Models with process error only

When considering statistical models with observation error, there is no need to distinguish between discrete and continuous time deterministic models because the fitting methods are the same for both. This is not the case when fitting statistical models with process error because the stochastic component influences discrete and continuous models differently. To introduce models with process error, it is easier to begin with discrete models. A discrete version of logistic growth is given by

where Nt is the algal abundance at time t, and Nt _ 1 is the abundance at time t_ 1. One form of a discrete logistic model with process error is

Table 2 Common search algorithms

Search algorithm Approach

Comments

Gauss / Newton-Raphson

Nelder-Mead simplex

Evolutionary algorithms/ simulated annealing

Gradient method that approximates the local derivative(s) of the likelihood surface using a truncated Taylor series. The Gauss method calculates the first derivative, and the Newton-Raphson method calculates both the first and second derivatives. Other similar algorithms are quasi-Newton and conjugate gradient.

Direct method that uses a geometric shape to move around the likelihood function. The algorithm iteratively moves the vertex with largest likelihood value into a new region that the other vertices suggest should have a lower likelihood.

Direct method that uses a population of surface solutions. The surface solutions are used to stochastically generate new points on the surface, with those that are 'less fit' being discarded. Simulated annealing is a form of evolutionary algorithm that uses only a single point.

Gradient methods work well when the likelihood surface is reasonably smooth, but can be unstable when the surface is rough. Gradient methods are also computationally slow on large dimensional problems.

The simplex method can take longer than gradient methods on smooth surfaces, but is very robust on rough surfaces.

Slower than gradient and simplex methods on smooth surfaces, but works well on surfaces with hills and valleys.

Table 3 Sequence for deriving the likelihood function for lognormal, gamma, and Poisson distributed observation error. The most likely parameter estimates are shown in Table 4

Likelihood function derivation

Lognorma/. Continuous, positive response variable.

If we assume that the observation process in unbiased, then the expectation E{} of the observed random variable should match the deterministic prediction. For the lognormal distribution E{f (Yt , o-2)} = exp^f + , and we can link the stochastic and deterministic components by writing = ln(Nt(r ,K)) -^ This gives f(Y .y,,„,<«>, .2) .ffiffiexp(- ^-ln("2(:2K)) + ^

/(r,K, j = 2In(2™2) + £ln<y,)+ — £ ln<y,)-ln(N,(r,K))+ ^ 2 f=1 2a f=1 \ 2 /

Gamma. Continuous, positive response variable.

where r(k) is the gamma function. Following the same approach as for the lognormal distribution, we use the expectation Eff(Y(|0(,k)}

= kö( to link the deterministic population dynamics to the distribution of observation. This gives 0( = N((r ,K) and k f (Y = ytNt(r,K) , k) = 1(k -1) exP( - 1tk=Nt (r,K)) , r(k)(Nt (r,K)/k)k

/(r,K , k|y) = n ln(r(k)) + (k ln(Nt(r, K)/k) + k^^ - (k-1 )ln(yt))

Po/'sson. Discrete, positive response variable with the mean equal to the variance. To illustrate how it can be used to fit count data, the raw algal densities are rounded to the nearest integer.

Following the same approach as for the lognormal distribution, we use the expectation Eff(Y(| A()} = A( to link the deterministic population dynamics to the distribution of observation error. This gives A( = N((r,K) and f (Y = yt|N( (r,K))= N ^ eXP(- N((r,K))

Table 4 Parameter estimates for the different statistical models fit to the algal dynamics in Figure 1

Model

Location of likelihood function

Parameter estimates

Continuous logistic growth with normal observation Equation 7 error

Continuous logistic growth with lognormal Table 3

observation error

Continuous logistic growth with gamma observation Table 3 error

Continuous logistic growth with Poisson Table 3

observation error

Discrete logistic growth with normal process error Equation 8 on the log scale

Continuous logistic growth with normal process Equation 9 error

Discrete Gompertz growth with normal process Equation 11 error and normal observation error on the log scale r = 1.70 K = 9.90 x 105 ¿f2 = 1.82 x 109

ä = 4.06 c = 0.70 cf2 = 7.47x10 - 2 f 2 =4.32x10 -

where Wt ~ f(0), with j(6) being a normal distribution with mean zero. The likelihood function is

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