЃў
2
пЂЅ C 0 пѓ— пѓЄ пѓ§пѓ§
пѓЄпѓ« пѓЁ пЃў пЂ« 2 пѓ— t
пЃЎ
пѓ¶
пѓ¦ пЃў
пѓ·пѓ· пЂ­ пѓ§пѓ§
пѓё
пѓЁпЃў пЂ«t
пѓ¶
пѓ·пѓ·
пѓё
2 пѓ—пЃЎ
пѓ№
пѓє
пѓєпѓ»
пЃ­r
2пѓ—пЃ­ r
пѓ©
2
2
пЃіr
пЃіr
пѓ¦
пѓ¶
пѓ¦
пѓ¶
пЂ­
пЃ­
пЂ­
пЃ­
пѓЄ
2
r
r
пѓ·
пѓ§
пѓ·
пЂЅ C 0 пѓ— пѓ§пѓ§
пЂ­
пѓ§ пЂ­ пЃ­ пЂ« пЃі 2 пѓ—t пѓ·
пѓЄ пЂ­ пЃ­ пЂ« 2 пѓ—пЃі 2 пѓ— t пѓ·
r
r
r
r
пѓё
пѓЁ
пѓё
пѓЄпѓЁ
пѓ«
2
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 24
2
пѓ№
пѓє
пѓє
пѓє
пѓ»
Bruce H. Stanley, Oct. 16, 2003
- Random Initial Concentration -
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 25
Bruce H. Stanley, Oct. 16, 2003
Variable Initial Concentration:
Product of Random Variables
пЃ›
E пЃ›C t пЃќ пЂЅ E C 0 пѓ— e
r пѓ—t
пЃќ пЂЅ E пЃ›C пЃќ пѓ— E пЃ›e пЃќ пЂ­ Cov пЂЁC
r пѓ—t
0
0
,e
r пѓ—t
пЂ©
пЃ› пЃќ
п‚» E пЃ›C 0 пЃќ пѓ— E e
Delta Method
r пѓ—t
пЃ› пЃќ пЂ« пЂЁE пЃ›e пЃќпЂ© пѓ— Var пЃ›C пЃќ пЂ« 2 пѓ— E пЃ›C пЃќ пѓ— E пЃ›e пЃќпѓ— Cov пЂЁC
пЂЁCov пЂЁC , e пЂ©пЂ© пЂ« E пЃ›пЂЁC пЂ­ E пЃ›C пЃќпЂ© пѓ— пЂЁe пЂ­ E пЃ›e пЃќпЂ© пЃќ
пЂ« 2 пѓ— E пЃ›e пЃќ пѓ— E пЃ›пЂЁC пЂ­ E пЃ›C пЃќпЂ© пѓ— пЂЁe пЂ­ E пЃ›e пЃќпЂ©пЃќ
пЂ« 2 пѓ— E пЃ›C пЃќ пѓ— E пЃ›пЂЁC пЂ­ E пЃ›C пЃќпЂ© пѓ— пЂЁe пЂ­ E пЃ›e пЃќпЂ© пЃќ
Var пЃ›C t пЃќ пЂЅ пЂЁ E пЃ›C 0 пЃќпЂ© пѓ— Var e
2
r пѓ—t
r пѓ—t
2
0
r пѓ—t
2
0
пЂ©
2
r пѓ—t
r пѓ—t
2
0
пЃ› пЃќ пЂ« пЂЁE пЃ›e пЃќпЂ©
п‚» пЂЁ E пЃ›C 0 пЃќпЂ© пѓ— Var e
2
r пѓ—t
r пѓ—t
0
r пѓ—t
0
r пѓ—t
0
,e
0
0
r пѓ—t
0
r пѓ—t
0
2
0
2
r пѓ—t
r пѓ—t
r пѓ—t
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
2
пѓ— Var пЃ›C 0 пЃќ
Slide 26
Delta Method
Bruce H. Stanley, Oct. 16, 2003
- Other “Non-Linear” Models -
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 27
Bruce H. Stanley, Oct. 16, 2003
Other “Non-linear” Models
• Bi- (or multi-) first-order model ………..…... C  C  e  ri t i
t
0
• Non-linear functions of time, …………..…… C  C  e  ri  x i
t
0
e.g., t = degree days or cum.
x пЂЅ пѓІ Degradatio n _ factor
rainfall (Nigg et al. 1977)
t
i
i
пЂЁпЃґ пЂ© пѓ— d пЃґ
0
• First-order with asymptote (Pree et al. 1976).. C t  C   C 0  C    e 
ri пѓ—t i
• Two-compartment first-order……………….. C t   1  e k i t   2  e k 2 t
C0
• Distributed loss rate…………………….…… C 
t
пЃЎ
(Gustafson and Holden 1990)
пЂЁ1 пЂ« пЃў пѓ— t пЂ©
C0
• Power-rate model (Hamaker 1972)………..… C t 
пЂЁ1 пЂ« пЃЎ пѓ— C
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 28
пЃЎ
0
пѓ—r пѓ—t
пЂ©
1
пЃЎ
Bruce H. Stanley, Oct. 16, 2003
First-order With Asymptote
C(t)
100.00
10.00
0
2
4
6
8
10
t
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 29
Bruce H. Stanley, Oct. 16, 2003
Two Compartment Model
C(t)
100.00
10.00
0
2
4
6
8
10
t
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 30
Bruce H. Stanley, Oct. 16, 2003
Distributed Loss Model
C(t)
100.00
10.00
0
2
4
6
8
10
t
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 31
Bruce H. Stanley, Oct. 16, 2003
Power Rate Model
C(t)
100.00
10.00
0
2
4
6
8
10
t
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 32
Bruce H. Stanley, Oct. 16, 2003
- Half-lives -
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 33
Bruce H. Stanley, Oct. 16, 2003
Half-lives for Various Models (p = 0.5)
• First-order*……………………….
tp пЂЅ
ln пЂЁ p пЂ©
r
ln пЂЁ p пЂ© пЂ­
 r
j
• Multi-first-order*…………………
tip пЂЅ
• First-order with asymptote ………
пѓ¦ p пѓ—C0 пЂ­ Cп‚Ґ
ln пѓ§пѓ§
пѓЁ C0 пЂ­ Cп‚Ґ
tp пЂЅ
r
• Two-compartment first-order ……
ri
ln пЂЁ p пЂ©
r fast
пѓ—t j пЂ©
jп‚№i
п‚Ј tp п‚Ј
пѓ¶
пѓ·
пѓ·
пѓё
ln пЂЁ p пЂ©
rslow
• Distributed loss rate ……………..
пЂ­пЃі r
пѓ¶
пЃ­ r пѓ¦пѓ§ пЃ­ r
tp пЂЅ 2 пѓ— p
пЂ­ 1пѓ·
пѓ·
пЃіr пѓ§
пѓЁ
пѓё
• Power-rate model ……………….
tp пЂЅ
2
1
пЃЎ
пЃЎ пѓ—C0 пѓ— r
пЂЁ
пѓ— p
пЂ­пЃЎ
пЂ­1
пЂ©
t
* Can substitute cumulative environmental factor for time, i.e., x пЂЅ пѓІ Degradatio n _ factor пЂЁпЃґ пЂ© пѓ— d пЃґ
0
Del. Chapter of ASA Meeting: MGF and
1st-Order
Dissipation
Slide 34
Bruce H. Stanley, Oct. 16, 2003
- References -
Del. Chapter of ASA Meeting: MGF and 1st-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 1st-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 1st-Order Dissipation
Slide 37
Bruce H. Stanley, Oct. 16, 2003
Questions?
Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 38
Bruce H. Stanley, Oct. 16, 2003
- Thank You! Del. Chapter of ASA Meeting: MGF and 1st-Order Dissipation
Slide 39
Bruce H. Stanley, Oct. 16, 2003
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