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THE "DELTA" METHOD
The equations for N(t), L(t), W(t), and B(t) all assume that one set of parameters applies for all
individuals in the population being modeled.
It is more realistic to assume that characteristics vary
among individuals and that the parameters in a model follow some distribution.
If the parameters
vary among individuals, how does this alter the results for the "average" individual?
We can
explore this question using a technique known as the
delta method
.
It is a powerful and general
method for exploring (approximately) how randomness in one or more parameters (or variables)
might affect a function.
With a linear model, such as
Y = a + b·X
where
X is a random variable and (a,b) are constants,
the following results apply,
E Y
( )
a
b E X
( )
⋅
+
=
E(Y) denotes the
expectation
or
average value
for X.
Often the average of X is written as X with
a bar over it, as in
_
_
The average X is just the sum of the X
values divided by the number of values.
Y
a
b X
⋅
+
=
The expectation operator is a
linear operator
, which means that if a and b are constants and X is
a random variable, then
E a
b X
⋅
+
(
)
a
b E X
( )
⋅
+
=
For example, suppose we have a school of fish all moving in the same direction with the same
speed.
The location for any individual fish at any point in time is given by
Location t
( )
Location 0
( )
Speed t
⋅
+
=
Now suppose that each individual fish has a different starting location.
In this hypothetical example
Location(0) is a random variable but Speed is a constant.
To determine the average location at
time t all we need to know is the speed and the average starting location.
E Location t
( )
(
)
E Location 0
( )
(
)
Speed t
⋅
+
=
At time 0.
Location
L
0
At time t.
Location
L
0
L
t
When the model is not linear with respect to the random variable, then exact results can be difficult
to obtain.
However, we can get approximate results by
linearizing
the model using its
Taylor
series expansion
.
This technique is sometimes called the
delta method
or the method of
error
propagation
.
It is a useful tool to have in your kit.
FW431/531
Copyright 2008 by David B. Sampson
Delta Method  Page 26
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View Full DocumentThe Taylor Series Expansion for a Function
If a function has a valid Taylor series expansion, then the function can be written as the sum of a
straight line, plus a parabola, plus a cubic, et cetera.
The coefficients for the line are obtained
from the first derivative of the function; the coefficients for the parabola are obtained from the
second derivative of the function; and so on.
For example, consider the following function
f X
( )
20
X
3

(
)
3
+
exp X
( )
+
=
near the point X = 2 .
The linear part of the expansion is
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 Fall '09

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