https://www.analyticsvidhya.com/blog/2015/12/complete-tutorial-time-series-modeling/
A stationary process (a.k.a. a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance also do not change over time.
Random Walk
we infer that the random walk is not a stationary process as it has a time variant variance.
X(t) = X(t-1) + Er(t)
Var[X(t)] = t * Var(Error) = Time dependent.
Dickey Fuller Test of Stationarity
X(t) = Rho * X(t-1) + Er(t)
=> X(t) - X(t-1) = (Rho - 1) X(t - 1) + Er(t)
We have to test if Rho – 1 is significantly different than zero or not. If the null hypothesis gets rejected, we’ll get a stationary time series.
Introduction to ARMA Time Series Modeling
AR stands for auto-regression and MA stands for moving average.AR or MA are not applicable on non-stationary series.
In case you get a non stationary series, you first need to stationarize the series (by taking difference / transformation) and then choose from the available time series models.
Auto-Regressive Time Series Model
x(t) = alpha * x(t – 1) + error (t)
AR(1) formulation denotes that the next instance is solely dependent on the previous instance. The alpha is a coefficient which we seek so as to minimize the error function. Notice that x(t- 1) is indeed linked to x(t-2) in the same fashion. Hence, any shock to x(t) will gradually fade off in future.
For instance, let’s say x(t) is the number of juice bottles sold in a city on a particular day. During winters, very few vendors purchased juice bottles. Suddenly, on a particular day, the temperature rose and the demand of juice bottles soared to 1000. However, after a few days, the climate became cold again. But, knowing that the people got used to drinking juice during the hot days, there were 50% of the people still drinking juice during the cold days. In following days, the proportion went down to 25% (50% of 50%) and then gradually to a small number after significant number of days.
Moving Average Time Series Model
x(t) = beta * error(t-1) + error (t)
A manufacturer produces a certain type of bag, which was readily available in the market. Being a competitive market, the sale of the bag stood at zero for many days. So, one day he did some experiment with the design and produced a different type of bag. This type of bag was not available anywhere in the market. Thus, he was able to sell the entire stock of 1000 bags (lets call this as x(t) ). The demand got so high that the bag ran out of stock. As a result, some 100 odd customers couldn’t purchase this bag. Lets call this gap as the error at that time point. With time, the bag had lost its woo factor. But still few customers were left who went empty handed the previous day.
Difference between AR and MA models
In MA model, noise / shock quickly vanishes with time. The AR model has a much lasting effect of the shock.
The primary difference between an AR and MA model is based on the correlation between time series objects at different time points. The correlation between x(t) and x(t-n) for n > order of MA is always zero. This directly flows from the fact that covariance between x(t) and x(t-n) is zero for MA models (something which we refer from the example taken in the previous section). However, the correlation of x(t) and x(t-n) gradually declines with n becoming larger in the AR model. This difference gets exploited irrespective of having the AR model or MA model. The correlation plot can give us the order of MA model.
Exploiting ACF and PACF plots
Q1. Is it an AR or MA process?
Total Correlation Chart (also known as Auto – correlation Function / ACF). ACF is a plot of total correlation between different lag functions. For instance, in GDP problem, the GDP at time point t is x(t). We are interested in the correlation of x(t) with x(t-1) , x(t-2) and so on.
partial correlation function (PACF)
Q2. What order of AR or MA process do we need to use?
In a moving average series of lag n, we will not get any correlation between x(t) and x(t – n -1) . Hence, the total correlation chart cuts off at nth lag. So it becomes simple to find the lag for a MA series. For an AR series this correlation will gradually go down without any cut off value. So what do we do if it is an AR series?
Here is the second trick. If we find out the partial correlation of each lag, it will cut off after the degree of AR series. For instance,if we have a AR(1) series, if we exclude the effect of 1st lag (x (t-1) ), our 2nd lag (x (t-2) ) is independent of x(t). Hence, the partial correlation function (PACF) will drop sharply after the 1st lag.