Parameter

For usage in computer science and programming, see parameter (computer science).

A parameter (L: auxiliary measure) is an argument of a mathematical function that is considered in some way incidental or secondary within a particular context of discussion. Thus, what is considered a principal argument and what is considered a parameter may change from one context to the next.

Example

In a section on frequently misused words in his book The Writer's Art, James J. Kilpatrick quoted a letter from a correspondent, giving examples to illustrate the correct use of the word parameter:

A parametric equaliser is an audio filter that allows the frequency of maximum cut or boost to be set by one control, and the size of the cut or boost by another. These settings, the frequency level of the peak or trough, are two of the parameters of a frequency response curve, and in a two-control equaliser they completely describe the curve. More elaborate parametric equalisers may allow other parameters to be varied, such as skew. These parameters each describe some aspect of the response curve seen as a whole, over all frequencies. A graphic equaliser provides individual level controls for various frequency bands, each of which acts only on that particular frequency band.

Types of parameter

Mathematical

In mathematics, the difference in meaning between a parameter and an argument of a function is that the parameters are the symbols that are part of the function's definition, while arguments are the symbols that are supplied to the function when it is used. The value or objects assigned to the parameters by the corresponding arguments of a function or system are not reassigned during the function's evaluation. So, parameters are effectively constants during the evaluation or processing of that function or system. The value of arguments can change outside of the function and between function usages. This distinction, the parameter's constancy, is a key part of the meaning of a parameter in any situation, often in usage beyond just mathematics.

In some informal situations people regard it as a matter of convention (and therefore a historical accident) whether some or all the arguments of a function are called parameters.

Computer science

When the terms formal parameter and actual parameter are used, they generally correspond with the definitions used in computer science. In the definition of a function such as

f(x) = x + 2,

x is a formal parameter. When the function is used as in

y = f(3) + 5,

3 is the actual parameter value that is used to solve the equation. These concepts are discussed in a more precise way in functional programming and its foundational disciplines, lambda calculus and combinatory logic.

In computing, the parameters passed to a function subroutine are more normally called arguments.

Logic

In logic, the parameters passed to (or operated on by) an open predicate are called parameters by some authors (e.g., Prawitz, "Natural Deduction"; Paulson, "Designing a theorem prover"). Parameters locally defined within the predicate are called variables. This extra distinction pays off when defining substitution (without this distinction special provision has to be made to avoid variable capture). Others (maybe most) just call parameters passed to (or operated on by) an open predicate variables, and when defining substitution have to distinguish between free variables and bound variables.

Engineering

In engineering (especially involving data acquisition) the term parameter sometimes loosely refers to an individual measured item. For example an airliner flight data recorder may record 88 different items, each termed a parameter. This usage isn't consistent, as sometimes the term channel refers to an individual measured item, with parameter referring to the setup information about that channel.

Analytic geometry

In analytic geometry, curves are often given as the image of some function. The argument of the function is invariably called "the parameter". A circle of radius 1 centered at the origin can be specified in more than one form:

implicit form

x^2+y^2=1

parametric form

(x,y)=(\cos t,\sin t)

where t is the parameter.

A somewhat more detailed description can be found at parametric equation.

Mathematical analysis

In mathematical analysis, one often considers "integrals dependent on a parameter". These are of the form

F(t)=\int_{x_0(t)}^{x_1(t)}f(x;t)\,dx.

In this formula, t is the argument of the function F on the left-hand side, and the parameter that the integral depends on, on the right-hand side. The quantity x is a dummy variable or variable (or parameter) of integration. Now, if we performed the substitution x=g(y), it would be called a change of variable.

Probability theory

In probability theory, one may describe the distribution of a random variable as belonging to a family of probability distributions, distinguished from each other by the values of a finite number of parameters. For example, one talks about "a Poisson distribution with mean value λ", or "a normal distribution with mean μ and variance σ²". The latter formulation and notation leaves some ambiguity whether σ or σ² is the second parameter; the distinction is not always relevant.

It is possible to use the sequence of moments (mean, mean square, ...) or cumulants (mean, variance, ...) as parameters for a probability distribution.

Statistics

In statistics, the probability framework above still holds, but attention shifts to estimating the parameters of a distribution based on observed data, or testing hypotheses about them. In classical estimation these parameters are considered "fixed but unknown", but in Bayesian estimation they are random variables with distributions of their own.

It is possible to make statistical inferences without assuming a particular parametric family of probability distributions. In that case, one speaks of non-parametric statistics as opposed to the parametric statistics described in the previous paragraph. For example, Spearman is a non-parametric test as it is computed from the order of the data regardless of the actual values, whereas Pearson is a parametric test as it is computed directly from the data and can be used to derive a mathematical relationship.

Statistics are mathematical characteristics of samples which can be used as estimates of parameters, mathematical characteristics of the populations from which the samples are drawn. For example, the sample mean ( $\overline X$ ) can be used as an estimate of the mean parameter (μ) of the population from which the sample was drawn.

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