Parameter
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{{For|usage in computer science and programming|Parameter (computer science)}}
{{For|the journal of the U.S. Army War College|Parameters (journal)}}
In [[mathematics]], [[statistics]], and the mathematical [[science]]s, a '''parameter''' ([[Greek language|G:]] ''auxiliary measure'') is a quantity that defines certain characteristics of systems or [[mathematical function|function]]s. Often represented by θ in general form, other symbols carry standard, specific meanings. When evaluating the function over a [[domain]] or determining the response of the system over a period of time, the [[independent variable]]s are varied, while the parameters are held constant. The function or system may then be reevaluated or reprocessed with different parameters, to give a function or system with different behavior.
However, in the special case of [[parametric equations]] the independent variable is called the parameter.
==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'':
{{cquote|W.M. Woods...a mathematician...writes... "...a variable is one of the many things a ''parameter'' is not." ... The dependent variable, the speed of the car, depends on the independent variable, the position of the gas pedal.}}
{{cquote|[Kilpatrick quoting Woods] "Now...the engineers...change the lever arms of the linkage...the speed of the car...will still depend on the pedal position...''but in a...different manner''. You have changed a parameter"}}
* A [[equalization|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 [[equalization|graphic equaliser]] provides individual level controls for various frequency bands, each of which acts only on that particular frequency band.
* If asked to imagine the graph of the relationship ''y'' = ''ax''<sup>2</sup>, one typically visualizes a range of values of ''x'', but only one value of ''a''. Of course a different value of ''a'' can be used, generating a different graphical appearance. The ''a'' can therefore be considered to be a parameter: less variable than the variable ''x'', but less constant than the constant 2.
* Suppose that you believe that the amount of money you are going to make next month is defined by the wage you receive per hour times the amount of hours you are going to work (''y'' = ''ax'')(Income=wage x hours that you work). You are able to influence the amount of hours you work next month (your boss is very flexible), but you think that you will not be able to change the height of your wage (your boss is a real pain in the ass when it comes to disccussing wages). Then:
**Your wage is the parameter
**The amount of hours you work is the variable.
**This is of course depending on whether you think you can or cannot change your wage.
Suppose that you and a friend are both working for the same boss making the same money working the same hours, that the formula would become:
*(''y'' = ''2ax'')(Income=2 x wage x hours that you work).
**the constant is 2
**the parameter is your wage
**the variable is the amount of hours you are going to work
==Parameters in various contexts in math and science==
===Mathematical functions===
Mathematical functions typically can have one or more variables and zero or more parameters. The two are often distinguished by being grouped separately in the list of [[Argument#Mathematics, science and linguistics|arguments]] that the function takes:
:<math>f(x_1, x_2, \dots; a_1, a_2, \dots) = \cdots\,</math>
The symbols before the semicolon in the function's definition, in this example the <math>x</math>'s, denote variables, while those after it, in this example the <math>a</math>'s, denote parameters.
Strictly speaking, parameters are denoted by the symbols that are part of the function's ''definition'', while arguments are the values that are supplied to the function when it is used. Thus, a parameter might be something like "the ratio of the cylinder's radius to its height", while the argument would be something like "2" or "0.1".
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.
===Analytic geometry===
In [[analytic geometry]], [[curve]]s 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
:<math>x^2+y^2=1</math>
*''parametric'' form
:<math>(x,y)=(\cos t,\sin t)</math>
: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
:<math>F(t)=\int_{x_0(t)}^{x_1(t)}f(x;t)\,dx.</math>
In this formula, ''t'' is on the left-hand side the argument of the function ''F'', and it is on the right-hand side the ''parameter'' that the integral depends on. When evaluating the integral, ''t'' is held constant, and so it considered a parameter. If we are interested in the value of ''F'' for different values of ''t'', then, we now consider it to be a variable. The quantity ''x'' is a ''dummy variable'' or ''variable of integration'' (confusingly, also sometimes called a ''parameter of integration'').
===Probability theory===
[[Image:Poisson distribution PMF.png|thumb|right|These traces all represent Poisson distributions, but with different values for the parameter λ]]
In [[probability theory]], one may describe the [[probability distribution|distribution]] of a [[random variable]] as belonging to a ''family'' of [[probability distribution]]s, 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". The function defining the distribution (the [[probability mass function]]) is:
:<math>f(k;\lambda)=\frac{e^{-\lambda} \lambda^k}{k!}.</math>
This example nicely illustrates the distinction between constants, parameters, and variables. ''e'' is [[Euler's Number]], a fundamental [[mathematical constant]]. The parameter λ is the [[mean]] number of observations of some phenomenon in question, a property characteristic of the system. ''k'' is a variable, in this case the number of occurrences of the phenomenon actually observed from a particular sample. If we want to know the probability of observing ''k<sub>1</sub>'' occurrences, we plug it into the function to get <math>f(k_1 ; \lambda)</math>. Without altering the system, we can take multiple samples, which will have a range of values of ''k'', but the system will always be characterized by the same λ.
For instance, suppose we have a [[radioactivity|radioactive]] sample that emits, on average, five particles every ten minutes. We take measurements of how many particles the sample emits over ten-minute periods. The measurements will exhibit different values of ''k'', and if the sample behaves according to Poisson statistics, then each value of ''k'' will come up in a proportion given by the probability mass function above. From measurement to measurement, however, λ remains constant at 5. If we do not alter the system, then the parameter λ is unchanged from measurement to measurement; if, on the other hand, we modulate the system by replacing the sample with a more radioactive one, then the parameter λ would increase.
Another common distribution is the [[normal distribution]], which has as parameters the mean μ and the variance σ².
It is possible to use the sequence of [[moment (mathematics)|moments]] (mean, mean square, ...) or [[cumulant]]s (mean, variance, ...) as parameters for a probability distribution.
===Statistics and econometrics===
In [[statistics]] and [[econometrics]], the probability framework above still holds, but attention shifts to [[statistical estimation|estimating]] the parameters of a distribution based on observed data, or [[Hypothesis testing|testing hypotheses]] about them. In [[classical statistics|classical estimation]] these parameters are considered "fixed but unknown", but in [[Bayesian probability|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's rank correlation coefficient|Spearman]] is a non-parametric test as it is computed from the order of the data regardless of the actual values, whereas [[Pearson product-moment correlation coefficient|Pearson]] is a parametric test as it is computed directly from the data and can be used to derive a mathematical relationship.
[[Statistic]]s 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'' (<math>\overline X</math>) can be used as an estimate of the ''mean'' parameter (μ) of the population from which the sample was drawn.
==Other fields==
Other fields use the term "parameter" as well, but with a different meaning.
===Logic===
In [[logic]], the parameters passed to (or operated on by) an ''open predicate'' are called ''parameters'' by some authors (e.g., [[Dag Prawitz|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===<!-- This section is linked from [[RLC circuit]] -->
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.
"Speaking generally, '''properties''' are those physical quantities which directly describe the physical attributes of the system; '''parameters''' are those combinations of the properties which suffice to determine the response of the system. Properties can have all sorts of dimensions, depending upon the system being considered; parameters are dimensionless, or have the dimension of time or its reciprocal."<ref>John D. Trimmer, 1950, Response of Physical Systems (New York: Wiley), p. 13</ref>
The term can also be used in engineering contexts, however, as it is typically used in the physical sciences.
===Computer science===
{{main|Parameter (computer science)}}
When the terms '''formal parameter''' and '''actual parameter''' are used, they generally correspond with the [[parameter (computer science)|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 or just the value of ''f''(3),
'''3''' is the '''actual parameter''' value that is substituted for ''x'', the formal parameter, in the function definition. These concepts are discussed in a more precise way in [[functional programming]] and its foundational disciplines, [[lambda calculus]] and [[combinatory logic]].
In [[computing]], parameters are often called ''arguments'', and the two words are used interchangeably. However, some computer languages such as C define ''argument'' to mean ''actual parameter'' (i.e., the value), and ''parameter'' to mean ''formal parameter''.
==Reference==
<div class="references-small"><references/></div>
==See also==
{{Wiktionary}}
*[[Parametrization]] (i.e., [[coordinate system]])
*[[Parametrization (climate)]]
*[[Parsimony]] (with regards to the trade-off of many or few parameters in data fitting)
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