Maximum-likelihood method for inverse modelling

As already mentioned, in this method an objective function is used for assessing the agreement between a given model and an observed data set. A general form of the objective function, for a given parameter (P) of the model, can be written as (Hollenbeck and Jensen, 1998):

[5.26]

where Dj represent the observed data values and Pj(P) are the corresponding values predicted using the model with the parameter p, qJj are weights for the residual values, and j is an integer varying for 1 to nD, the number of elements in the data set.

Equation [5.26] is the generalised weighted least-squares estimator. With ((Ji set to one, it represents the ordinary least squares estimator, while maximum-likelihood estimator is obtained by setting ((Ji based on the information of the measurement errors (Hollenbeck and Jensen, 1998).

Searching for the minimal discrepancies between an observed data set

and a model's predictions is also the modus operandi of the maximum­

likelihood estimator. However the discrepancies are now quantified taking into account the uncertainty of the observations. This gives meaning to the

variations in the objective function. Here the inverse variance

(

(Yo 2) in the

observations is used:

1

cp. =}

-

(Y2 .

O,) [5.27]

Setting the weighting factor equal to the reciprocal of the data variances formalises the principle that the precision of the parameter estimate should be related to the precision of the data. Using Equation [5.27] as weighting factor

also normalises the squared differences in the objective function to the unit variance. This means that their sum will present a X2 probability distribution (Hollenbeck and Jensen, 1998) . This property is stronger the larger the data set

is, as compared to the number of parameters.

The characteristics of the variations in <l>(P) represent the information

about the parameter p. This can be visualised as in Figure 5.19. The shape of

the objective function provides information about the degree of certainty in the parameter estimate. The bigger the curvature of <l>(p), the smaller the certainty. That is, a deep incised valley-form for <l>(P) is better than that of a shallow valley-form. Also the objective function provides information on the

adequacy of the model. The value of <l>(P) should be 11 small" when minimised,

. -

which means that only random variations, such a s measurement imprecision or natural variability, are responsible for the non-perfect match between the model and the data (a perfect match would have <l>(P) = 0) . The rate at which

the value of <l>(P) changes when p varies indicates the sensitivity of the model to that parameter, and provides information about the confidence intervals for the optimum value of p .

.. _.- ... ... ... -- - -- "- - . ... ... ... .. ... ... --

G"p Poptimum

p

Figure 5.19. Representation of the information about the parameter estimate

Equation [5.27] is the representation for one parameter, but ct>(p) can be extended to any number of unknown parameters. In this case the graphical representation is no longer a single curve, ct>(p) is a surface with as many dimensions as the number of parameters. The minimisation of the objective function with a large number of parameters requires an optimisation algorithm and a good set of first " guesses" to initiate the procedure, because the presence of local minima is likely for complex models. Commonly a visual inspection a priori is employed for narrowing down the range of the parameter's value and to investigate the relation between the different parameters. These procedures are often time consuming and can become quite chaotic with a large number of parameters, especially if they are correlated.

Once the global minimum value of ct>(p) is found, the adequacy of the model and the confidence intervals for the parameters can be evaluated. As the maximum-likelihood estimator follows a X2 probability distribution (Press et al., 2002), the probability of model adequacy

(PAdeq)

can be found by:

[5.28]

where 'D is the X2 cumulative density function for ct>(p) minimum and no

-

np

degrees of freedom, and np represents the number of parameters.

The confidence intervals of the estimate of p can also be found using the

relationship of the objective function with the X2 distribution. Defining the

probability level

(Peanj),

the variation of the objective function is given by:

[5.29]

where 'D -1 is the inverse cumulative X2 distribution. The variation in ct>(p) represents a variation in p (see Figure 5.19). The standard deviations of the estimate of p

(

o-p

)

can be found by defining

Peanj

as 68.3% (assuming a normal probability distribution).

However, the description of the confidence intervals of a parameter estimate in models with more than one parameter is more complex. In such

cases the confidence intervals of the parameters are interrelated and a multivariate confidence region should be considered. This confidence region is determined by Equation [5.29] for all possible parameter combinations, therefore it has

np

dimensions. The graphical representation is impossible for more than three parameters and its visualisation is already difficult for more than two (Hollenbeck and Jensen, 1998; Friedel, 2005; Hupet et al., 2005).

Common alternatives include calculating O'p for each parameter using Equation

[5.29] with optimum values for the other parameters, or showing the confidence ellipsoids for various pairs of parameters.

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