The previous post established what stationarity means and how to recognise when a series lacks it. The airline passenger series fails on both counts: a clear upward trend and a repeating seasonal cycle, confirmed by an ACF that barely drops across 40 lags. This post is about what to do about it. The main tool... Continue Reading →
Stationarity
The previous posts built up a set of forecasting tools; SES, Holt-Winters, ARIMA, SARIMA, and each one quietly assumed something about the series it was working with. That assumption is stationarity. This post is about making that assumption explicit: what stationarity means, how to spot it, and what happens when a series violates it. Time... Continue Reading →
SARIMA (p,d,q)(P,D,Q)m: Adding Seasonality to ARIMA
ARIMA captured the trend but left the seasonal cycle untouched. To handle both at once, SARIMA (Seasonal ARIMA) extends the model with a second set of parameters that operate at the seasonal lag rather than the observation lag. The result is a unified framework that handles trend and seasonality simultaneously. The parameters SARIMA(p,d,q)(P,D,Q)m has two... Continue Reading →
ARIMA (p, d, q): Autoregressive Integrated Moving Average
The previous post showed that AR, MA, and ARMA break down on the airline series because they assume stationarity. The common fix is to difference the series manually and then fit ARMA on the result. ARIMA formalises exactly this. The โIโ stands for integrated, and it refers to the differencing step being built directly into... Continue Reading →
AR, MA, and ARMA
The smoothing methods from the previous posts; SES, DES, Holt-Winters work by taking weighted averages of past observations. This post introduces a different family: AR, MA, and ARMA models. Instead of smoothing, these treat forecasting as a regression problem. The predictors are either past observed values, past forecast errors, or a combination of both. The... Continue Reading →
Holt-Winters: Adding Seasonality to the Mix
SES models the level. DES adds trend. The natural question is: what about seasonality? Thatโs what Triple Exponential Smoothing, better known as the Holt-Winters method, does. It extends the DES framework with a third smoothed component that tracks the seasonal pattern, updating it over time just like the level and trend. Before getting into the... Continue Reading →
Double Exponential Smoothing
The previous posts showed that SES, no matter how well you tune ฮฑ, always produces a flat forecast. The model only tracks the current level, and when the series is trending, that flat line consistently falls behind. Double Exponential Smoothing (DES), also known as Holtโs linear method, fixes exactly this by adding a second component:... Continue Reading →
SES in Practice: Testing, Fitting, and Measuring
The previous post covered what Simple Exponential Smoothing is and how the formula works. This post is about actually applying it, checking the series structure, splitting the data, fitting SES, and measuring how well it did. Checking the structure of the series Before fitting anything, it helps to confirm what youโre working with. The airline... Continue Reading →
Simple Exponential Smoothing
Up to this point, the methods weโve looked at; decomposition, regression with Fourier terms, have all been about understanding the structure of a time series: pulling apart trend, seasonality, and remainder into interpretable pieces. Those methods are useful for analysis, and they can be extended to produce forecasts. But thereโs a separate family of forecasting... Continue Reading →
Fourier Terms for Seasonality
In the previous posts we used dummy variables to handle seasonality in regression: one binary column per month, one per day of the week, and so on. That works well for clean cases, but it runs into real problems as soon as the data gets more complex. An alternative is to model seasonality using Fourier... Continue Reading →