In mathematics, statistics, and the mathematical sciences, a parameter (G: auxiliary measure) is a quantity that defines certain characteristics of systems or functions. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function 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 variables are varied, while the parameters are held constant. Dependent variables and independent variables refer to values that change in relationship to each other The function or system may then be reevaluated or reprocessed with different parameters, to give a function or system with different behavior.

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. Equalization (or equalisation, EQ) is the process of changing the frequency envelope of a sound in Audio processing. An audio filter is a type of filter used for processing Sound signals. Frequency is a measure of the number of occurrences of a repeating event per unit Time. 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. Equalization (or equalisation, EQ) is the process of changing the frequency envelope of a sound in Audio processing.

• If asked to imagine the graph of the relationship y = ax², 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.

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 arguments that the function takes:

The symbols before the semicolon in the function's definition, in this example the x's, denote variables, while those after it, in this example the a's, denote parameters. In Logic, an argument is a Set of one or more Declarative sentences (or "propositions") known as the Premises along

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, curves are often given as the image of some function. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object 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

x² + y² = 1

• parametric form

(x,y) = (cost,sint)

where t is the parameter.

A somewhat more detailed description can be found at parametric equation. In Mathematics, parametric equations are a method of defining a curve

Mathematical analysis

In mathematical analysis, one often considers "integrals dependent on a parameter". Analysis has its beginnings in the rigorous formulation of Calculus. These are of the form

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

These traces all represent Poisson distributions, but with different values for the parameter λ

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. Probability theory is the branch of Mathematics concerned with analysis of random phenomena In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable For example, one talks about "a Poisson distribution with mean value λ". In Probability theory and Statistics, the Poisson distribution is a Discrete probability distribution that expresses the probability of a number of events The function defining the distribution (the probability mass function) is:

This example nicely illustrates the distinction between constants, parameters, and variables. In Probability theory, a probability mass function (abbreviated pmf) is a function that gives the probability that a discrete Random variable e is Euler's Number, a fundamental mathematical constant. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line A mathematical constant is a number usually a Real number, that arises naturally in Mathematics. The parameter λ is the mean number of observations of some phenomenon in question, a property characteristic of the system. In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean 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 k₁ occurrences, we plug it into the function to get f(k₁;λ). 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 radioactive sample that emits, on average, five particles every ten minutes. Radioactive decay is the process in which an unstable Atomic nucleus loses energy by emitting ionizing particles and Radiation. 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. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields

It is possible to use the sequence of moments (mean, mean square, . . . ) or cumulants (mean, variance, . In Probability theory and Statistics, a Random variable X has an Expected value μ = E ( X) and a Variance σ2 . . ) as parameters for a probability distribution.

Statistics and econometrics

In statistics and econometrics, 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. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Econometrics is concerned with the tasks of developing and applying Quantitative or Statistical methods to the study and elucidation of economic principles Estimation theory is a branch of Statistics and Signal processing that deals with estimating the values of parameters based on measured/empirical data A statistical hypothesis test is a method of making statistical decisions using experimental data In classical estimation these parameters are considered "fixed but unknown", but in Bayesian estimation they are random variables with distributions of their own. Frequency probability is the interpretation of probability that defines an event's Probability as the limit of its relative frequency in a large Bayesian probability interprets the concept of Probability as 'a measure of a state of knowledge'.

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. Non-parametric statistics is a branch of Statistics concerned with non-parametric Statistical models and non-parametric inference, including non-parametric Parametric statistics are statistics where the population is assumed to fit any parametrized distributions (most typically the Normal distribution) 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. In Statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter \rho In Statistics, the Pearson product-moment correlation coefficient (sometimes referred to as the MCV or PMCC, and typically denoted by r

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. A statistic (singular is the result of applying a function (statistical Algorithm) to a set of data. For example, the sample mean () 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. Logic is the study of the principles of valid demonstration and Inference. g. , Prawitz, "Natural Deduction"; Paulson, "Designing a theorem prover"). Dag Prawitz (born 1936 is a Swedish Philosopher and Logician. 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. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and For example an airliner flight data recorder may record 88 different items, each termed a parameter. The flight data recorder ( FDR) or Black Box is a Flight recorder used to record specific Aircraft performance parameters 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. " John D. Trimmer, 1950, Response of Physical Systems (New York: Wiley), p. 13.

The term can also be used in engineering contexts, however, as it is typically used in the physical sciences.

Computer science

Main article: Parameter (computer science)

When the terms formal parameter and actual parameter are used, they generally correspond with the definitions used in computer science. In Computer programming, a parameter is a variable which takes on the meaning of a corresponding Argument (computer science is same article--> argument In Computer programming, a parameter is a variable which takes on the meaning of a corresponding Argument (computer science is same article--> argument 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 Computer science, functional programming is a Programming paradigm that treats Computation as the evaluation of mathematical functions and In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for Variables in Mathematical logic

In computing, parameters are often called arguments, and the two words are used interchangeably. Computing is usually defined like the activity of using and developing Computer technology Computer hardware and software. However, some computer languages such as C define argument to mean actual parameter (i. e. , the value), and parameter to mean formal parameter.

See also

• Parametrization (i. Parameterization (or parametrization parameterisation in British English) is the process of defining or deciding the Parameters - usually of some model - that are e. , coordinate system)
• Parametrization (climate)
• Parsimony (with regards to the trade-off of many or few parameters in data fitting)