Continuous or discrete variable: Difference between revisions

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{{Short description|Types of quantitativenumerical variables in mathematics}}

{{distinguishDistinguish|Discrete-time and continuous-time variables}}

[[File:Continuous and discrete variables.png|thumb|upright=1.3|right|Variables can be divided into two main categories: [[Categorical variable|qualitative (categorical)]] and quantitative (numerical). Continuous and discrete variables are subcategories of quantitative variables. Note that this schematic is not exhaustive in terms of the types of variables.]]

{{Probability fundamentals}}

In [[mathematics]] and [[statistics]], a quantitative [[variable (mathematics)|variable]] may be '''continuous''' or '''discrete''' if they are typically obtained by ''measuring'' or ''[[counting]]'', respectively.<ref>{{cite journal |last1=Ali |first1=Zulfiqar |last2=Bhaskar |first2=S. Bala |title=Basic statistical tools in research and data analysis |journal=Indian Journal of Anaesthesia |date=September 2016 |volume=60 |issue=9 |page=662–669 |doi=10.4103/0019-5049.190623|doi-access=free}}</ref> If it can take on two particular [[real number|real]] values such that it can also take onand all realthe values between them (including values that are arbitrarily or infinitesimally close together), the variable is continuous in that [[Interval (mathematics)|interval]].<ref>{{cite journal |last1=Kaliyadan |first1=Feroze |last2=Kulkarni |first2=Vinay |title=Types of Variables, Descriptive Statistics, and Sample Size |journal=Indian Dermatology Online Journal |date=January 2019 |volume=10 |issue=1 |pages=82–86 |doi=10.4103/idoj.IDOJ_468_18 |pmid=30775310 |pmc=6362742 |doi-access=free }}</ref> If it can take on a value such that there is a non-[[infinitesimal]] gap on each side of it containing no values that the variable can take on, then it is discrete around that value.<ref>K.D. Joshi, ''Foundations of Discrete Mathematics'', 1989, New Age International Limited, [https://books.google.com/books?id=RM1D3mFw2u0C&dq=continuous+discrete+variable+math&pg=PA7], page 7.</ref> In some contexts, a variable can be discrete in some ranges of the [[number line]] and continuous in others. In statistics, continuous and discrete variables are distinct [[Statistical data type|statistical data types]] which are described with different [[probability distribution]]s.

[[File:Continuous and discrete variables.png|thumb|upright=1.3|right|Variables can be divided into two main categories: qualitative (categorical) and quantitative (numerical). Continuous and discrete variables are subcategories of quantitative variables. Note that this schematic is not exhaustive in terms of the types of variables.]]

==Continuous variable==

A '''continuous variable''' is a variable such that there are possible values between any two values.

For example, a variable over a non-empty range of the [[real number]]s is continuous, if it can take on any value in that range. The reason is that any range of real numbers between <math>a</math> and <math>b</math> with <math>a, b \in \mathbb{R}; a \neq b</math> is uncountable, with infinitely many values within the range.<ref>{{cite journal |last1=Brzychczy |first1=Stanisaw |last2=Gorniewicz |first2=Lech |title=Continuous and discrete models of neural systems in infinite-dimensional abstract spaces |journal=Neurocomputing |date=2011 |volume=74 |issue=17 |pagepages=2711-27152711–2715 |doi=10.1016/j.neucom.2010.11.005}}</ref>

A '''continuous variable''' is a variable whose value is obtained by measuring, i.e., one which can take on an [[uncountable set]] of values.

Methods of [[calculus]] are often used in problems in which the variables are continuous, for example in continuous [[optimization]] problems.<ref>{{Cite book |last1=Griva |first1=Igor |url=https://www.worldcat.org/oclc/236082842 |title=Linear and nonlinear optimization |last2=Nash |first2=Stephen |last3=Sofer |first3=Ariela|author3-link= Ariela Sofer |publisher=Society for Industrial and Applied Mathematics |year=2009 |isbn=978-0-89871-661-0 |edition=2nd |___location=Philadelphia |pages=7 |language=en |oclc=236082842}}</ref>

For example, a variable over a non-empty range of the [[real number]]s is continuous, if it can take on any value in that range. The reason is that any range of real numbers between <math>a</math> and <math>b</math> with <math>a, b \in \mathbb{R}; a \neq b</math> is uncountable, with infinitely many values within the range.<ref>{{cite journal |last1=Brzychczy |first1=Stanisaw |last2=Gorniewicz |first2=Lech |title=Continuous and discrete models of neural systems in infinite-dimensional abstract spaces |journal=Neurocomputing |date=2011 |volume=74 |issue=17 |page=2711-2715 |doi=10.1016/j.neucom.2010.11.005}}</ref>

In [[statistics|statistical theory]], the [[probability distribution]]s of continuous variables can be expressed in terms of [[probability density function]]s. <ref name="Springer Texts in Statistics">{{cite journal |last1=Dekking |first1=Frederik Michel |last2=Kraaikamp |first2=Cornelis |last3=Lopuhaä |first3=Hendrik Paul |last4=Meester |first4=Ludolf Erwin |date=2005 |title=A Modern Introduction to Probability and Statistics |url=https://doi.org/10.1007/1-84628-168-7 |journal=Springer Texts in Statistics |language=en |doi=10.1007/1-84628-168-7 |isbn=978-1-85233-896-1 |issn=1431-875X|url-access=subscription }}</ref>

Methods of [[calculus]] are often used in problems in which the variables are continuous, for example in continuous [[optimization]] problems.<ref>{{Cite book |last1=Griva |first1=Igor |url=https://www.worldcat.org/oclc/236082842 |title=Linear and nonlinear optimization |last2=Nash |first2=Stephen |last3=Sofer |first3=Ariela|author3-link= Ariela Sofer |publisher=Society for Industrial and Applied Mathematics |year=2009 |isbn=978-0-89871-661-0 |edition=2nd |___location=Philadelphia |pages=7 |language=en |oclc=236082842}}</ref>

In [[continuous time|continuous-time]] [[dynamical system|dynamics]], the variable ''time'' is treated as continuous, and the equation describing the evolution of some variable over time is a [[differential equation]].<ref>{{cite journal |last1=Poyton |first1=A. A. |last2=Varziri |first2=Mohammad Saeed |last3=McAuley |first3=Kimberley B. |last4=MclellanPat James |first4=Pat James |last5=Ramsay |first5=James O. |title=Parameter estimation in continuous-time dynamic models using principal differential analysis |journal=Computers & Chemical Engineering |date=February 15, 2006 |volume=30 |issue=4 |pagepages=698-708698–708 |doi=10.1016/j.compchemeng.2005.11.008}}</ref> The [[instantaneous rate of change]] is a well-defined concept that takes the ratio of the change in the dependent variable to the independent variable at a specific instant.

In [[statistics|statistical theory]], the [[probability distribution]]s of continuous variables can be expressed in terms of [[probability density function]]s. <ref name="Springer Texts in Statistics">{{cite journal |last1=Dekking |first1=Frederik Michel |last2=Kraaikamp |first2=Cornelis |last3=Lopuhaä |first3=Hendrik Paul |last4=Meester |first4=Ludolf Erwin |date=2005 |title=A Modern Introduction to Probability and Statistics |url=https://doi.org/10.1007/1-84628-168-7 |journal=Springer Texts in Statistics |language=en |doi=10.1007/1-84628-168-7 |isbn=978-1-85233-896-1 |issn=1431-875X}}</ref>

In [[continuous time|continuous-time]] [[dynamical system|dynamics]], the variable ''time'' is treated as continuous, and the equation describing the evolution of some variable over time is a [[differential equation]].<ref>{{cite journal |last1=Poyton |first1=A. A. |last2=Varziri |first2=Mohammad Saeed |last3=McAuley |first3=Kimberley B. |last4=MclellanPat James |first4=Pat James |last5=Ramsay |first5=James O. |title=Parameter estimation in continuous-time dynamic models using principal differential analysis |journal=Computers & Chemical Engineering |date=February 15, 2006 |volume=30 |issue=4 |page=698-708 |doi=10.1016/j.compchemeng.2005.11.008}}</ref> The [[instantaneous rate of change]] is a well-defined concept that takes the ratio of the change in the dependent variable to the independent variable at a specific instant.

[[File: Continuous or discrete variables example.jpg |thumb|upright=1.3|right|This is an image of vials with different amounts of liquid. A continuous variable could be the volume of liquid in the vials. A discrete variable could be the number of vials.]]

==Discrete variable==

In contrast, a variable is a '''discrete variable''' if and only if there exists a one-to-one correspondence between this variable and a subset of <math>\mathbb{N}</math>, the set of [[natural numbers]].<ref>{{cite book |last1=Odifreddi |first1=Piergiorgio |title=Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers |date=February 18, 1992 |publisher=North Holland Publishing Company |isbn=978-0444894830 |page=18}}</ref> In other words;, a discrete variable over a particular interval of real values is one for which, for any value in the range that the variable is permitted to take on, there is a positive minimum distance to the nearest other permissible value. The value of a discrete variable can be obtained by counting, and the number of permitted values is either finite or [[countably infinite]]. Common examples are variables that must be [[Integer|integersinteger]]s, non-negative integers, positive integers, or only the integers 0 and 1.<ref>{{cite book |last1=van Douwen |first1=Eric |title=Handbook of Set-Theoretic Topology |date=1984 |publisher=Elsevier |___location=North Holland |isbn=978-0-444-86580-9 |pages=113-167113–167}}</ref>

Methods of calculus do not readily lend themselves to problems involving discrete variables. Especially in multivariable calculus, many models rely on the assumption of continuity.<ref>{{cite book |last1=Clogg |first1=Clifford C. |last2=Shockey |first2=James W. |title=Handbook of Multivariate Experimental Psychology |date=1988 |publisher=Springer Publishing Company |___location=Boston, Massachusetts |isbn=978-1-4613-0893-5 |pagepages=337-365337–365}}</ref> Examples of problems involving discrete variables include [[integer programming]].

In statistics, the probability distributions of discrete variables can be expressed in terms of [[probability mass function]]s.<ref name="Springer Texts in Statistics" />

In [[discrete time]] dynamics, the variable ''time'' is treated as discrete, and the equation of evolution of some variable over time is called a [[difference equation]].<ref>{{cite book |last1=Thyagarajan |first1=K.S. |title=Introduction to Digital Signal Processing Using MATLAB with Application to Digital Communications |date=2019 |publisher=Springer Publishing Company |isbn=978-3319760285 |pagepages=21-6321–63 |edition=1}}</ref> For certain discrete-time dynamical systems, the system response can be modeledmodelled by solving the difference equation for an analytical solution.

In [[econometrics]] and more generally in [[regression analysis]], sometimes some of the variables being [[empirical]]ly related to each other are 0-1 variables, being permitted to take on only those two values.<ref>{{cite journal |last1=Miller |first1=Jerry L.L. |last2=Erickson |first2=Maynard L. |title=On Dummy Variable Regression Analysis |journal=Sociological Methods & Research |date=May 1974 |volume=2 |issue=4 |pagepages=395-519395–519 |doi=10.1177/004912417400200402}}</ref> The purpose of the discrete values of 0 and 1 is to use the dummy variable as a ‘switch’ that can ‘turn on’ and ‘turn off’ by assigning the two values to different parameters in an equation. A variable of this type is called a [[dummy variable (statistics)|dummy variable]]. If the [[dependent variable]] is a dummy variable, then [[logistic regression]] or [[probit regression]] is commonly employed. In the case of regression analysis, a dummy variable can be used to represent subgroups of the sample in a study (e.g. the value 0 corresponding to a constituent of the control group). <ref>{{cite book |last1=Hardy |first1=Melissa A. |title=Regression with Dummy Variables (Quantitative Applications in the Social Sciences) |date=February 25, 1993 |publisher=Sage Publications, Inc. |___location=Newbury Park |isbn=0803951280 |page=v |edition=1st}}</ref>

==Mixture of continuous and discrete variables==

A mixed multivariate model can contain both discrete and continuous variables. For instance, a simple mixed multivariate model could have a discrete variable <math>x</math>, which only takes on values 0 or 1, and a continuous variable <math>y</math>.<ref>{{cite journal |last1=Olkin |first1=Ingram |last2=Tate |first2=Robert |title=Multivariate Correlation Models with Mixed Discrete and Continuous Variables |journal=The Annals of Mathematical Statistics |date=June 1961 |volume=32 |issue=2 |pagepages=448-465448–465 |doi=10.1214/aoms/1177705052|doi-access=free }}</ref> An example of a mixed model could be a research study on the risk of psychological disorders based on one binary measure of psychiatric symptoms and one continuous measure of cognitive performance.<ref>{{cite journal |last1=Fitzmaurice |first1=Garrett M. |last2=Laird |first2=Nan M. |title=Regression Models for Mixed Discrete and Continuous Responses with Potentially Missing Values |journal=Biometrics |date=March 1997 |volume=53 |issue=1 |pagepages=110-122110–122 |doi=10.2307/2533101|jstor=2533101 }}</ref> Mixed models may also involve a single variable that is discrete over some range of the number line and continuous at another range.

In probability theory and statistics, the probability distribution of a mixed random variable consists of both discrete and continuous components. A mixed random variable does not have a [[Cumulative distribution function|cumulative distribution function]] that is discrete or everywhere-continuous. An example of a mixed type random variable is the probability of wait time in a queue. The likelihood of a customer experiencing a zero wait time is discrete, while non-zero wait times are evaluated on a continuous time scale.<ref>{{cite journal |last1=Sharma |first1=Shalendra D. |title=On a Continuous/Discrete Time Queueing System with Arrivals in Batches of Variable Size and Correlated Departures |journal=Journal of Applied Probability |date=March 1975 |volume=12 |issue=1 |pagepages=115-129115–129 |doi=10.2307/3212413|jstor=3212413 }}</ref> In physics (particularly quantum mechanics, where this sort of distribution often arises), [[Dirac delta function]]s are often used to treat continuous and discrete components in a unified manner. For example, the previous example might be described by a probability density <math>p(t)=\alpha \delta (t) + g(t)</math>, such that <math>P(t>0)=\int_0^\infty g(t)=1-\alpha</math>, and <math>P(t=0)=\alpha</math>.

==See also==

{{div col|colwidth=22em}}

* [[Continuous-time stochastic process]]

* [[Continuous geometryfunction]]

* [[Continuous modellinggeometry]]

* [[DiscreteContinuous modelling]]

* [[Continuous or discrete spectrum]]

* [[Continuous functionspectrum]]

* [[Count data]]

* [[Discrete-time stochastic process]]

* [[Discrete geometry]]

* [[Discrete mathematics]]

* [[ContinuousDiscrete spectrummeasure]]

* [[Discrete measuremodelling]]

* [[Discrete series representation]]

* [[Discrete space]]

* [[Discrete spectrum]]

* [[Discrete time and continuous time]]

* [[Continuous-time stochastic process]]

* [[Discrete-time stochastic process]]

* [[Continuous modelling]]

* [[Discrete modelling]]

* [[Continuous geometry]]

* [[Discrete geometry]]

* [[Continuous series representation]]

* [[Discrete series representation]]

* [[Discretization]]

* [[Interpolation]]

* [[Principal series representation]] (continuous series representation)

* [[Discrete measure]]

* [[Discrete space]]

{{div col end}}

==References==

{{reflistReflist}}

[[Category:Mathematical terminology]]

[[Category:Statistical data types]]