Encyclopedia
In
mathematics,
statistics, and the mathematical
sciences,
parameters are quantities that define certain characteristics of systems or
functions. When evaluating the function over a domain or determining the response of the system over a period of time, the independent variables are modulated, 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. Loosely speaking, then, parameters are constants in a narrow context but are variables in a larger context. Moreover, whether a quantity is a parameter or a variable is generally determined by its role in a particular system or function, rather than by anything intrinsic to the quantity.
In non-technical contexts or in jargon,
parameter may also simply be a synonym for .
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.
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
xs, denote variables, while those after it, in this example the
as, 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 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. 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
- parametric form
- 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
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 .
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". The function defining the distribution is:
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
k1 occurrences, we plug it into the function to get
f. 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 &lambda.
For instance, suppose we have a
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 statitstics, 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 σ
2. The latter formulation and notation leaves some ambiguity whether σ or σ
2 is the second parameter; the distinction is not always relevant.
It is possible to use the sequence of moments or cumulants 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 can be used as an estimate of the
mean parameter 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 an
open predicate are called
parameters by some authors . Parameters locally defined within the predicate are called
variables. This extra distinction pays off when defining substitution . Others just call parameters passed to an open predicate
variables, and when defining substitution have to distinguish between
free variables and
bound variables.
Engineering
In engineering 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." John D. Trimmer, 1950, Response of Physical Systems , p. 13.
The term can also be used in engineering contexts, however, as it is typically used in the physical sciences.
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 + 2,
x is a formal parameter. When the function is used as in
- y = f + 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.
See also
- Parametrization
- Parametrization
- Parsimony
