Generally, a model for time-series forecasting can be written as

where yₜ is the variables to be forecasted (dependent variable, or response variable), t is the time at which the forecast is made, h is the forecast horizon, Xₜ is the variables used at time t to make forecast (independent variable), θ is a vector of parameters in function *g*, and εₜ₊ₕ denotes errors. It is worth noting that the observed data is *uniquely orderly according to the time of observation*, but it doesn’t have to be dependent on time, i.e. time (index of the observations) doesn’t have to be one of the independent variables.

## Some possible properties of time series

**Stationarity**: a stationary process is *a stochastic process, whose mean, variance and autocorrelation structure do not change over time. *It can also be defined formally using mathematical terms, but in this article, it’s not necessary. Intuitively, if a time series is stationary, we look at some parts of them, they should be very similar — the time series is flat looking and the shape doesn’t depend on the shift of time. (A quick check of knowledge: is f(t) = sin(t) a stationary process? It surely isn’t, since it’s not stochastic, stationarity is not one of its properties)

Figure 1.1 shows the simplest example of a stationary process — white noise.

The above image Figure 1.2 shows a non-stationary time series. Why is it so? We can see the obvious trend, it means that the variance changes over time. But if we use linear regression to fit a line to it (to capture the trend) and remove the trend the data now has a constant location and variance, but it’s still not stationary because of periodic behavior, which is not stochastic.

When using ARMA to model a time series, one of the assumptions is that the data is stationary.

**Seasonality**: Seasonality is *the property of showing certain variations in a specific time interval that is shorter than a year* (it can be over a different period of course. If we are observing the hourly temperature in a day and collect data over a couple of days, then the period is a day and it can also have seasonality — peaks might appear at 2 or 3 pm. This means we…

Continue reading: https://towardsdatascience.com/deep-understanding-of-the-arima-model-d3f0751fc709?source=rss—-7f60cf5620c9—4

Source: towardsdatascience.com

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