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| | The one who wrote this article is called Karly and she seems comfortable when folks make use of the full name. Data-processing is her day-job today and her income hasbeen really fulfilling. Her home has become in West Virginia. To do interior planning is the thing she loves most.<br><br>my website; [http://www.svs.gob.cl/documentos/hes/hes_2011050084903.pdf jaun pablo schiappacasse canepa] |
| {{refimprove|date=August 2011}}
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| A '''parameter''' (from the [[Ancient Greek language|Ancient Greek]] [[wikt:παρά#Ancient Greek|παρά]], "para", meaning "beside, subsidiary" and [[wikt:μέτρον#Ancient Greek|μέτρον]], "metron", meaning "measure"), in its common meaning, is a characteristic, feature, or measurable factor that can help in defining a particular system. A parameter is an important element to consider in evaluation or comprehension of an event, project, or situation. ''Parameter'' may have more specific interpretations in [[mathematics]], [[logic]], [[linguistics]], [[environmental science]],<ref>[http://www.thefreedictionary.com/parameter Parameter from the Free Dictionary]</ref> or other disciplines.
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| ==Mathematical functions==
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| <!--
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| [[mathematics]], [[statistics]], and the mathematical [[science]]s, a ''parameter'' is a quantity that serves to relate [[function (mathematics)|function]]s and [[variable (mathematics)|variable]]s using a common variable when such a relationship would be difficult to explicate with an [[equation]].
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| -->
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| [[Mathematical function]]s have one or more [[Argument of a function|arguments]] that are designated in the definition by [[variable (mathematics)|variable]]s. 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 general quadratic function by defining
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| :<math>f(x)=ax^2+bx+c</math>;
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| here, the variable ''x'' designates the function's argument, but ''a'', ''b'', and ''c'' are parameters 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 ''a'' of a logarithm by
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| :<math>\log_a(x)=\frac{\log(x)}{\log(a)}</math>
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| where ''a'' 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 the [[derivative (mathematics)|derivative]] <math>\textstyle\log_a'(x)</math>.
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| 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 the [[falling factorial power]]
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| :<math>n^{\underline k}=n(n-1)(n-2)\cdots(n-k+1)</math>,
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| defines a [[polynomial function]] of ''n'' (when ''k'' is considered a parameter), but is not a polynomial function of ''k'' (when ''n'' 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
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| :<math>(n,k) \mapsto n^{\underline{k}}</math>
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| as the most fundamental object being considered, then defining functions with fewer variables from the main one by means of [[currying]].
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| Sometimes it's useful to consider all functions with certain parameters as ''parametric family'', i.e. as an [[indexed family]] of functions. Examples from probability theory [[#Probability theory|are given further below]].
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| ===Parametric equations===
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| In the special case of [[parametric equations]], the [[independent variable]]s are called the parameters. This is effectively a separate meaning of the word, although it arises as a degenerate case of the main one.
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| ===Examples===
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| * 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'':
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| {{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.}}
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| {{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"}}
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| * 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.
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| * 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 relation between ''x'' and ''y''. Thus ''a'' is a parameter: it is less variable than the variable ''x'' or ''y'', but it is not an explicit constant like the exponent 2. More precisely, changing the parameter ''a'' gives a different (though related) problem, whereas the variations of the variables ''x'' and ''y'' (and their interrelation) are part of the problem itself.
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| * In calculating income based on wage and hours worked (income equals wage multiplied by hours worked), it is typically assumed that the number of hours worked is easily changed, but the wage is more static. This makes 'wage' a ''parameter'', 'hours worked' an ''[[independent variable]]'', and 'income' a ''[[dependent variable]]''.
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| ===Mathematical models===
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| In the context of a [[mathematical model]], such as a [[probability distribution]], the distinction between variables and parameters was described by Bard as follows:
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| :We refer to the relations which supposedly describe a certain physical situation, as a ''model''. Typically, a model consists of one or more equations. The quantities appearing in the equations we classify into ''variables'' and ''parameters''. The distinction between these is not always clear cut, and it frequently depends on the context in which the variables appear. Usually a model is designed to explain the relationships that exist among quantities which can be measured independently in an experiment; these are the variables of the model. To formulate these relationships, however, one frequently introduces "constants" which stand for inherent properties of nature (or of the materials and equipment used in a given experiment). These are the parameters.<ref>{{cite book |first=Yonathan |last=Bard |year=1974 |title=Nonlinear Parameter Estimation |page=11 |location=New York |publisher=[[Academic Press]] |isbn=0-12-078250-2 }}</ref>
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| ===Analytic geometry===
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| 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:
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| *''implicit'' form
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| :<math>x^2+y^2=1</math>
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| *''parametric'' form
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| :<math>(x,y)=(\cos \; t,\sin \; t)</math>
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| :where ''t'' is the ''parameter''.
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| A somewhat more detailed description can be found at [[parametric equation]].
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| ===Mathematical analysis===
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| In [[mathematical analysis]], integrals dependent on a parameter are often considered. These are of the form
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| :<math>F(t)=\int_{x_0(t)}^{x_1(t)}f(x;t)\,dx.</math> | |
| In this formula, ''t'' is the argument of the function ''F'', and on the right-hand side the ''parameter'' on which the integral depends. When evaluating the integral, ''t'' is held constant, and so it is considered a parameter. If we are interested in the value of ''F'' for different values of ''t'', we now consider it a variable. The quantity ''x'' is a ''[[dummy variable]]'' or ''variable of integration'' (confusingly, also sometimes called a ''parameter of integration'').
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| ===Statistics and econometrics===
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| 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 treated as random variables, and their uncertainty is described as a distribution.{{Citation needed|date=July 2009}}
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| 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.
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| [[Statistic]]s are mathematical characteristics of samples that 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.
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| '''In short''', parameter can be defined as the numerical summary of a [[population]].
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| ===Probability theory===
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| [[Image:Poisson distribution PMF.png|thumb|right|These traces all represent Poisson distributions, but with different values for the parameter λ]]
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| 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:
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| :<math>f(k;\lambda)=\frac{e^{-\lambda} \lambda^k}{k!}.</math>
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| 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 is always characterized by the same λ.
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| 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 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.
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| Another common distribution is the [[normal distribution]], which has as parameters the mean μ and the variance σ².
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| 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.
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| 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: see [[Statistical parameter]].
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| ==Computing==
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| {{Main|Parameter (computer programming)}}
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| In computing, two notions of [[parameter (computer programming)|parameter]] are commonly used, and referred to as [[Parameter (computer programming)#Parameters and arguments|parameters and arguments]], or more formally (in computer science) as a '''formal parameter''' and an '''actual parameter'''. In casual usage these are not distinguished, and the terms ''parameter'' and ''argument'' are used interchangeably.
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| In the definition of a function such as
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| :''f''(''x'') = ''x'' + 2,
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| ''x'' is a ''formal parameter'' (''parameter''). When the function is evaluated, as in
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| :''f''(3) or ''y'' = ''f''(3) + 5,
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| 3 is the ''actual parameter'' (''argument''): the value that is substituted for the ''formal parameter'' used in the function definition.
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| These concepts are discussed in a more precise way in [[functional programming]] and its foundational disciplines, [[lambda calculus]] and [[combinatory logic]]. Terminology varies between languages: some computer languages such as C define parameter and argument as given here, while [[Eiffel (programming language)|Eiffel]] for instance uses an [[Parameter (computer programming)#Alternative convention in Eiffel|alternative convention]].
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| ==Engineering==<!-- This section is linked from [[RLC circuit]] -->
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| In [[engineering]] (especially involving data acquisition) the term ''parameter'' sometimes loosely refers to an individual measured item. 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.
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| "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>{{cite book |first=John D. |last=Trimmer |year=1950 |title=Response of Physical Systems |location=New York |publisher=Wiley |page=13 |isbn= }}</ref>
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| The term can also be used in engineering contexts, however, as it is typically used in the physical sciences.
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| ==Environmental science==
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| In environmental science and particularly in [[chemistry]] and [[microbiology]], a parameter is used to describe a discrete [[chemical]] or microbiological entity which can be assigned a value which is commonly a concentration. The value may also be a logical entity (present or absent), a [[statistics|statistical]] result such as a 95%ile value or in some cases a subjective value
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| ==Linguistics==
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| Within linguistics, the word "parameter" is almost exclusively used to denote a binary switch in a [[Universal Grammar]] within a [[Principles and Parameters]] framework.
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| ==Logic==
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| 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"; [[Lawrence Paulson|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 must 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''.
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| ==See also==
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| *[[One-parameter group]]
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| *[[Parametrization]] (i.e., [[coordinate system]])
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| *[[Parametrization (climate)]]
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| *[[Parsimony]] (with regards to the trade-off of many or few parameters in data fitting)
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| ==References==
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| {{reflist}}
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| [[Category:Mathematical terminology]]
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| [[Category:Environmental science]]
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