Principles of System Identification

7

Random Processes

This is an introductory and foundational chapter for the theory of random processes. The

objective is to review concepts and definitions pertaining to random variables and random

processes. It is recommended to pay special attention to the concepts of expectation operator,

correlation, stationarity and ergodicity. Interpretations of these concepts in the context of

prediction are provided.

7.1 INTRODUCTORY REMARKS

Chapters 1 and 2 taught us that models developed in identification should ideally explain both the

deterministic and stochastic parts of a process. The theory for modeling deterministic processes was

expounded in Part I. Our journey into the modeling of random processes gets underway with this

chapter. The remaining chapters of this part (Part II) describe the measures, models and representa-

tions of random processes.

Throughout this part, we shall assume that the exogenous inputs are switched off with the ob-

jective of solely focusing on the stochastic effects. In Part IV, which is exclusively concerned with

system identification, we shall superimpose the deterministic and stochastic models under the addi-

tivity assumption.

Recall the early discussions in Chapter 1. Uncertainty is one of the inescapable truths in the

analysis of processes and measurements. The sources of uncertainty in data-driven modeling are

primarily (i) insufficient understanding of the process (modeling errors), (ii) measurement errors

and (iii) effects of unmeasured causes. The second source, is in fact, a manifestation of our inability

to perfectly design and understand the instrumentation and its characteristics. All sources of uncer-

tainty in a process lead to the same repercussion - that it is not possible to accurately predict that

process. Then we have what is known as a random process. To complement this definition, it is

useful to define, a perfectly predictable process (within the walls of theoretical analysis), known as

the deterministic process that generates a deterministic signal. Strictly such processes are never en-

countered in practice. However, the conception is useful in handling situations where the measured

response is due to a mix of known and unknown causes, such as in system identification.

The goal of probability theory and random process modeling is essentially to exploit the pre-

dictability of a random process to the maximum extent possible. With uncertainty reining the

workhorses of prediction, the simplest recourse is to list all possible predictions (outcomes) and

postulate the chance of each of those outcomes.

Example 7.1: Rainfall Prediction

It is a well-known fact that the occurrence of rainfall in any geographical location can never

be accurately predicted. The natural solution is to list both possibilities (“yes” or “no”) and

associate each outcome with their respective chances (probabilities).

Naturally, for events described by continuous-valued variables (e.g., reactor temperature, wind

speed), it is not possible to specify the chance associated with each outcome. Instead we work

with infinitesimal intervals of outcomes and what are known as probability density functions. The

cornerstones of theory of random processes are the concepts of probability and random variables. It

is fitting therefore that we begin this chapter with a review of the same.

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