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Aparameter (from Ancient Greek παρά (pará) 'beside, subsidiary' and μέτρον (métron) 'measure'), generally, is any characteristic that can help in defining or classifying a particularsystem (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc.
Parameter has more specific meanings within various disciplines, includingmathematics,computer programming,engineering,statistics,logic,linguistics, and electronic musical composition.
In addition to its technical uses, there are also extended uses, especially in non-scientific contexts, where it is used to mean defining characteristics or boundaries, as in the phrases 'test parameters' or 'game play parameters'.[citation needed]
When asystem is modeled byequations, the values that describe the system are calledparameters. For example, inmechanics, the masses, the dimensions and shapes (for solid bodies), the densities and the viscosities (for fluids), appear as parameters in the equations modeling movements. There are often several choices for the parameters, and choosing a convenient set of parameters is calledparametrization.
For example, if one were considering the movement of an object on the surface of a sphere much larger than the object (e.g. the Earth), there are two commonly used parametrizations of its position: angular coordinates (like latitude/longitude), which neatly describe large movements along circles on the sphere, and directional distance from a known point (e.g. "10km NNW of Toronto" or equivalently "8km due North, and then 6km due West, from Toronto" ), which are often simpler for movement confined to a (relatively) small area, like within a particular country or region. Such parametrizations are also relevant to the modelization of geographic areas (i.e.map drawing).
Mathematical functions have one or morearguments that are designated in the definition byvariables. A function definition can also contain parameters, but unlike variables, parameters are not listed among the arguments that the function takes. When parameters are present, the definition actually defines a whole family of functions, one for every valid set of values of the parameters. For instance, one could define a generalquadratic function by declaring
Here, the variablex designates the function's argument, buta,b, andc are parameters (in this instance, also calledcoefficients) that determine which particular quadratic function is being considered. A parameter could be incorporated into the function name to indicate its dependence on the parameter. For instance, one may define the base-b logarithm by the formula
whereb is a parameter that indicates which logarithmic function is being used. It is not an argument of the function, and will, for instance, be a constant when considering thederivative.
In some informal situations it is a matter of convention (or historical accident) whether some or all of the symbols in a function definition are called parameters. However, changing the status of symbols between parameter and variable changes the function as a mathematical object. For instance, the notation for thefalling factorial power
defines apolynomial function ofn (whenk is considered a parameter), but is not a polynomial function ofk (whenn is considered a parameter). Indeed, in the latter case, it is only defined for non-negative integer arguments. More formal presentations of such situations typically start out with a function of several variables (including all those that might sometimes be called "parameters") such as
as the most fundamental object being considered, then defining functions with fewer variables from the main one by means ofcurrying.
Sometimes it is useful to consider all functions with certain parameters asparametric family, i.e. as anindexed family of functions. Examples from probability theoryare given further below.
W.M. Woods ... a mathematician ... writes ... "... a variable is one of the many things aparameter is not." ... The dependent variable, the speed of the car, depends on the independent variable, the position of the gas pedal.
[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"
In the context of amathematical model, such as aprobability distribution, the distinction between variables and parameters was described by Bard as follows:
Inanalytic geometry, acurve can be described as the image of a function whose argument, typically called theparameter, lies in areal interval.
For example, theunit circle can be specified in the following two ways:
with parameter As aparametric equation this can be written
The parametert in this equation would elsewhere in mathematics be called theindependent variable.
Inmathematical analysis, integrals dependent on a parameter are often considered. These are of the form
In this formula,t is the argument of the functionF, and on the right-hand side theparameter on which the integral depends. When evaluating the integral,t is held constant, and so it is considered to be a parameter. If we are interested in the value ofF for different values oft, we then considert to be a variable. The quantityx is adummy variable orvariable of integration (confusingly, also sometimes called aparameter of integration).
Instatistics andeconometrics, the probability framework above still holds, but attention shifts toestimating the parameters of a distribution based on observed data, ortesting hypotheses about them. Infrequentist estimation parameters are considered "fixed but unknown", whereas inBayesian estimation they are treated as random variables, and their uncertainty is described as a distribution.[citation needed][2]
Inestimation theory of statistics, "statistic" orestimator refers to samples, whereas "parameter" orestimand refers to populations, where the samples are taken from. Astatistic is a numerical characteristic of a sample that can be used as an estimate of the corresponding parameter, the numerical characteristic of thepopulation from which the sample was drawn.
For example, thesample mean (estimator), denoted, can be used as an estimate of themean parameter (estimand), denotedμ, of the population from which the sample was drawn. Similarly, thesample variance (estimator), denotedS2, can be used to estimate thevariance parameter (estimand), denotedσ2, of the population from which the sample was drawn. (Note that the sample standard deviation (S) is not an unbiased estimate of the population standard deviation (σ): seeUnbiased estimation of standard deviation.)
It is possible to make statistical inferences without assuming a particular parametric family ofprobability distributions. In that case, one speaks ofnon-parametric statistics as opposed to theparametric statistics just described. For example, a test based onSpearman's rank correlation coefficient would be called non-parametric since the statistic is computed from the rank-order of the data disregarding their actual values (and thus regardless of the distribution they were sampled from), whereas those based on thePearson product-moment correlation coefficient are parametric tests since it is computed directly from the data values and thus estimates the parameter known as thepopulation correlation.
Inprobability theory, one may describe thedistribution of arandom variable as belonging to afamily ofprobability distributions, distinguished from each other by the values of a finite number ofparameters. For example, one talks about "aPoisson distribution with mean value λ". The function defining the distribution (theprobability mass function) is:
This example nicely illustrates the distinction between constants, parameters, and variables.e isEuler's number, a fundamentalmathematical constant. The parameter λ is themean 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 observingk1 occurrences, we plug it into the function to get. Without altering the system, we can take multiple samples, which will have a range of values ofk, but the system is always characterized by the same λ.
For instance, suppose we have aradioactive 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 exhibit different values ofk, and if the sample behaves according to Poisson statistics, then each value ofk 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 thenormal distribution, which has as parameters the mean μ and the variance σ².
In these above examples, the distributions of the random variables are completely specified by the type of distribution, i.e. Poisson or normal, and the parameter values, i.e. mean and variance. In such a case, we have a parameterized distribution.
It is possible to use the sequence ofmoments (mean, mean square, ...) orcumulants (mean, variance, ...) as parameters for a probability distribution: seeStatistical parameter.
Incomputer programming, two notions ofparameter are commonly used, and are referred to asparameters and arguments—or more formally as aformal parameter and anactual parameter.
For example, in the definition of a function such as
x is theformal parameter (theparameter) of the defined function.
When the function is evaluated for a given value, as in
3 is theactual parameter (theargument) for evaluation by the defined function; it is a given value (actual value) that is substituted for theformal parameter of the defined function. (In casual usage the termsparameter andargument might inadvertently be interchanged, and thereby used incorrectly.)
These concepts are discussed in a more precise way infunctional programming and its foundational disciplines,lambda calculus andcombinatory logic. Terminology varies between languages; some computer languages such asC define parameter and argument as given here, whileEiffel uses analternative convention.
Inartificial intelligence, amodel describes the probability that something will occur. Parameters in a model are the weight of the various probabilities. Tiernan Ray, in an article on GPT-3, described parameters this way:
A parameter is a calculation in a neural network that applies a great or lesser weighting to some aspect of the data, to give that aspect greater or lesser prominence in the overall calculation of the data. It is these weights that give shape to the data, and give the neural network a learned perspective on the data.[3]
Inengineering (especially involving data acquisition) the termparameter sometimes loosely refers to an individual measured item. This usage is not consistent, as sometimes the termchannel refers to an individual measured item, withparameter 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."[4]
The term can also be used in engineering contexts, however, as it is typically used in the physical sciences.
Inenvironmental science and particularly inchemistry andmicrobiology, a parameter is used to describe a discrete chemical or microbiological entity that can be assigned a value: commonly a concentration, but may also be a logical entity (present or absent), astatistical result such as a95 percentile value or in some cases a subjective value.
Within linguistics, the word "parameter" is almost exclusively used to denote a binary switch in aUniversal Grammar within aPrinciples and Parameters framework.
Inlogic, the parameters passed to (or operated on by) anopen predicate are calledparameters by some authors (e.g.,Prawitz'sNatural Deduction;[5]Paulson'sDesigning a theorem prover). Parameters locally defined within the predicate are calledvariables. This extra distinction pays off when defining substitution (without this distinction special provision must be made to avoid variable capture). Others (maybe most) just call parameters passed to (or operated on by) an open predicatevariables, and when defining substitution have to distinguish betweenfree variables andbound variables.
In music theory, a parameter denotes an element which may be manipulated (composed), separately from the other elements. The term is used particularly forpitch,loudness,duration, andtimbre, though theorists or composers have sometimes considered other musical aspects as parameters. The term is particularly used inserial music, where each parameter may follow some specified series.Paul Lansky andGeorge Perle criticized the extension of the word "parameter" to this sense, since it is not closely related to its mathematical sense,[6] but it remains common. The term is also common in music production, as the functions of audio processing units (such as the attack, release, ratio, threshold, and other variables on a compressor) are defined by parameters specific to the type of unit (compressor, equalizer, delay, etc.).
