Discrete-Time Process

A Discrete-Time Process is a process whose events occur on discrete time intervals.

• Context:
  • It can (often) be modeled by a Discrete-Time Model (a random variable sequence that ...).
  • It can range from being a Discrete-State Discrete-Time Process to being a Continuous-State Discrete-Time Process.
  • It can range from being a Fully-Observable Discrete-Time Process to being a Partially-Observable Discrete-Time Process.
  • It can range from being a First-Order Discrete-Time Process to being a Higher-Order Discrete-Time Process.
  • It can range from being a Stochastic Discrete-Time Process to being a Deterministic Discrete-Time Process.
  • …
• Example(s):
  • A Deterministic Discrete-Time Process, such as:
    • The number sequence generated by the Fibonacci series (can be modeled as F(n) = F(n-1) + F(n-2), where F(n) is the nth number in the sequence, and F(0)=0, F(1)=1).
    • The output of a machine that produces a constant number of parts per hour, given a stable input and settings.
    • …
  • A Stochastic Discrete-Time Process, such as:
    • The number of cars that pass through an intersection in a given hour (can be modeled with random variables: X(0), X(1), X(2), ..., where X(n) is the number of cars that pass through the intersection in the nth hour).
  • …
• Counter-Example(s):
  • a Continuous-Time Process.
• See: Temporal Bayesian Model, Markov Decision Process, Continuous-Time Process.

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References

2023

• GBard
  • A discrete-time process is a sequence of random variables, where each random variable represents the value of the process at a specific point in time. The time intervals between the successive points in time are typically equal, but they do not have to be.

Discrete-time processes are used in many different fields, including engineering, science, and economics. Some examples of discrete-time processes include:

   The number of customers arriving at a store in a given hour.
   The stock market prices of a particular company at the end of each trading day.
   The temperature readings from a weather station every minute.
   The audio signal from a digital recording.

Discrete-time processes can be analyzed using a variety of mathematical tools, including probability theory, statistics, and signal processing.

Time intervals: The time intervals between successive hours, which are equal to one hour in this case.

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