Parameters of Discrete Time Models of Detection of Change

A discrete time detection of change (DC) process is characterized by a state S₀ that at some stage t turns into state S₁. Either of two decisions is made at each stage τ: W—take another observation, or D—S₁ is the true state. In the former case, the result of each observation is a random variable x, which has a probability density function f_o(x) if t > τ or f₁(x) if t ≦ τ. In the latter case, if t > τ, an error loss is incurred, the knowledge that t > τ is gained, and the process continues, whereas if t ≦ τ the process terminates with a delay loss proportional to τ − t. In the modified detection of change (MDC) process D is a terminal decision.

Equations are presented for recursively computing useful parameters, such as the probability distributions of the number of observations and of the number of errors in the DC process. The relationships between the two processes are examined, yielding an alternative method for determining the minimum expected loss.