Stability in Time Series Analysis 📂Statistical Analysis

Stability in Time Series Analysis

Definition ¹

Time series data is said to have stationarity when its mean and variance are constant over time.

Description

It’s not normal正常 as in standard, but stationarity定常.

The fact that data is stationary means that its mean and variance are stabilized, making it easier to analyze. If the data is not stationary, it becomes difficult to analyze, hence preprocessing is carried out to induce stationarity. Usually, if the mean is not constant, differencing is applied, and if the variance is not constant, a transformation is applied.

Let’s look at the following four graphs.

Difficult Data: Not only does it show a complex fluctuation pattern, but the values tend to increase over time, and the extent of this increase becomes more significant. It’s not hard to predict future trends, but it’s very difficult to express them neatly in a formula.
Constant Mean: It’s not difficult because it spreads around a fixed 0, but the problem is the widening range.
Constant Variance: Each pattern has a consistent shape, but one needs to explain the trend of increasing values over time.
Stationary Data: Since both mean and variance are constant, one only needs to well explain the recurring fluctuations.

Thus, having stationary data is a very good thing, and in fact, it can be said to be a prerequisite for using time series analysis.

Meanwhile, if for every point in time $t_{1} , t_{2} , \cdots , t_{n}$ and for every time lag $k$, $Y_{t_{1}} , Y_{t_{2}} , \cdots , Y_{t_{n}}$ and $Y_{t_{1} - k} , Y_{t_{2} - k} , \cdots , Y_{t_{n} - k}$ have the same joint distribution, then the stochastic process $\left\{ Y_{t} \right\}$ is said to be strictly stationary. However, this condition is too ideal, and thus it’s not commonly mentioned.

Code

win.graph(); par(mfrow=c(2,2))
plot(AirPassengers,main='다루기 어려운 데이터')
plot(diff(AirPassengers),main='일정한 평균')
plot(log(AirPassengers),main='일정한 분산')
plot(diff(log(AirPassengers)),main='정상적 데이터')