Zero-truncated Poisson distribution: Difference between revisions

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Edited equation 1 to keep conventions consistent within the article (f(0;lambda) for F(0))
 
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In [[mathematics]] and in particular [[Dynamical system|mathematical dynamics]], '''discrete time and continuous time''' are two alternative frameworks within which to model [[Variable (mathematics)|variables]] that evolve over time.  
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==Discrete time==
 
'''Discrete time''' views values of variables as occurring at distinct, separate "points in time", or equivalently as being unchanged throughout each non-zero region of time ("time period"). Thus a variable jumps from one value to another as time moves from time period to the next. This view of time corresponds to a digital clock that gives a fixed reading of 10:37 for a while, and then jumps to a new fixed reading of 10:38, etc. In this framework, each variable of interest is measured once at each time period. The number of measurements between any two time periods is finite. Measurements are typically made at sequential [[integer]] values of the variable "time".
 
==Continuous time==
 
In contrast, '''continuous time''' views variables as having a particular value for potentially only an [[infinitesimal]]ly short amount of time. Between any two points in time there are an [[infinity|infinite]] number of other points in time. The variable "time" ranges over the entire [[real number line]], or depending on the context, over some subset of it such as the non-negative reals.
 
==Relevant contexts==
 
Discrete time is often employed when [[empirical]] [[measurement]]s are involved, because normally it is only possible to measure variables sequentially. For example, while [[economic activity]] actually occurs continuously, there being no moment when the economy is totally in a pause, it is nevertheless possible to measure economic activity only discretely. Hence published data on, for example, [[gross domestic product]] will show a sequence of [[Calendar year#Quarters|quarterly]] values.  
 
When one attempts to [[empirical]]ly explain such variables in terms of other variables and/or their own prior values, one uses [[time series]] or [[regression analysis|regression]] methods in which variables are indexed with a subscript indicating the time period in which the observation occurred. For example, ''y''<sub>''t''</sub> might refer to the value of [[income]] observed in unspecified time period ''t'', ''y''<sub>''3''</sub> to the value of income observed in the third time period, etc.
 
Moreover, when a researcher attempts to develop a theory to explain what is observed in discrete time, often the theory itself is expressed in discrete time in order to facilitate the development of a time series or regression model.
 
On the other hand, it is often more mathematically [[Closed form solution|tractable]] to construct [[Scientific theory|theoretical model]]s in continuous time, and often in areas such as [[physics]] an exact description requires the use of continuous time. In a continuous-time context the value of a variable ''y'' at an unspecified point in time is denoted as ''y''(''t'') or, when the meaning is clear, simply as ''y''.
 
==Types of equations==
 
===Discrete time===
 
Discrete time makes use of [[difference equation]]s, also known as recurrence relations. An example, known as the [[logistic map]] or logistic equation, is  
 
:<math> x_{t+1} = rx_t(1-x_t),</math>
 
in which ''r'' is a [[Parameter#Mathematical functions|parameter]] in the range from 2 to 4 inclusive, and ''x'' is a variable in the range from 0 to 1 inclusive whose value in period ''t'' [[nonlinearity|nonlinearly]] affects its value in the next period, ''t''+1. For example, if <math>r=4</math> and <math>x_1 = 1/3</math>, then for ''t''=1 we have <math>x_2=4(1/3)(2/3)=8/9</math>, and for ''t''=2 we have <math>x_3=4(8/9)(1/9)=32/81</math>.
 
Another example models the adjustment of a [[price]] ''P'' in response to non-zero [[excess demand]] for a product as
 
:<math>P_{t+1} = P_t + \delta \cdot f(P_t,...)</math>
 
where <math>\delta</math> is the positive speed-of-adjustment parameter which is less than or equal to 1, and where <math>f</math> is the [[excess demand function]].
 
===Continuous time===
 
Continuous time makes use of [[differential equation]]s. For example, the adjustment of a price ''P'' in response to non-zero excess demand for a product can be modeled in continuous time as
 
:<math>\frac{dP}{dt}=\lambda \cdot f(P,...)</math>
 
where the left side is the [[first derivative]] of the price with respect to time (that is, the rate of change of the price), <math>\lambda</math> is the speed-of-adjustment parameter which can be any positive finite number, and <math>f</math> is again the excess demand function.
 
==Graphical depiction==
 
The values of a variable measured in discrete time can be plotted as a [[step function]], in which each time period is given a region on the [[horizontal axis]] of the same length as every other time period, and the measured variable is plotted as a height that stays constant throughout the region of the time period. In this graphical technique, the graph appears as a sequence of horizontal steps. Alternatively, each time period can be viewed as a detached point in time, usually at an integer value on the horizontal axis, and the measured variable is plotted as a height above that time-axis point. In this technique, the graph appears as a set of dots.
 
The values of a variable measured in continuous time are plotted as a [[continuous function]], since the domain of time is considered to be the entire real axis or at least some connected portion of it.
 
==See also==
*[[Discrete calculus]]
 
[[Category:Time]]
[[Category:Dynamical systems]]

Latest revision as of 20:21, 31 October 2014

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