📂Statistical Analysis

Step Function and Pulse Function

Definition ¹

1. The function defined as follows $S_{t}^{(T)}$ is called a step function. $$ S_{t}^{(T)} := \begin{cases} 1 & , t \le T \\ 0 & , \text{otherwise} \end{cases} $$
2. The function defined as follows $P_{t}^{(T)}$ is called a pulse function. $$ \begin{align*} P_{t}^{(T)} :=& \nabla S_{t}^{(T)} \\ =& S_{t}^{(T)} - S_{t-1}^{(T)} \end{align*} $$

Description

Step functions and pulse functions are useful for representing equations used in intervention analysis, and their properties per se don’t have significant meanings. The step function is named for its stair-like appearance on the graph, and the pulse function represents a short moment of impact graphically. [ NOTE: It’s interesting that such shapes and concepts also appear in mathematical physics. ]

In the form of $Y_{t} = m_{t} + N_{t}$ in intervention analysis, the term intervening $m_{t}$ can be represented by these functions. If the analysis drastically changes at some point $T$, a step function can be used, or a pulse function can be used to handle just one exception. For example, it can be as follows: $$ m_{t} = \omega S_{t}^{(T)} $$

$$ m_{t} = \omega P_{t}^{(T)} $$ Here, $\omega$ is a coefficient. Since the function values of step and pulse functions are $0$ or $1$, such correction is needed. $m_{t}$ can be used more freely than you might think. For instance, $m_{t}$ itself can be assumed to follow some ARIMA model. The following resembles the form of $ARMA(1,1)$: $$ m_{t} = \delta m_{t-1} + \omega P_{t-1}^{(T)} $$ Similarly, $\delta$ is a coefficient. This representation is interesting because by using the backshift $B$, the following equation manipulation can be done: $$ \begin{align*} m_{t} =& \delta m_{t-1} + \omega P_{t-1}^{(T)} \\ =& \delta B m_{t} + \omega B P_{t}^{(T)} \end{align*} $$ If we move $\delta B m_{t}$ to the left side, $$ m_{t} - \delta B m_{t} = \omega B P_{t}^{(T)} $$ and divide both sides by $(1-\delta B)$, $$ m_{t} = {{\omega B} \over {1-\delta B}} P_{t}^{(T)} $$ that is, if $m_{t}$ is not extremely complicated, it can be represented in a clean form like $m_{t} = {{\omega ( B) } \over { \delta (B) }} P_{t}^{(T)}$. In this manner, we can also obtain the useful relationship, $$ S_{t}^{(T)} = {{1} \over {1 - B}} P_{t}^{(T)} $$ allowing us to develop equations more freely. Although in actual analysis there might not be a real occasion to use this, at least understanding that the $m_{t}$ in intervention analysis is derived in this fashion and form is necessary.