PLAN DE ACCIÓN

The most straightforward of the kinds of error that arise in model construction and analysis is error arising from measured quantities appearing in a model. Whenever a measured quantity appears in a constraint, it is, strictly speaking, automatically an empirical quantity. This means that it will be capable of variation, and we will have to determine the manner in which it varies.37 _{Unless there are clear theoretical reasons}

determining how a quantity varies, there is an element of choice for the modeler in terms of how measured quantities are understood to vary.

In order to discuss how error is introduced into a model we need to discuss the manner in which algebro-geometric constraints can vary. Strictly speaking, error will be introduced into any mathematical constraint that has an empirical meaning. In the case of error in algebro-geometric constraints, we must consider the variation of not only specification of values and spaces of the objects of the constraint, but also the algebraic form of the constraint. There are a number of different ways that a mathematical constraint could be varied, which relate to different kinds of error. For the purposes of modeling measurement error we only need a very simple kind of variation. This is a variation of a constraint that leaves the function defining the constraint, denoted F as in equation (3.1), fixed and allows the value-specifications of objects related by F to vary.

To see how measurements are treated, let us consider the simple pendulum model. The constraint equation C of the ideal simple pendulum model, viz.,

¨

θ+g

`sinθ=0,

36_{If∥ ⋅ ∥defines a norm on a spaceM, then the function}

d(x, y) = ∥x−y∥

defines a metric onM.

37_{Generally, we will assume that mathematical quantities are continuously varying, but it is possi-}

ble to consider the structure of the variation of mathematical quantities to only allow discrete jumps to other specific values.

relates three objects: θ(t),g and `. Let us suppose that the value for g is exact and that θ(t) is a definite function, so that g and θ(t) are fixed exactly. On the other hand suppose that the value of`is to be measured, in which case`, strictly speaking, must be modeled as empirical and consequently has a tolerance ε.

Now, consider a measurement situation. Suppose that the measured value of ` is

`µ and the measurement error isε`. Then, we set the identified value ` to be `µ and the specified tolerance ε to beε`. Then, within the mathematical model, any length ˜

` in the range∣`µ−`˜∣ ≲ε` is a valid value for the length`. Measuring the value of the length implies that `=`µ, which has the effect of adding the constraint

`−`µ=0

to the model constraint, forming the constraint system (θ¨+ (g/`)sinθ=0, `−`µ=0), which we can present in a reduced form as(θ¨+ (g/`µ)sinθ=0). But since the actual value of the length need not be equal to the measured value, and if we are dealing with real numbers it (almost) never will be, the actual length `α will be different from `µ by some amount δ`, viz., `α−`µ =δ`. Since we have now allowed ` from the model constraint to vary, we need to consider the effect of the variation on the model constraint.

Since making the length an empirical quantity allows the constraint equation of the model to vary, the constraint equation also becomes empirical. Thus, the error in the empirical quantity is translated to the constraint. Before considering the effect of error in `, let us denote the vector field of the model constraint by f(θ, `) = −g_`sinθ,

where we are now regarding ` as variable. Then the model constraint C takes the form

¨

θ−f(θ, `) =0.

We then begin by substituting `µ+δ` for `α in the model constraint equation. After some algebraic manipulation we find that

¨

θ−f(θ, `µ) = −

δ`

`µ+δ`

f(θ, `µ).

modified constraint is

δ_C = −δ` `µ

f(θ, `µ),

which will be small compared to f(θ, `µ) if the relative error δ`/`µ is small, which it will be for precise measurements.

We see from this that the size of δ_C compared to the size of f(θ, `µ) is thus just the relative error δ`/`µ. This is what we want to know, viz., that the measurement error produces a small relative error in the dynamics of the model. This means that small relative measurement error will not affect the dynamics of the model in a significant way. In this case we say that the model is well-conditioned with respect to measurement error, meaning that as long as the model is descriptively stable, i.e., it accurately describes the dynamics of the real pendulum, then inferences about the dynamics drawn on the basis of the model will also apply to the dynamics of the real pendulum. Since we are assuming that the model for the exact length `α is an exact description, we conclude that the model is well-conditioned with respect to measurement error. This does not come as a surprise, since we would not expect small error in the measurement of the length to have a significant effect on the model. But this is not true in general, and we therefore see that an advantage of this approach to analyzing measurement error by considering the constraint to be variable is that we can prove whether or not the model is stable under small errors in measurement. There is one additional issue for measurement error that arises in the NEO model- ing case. Since the model problem requires finding values of the six orbital parameters in order to fix the orbit of the object, the initial conditions and any condition com- puted from them are empirical. In the context of NEO modeling they do not simply consider tolerances, i.e., measurement error, for these quantities but also a probabil- ity distribution over the possible values of the quantities. In terms of the empirical quantities we are using, a gaussian probability distribution is defined over the range of valid values of each quantity with the identified value as the mean. Mathematically, then, the vector k of orbital parameters is treated as a random vector and the vari- ance is given by the covariance matrix of the joint probability distribution of the six orbital parameters. We may see that this case is compatible with the definition of an empirical A-G vector quantity, since it makes sense to add a probability distribution over valid values and there is no conflict because the definition does not require any

distribution at all. In fact,Stetter(2004) intentionally does not include a probability distribution in his definition of valid value of an empirical quantity. Part of the reason is that leaving out probability makes for a simpler theory. But he also argues that judgements of validity in practice are typically not precise enough to justify the choice of a probability distribution. Consequently Stetter opts for an alternative approach which models variability in the data in terms of variable intervals.

For our purposes here, this debate is significant for its implications for methodol- ogy avoidance strategies in epistemological modeling. The significance of this debate about how to model the variability in data is then that it provides an example of a

functionally defined object that can be implemented in a number of different ways. If we wanted to add a schematic way of talking about the variability of a quantity into our epistemological model, then we could require that this be treated in terms of some formal treatment of thedistribution measure of valid values around the identified value. This condition on the distribution measure is the functionally defined object, it has to function in such a way as to measure the distribution of valid values around the identified value. Stetter employs a system of variable intervals and a “validity value” to do this in the context of numerical algebra. The difference between this approach and the standard probabilistic modeling of error, then, is that a single functionally defined quantity, the distribution measure, can be implemented in different ways in different contexts. Consequently, the general concept of a distribution measure can be implemented in certain mathematical modeling contexts as a probability distribution and associated variance, and in numerical algebra in terms of variable intervals. This is an example of modularity in an A-G methodology avoidance strategy associated with variable implementation of a functionally defined type, which we will see more of in the following chapter.