Forecast a time series

Status

What other tools call it

Meridian's classification; an independent blind rating agreed on 86.2% of 312 rows (exact class, version 2).

Background

When to use it

Forecasting is required in many situations, such as scheduling call-centre staff, which needs forecasts of call volumes, or stocking an inventory, which needs forecasts of stock requirements.^[1] Quantitative forecasting can be applied when numerical information about the past is available and it is reasonable to assume that some aspects of past patterns will continue into the future.^[1] How predictable a quantity is depends on how well the factors behind it are understood, how much data is available, how similar the future is to the past, and whether forecasts can affect the thing being forecast.^[1] Forecasts of a currency exchange rate have a direct effect on the rate itself, so forecasting whether it will rise or fall tomorrow is about as predictable as a tossed coin.^[1] Before choosing a model, check the series against those four conditions for predictability, and expect little from a forecast of a quantity that reacts to forecasts of it. (Meridian's view)

The simplest time series forecasting methods use only the variable being forecast, so they extrapolate trend and seasonal patterns but ignore other information such as marketing initiatives or competitor activity.^[1] A forecaster may choose a time series model over one with predictor variables because the system is not understood, because the predictors' future values would themselves have to be forecast, or because only what will happen matters, not why.^[1] Models that combine a series' own past values with predictor variables are known by various names, including dynamic regression models and transfer function models.^[1]

When choosing forecasting models, it is common practice to split the data into training data, used to estimate a method's parameters, and test data, used to evaluate its accuracy.^[3] The test set is typically about 20% of the sample, and should ideally be at least as large as the furthest ahead the forecasts are required to reach.^[3] A model that fits the training data well will not necessarily forecast well, and a perfect fit can always be obtained by using a model with enough parameters.^[3] Time series cross-validation uses a series of test sets, each with a training set holding only the observations that came before it, and averages the forecast accuracy over the test sets.^[3] Judge a forecasting model on data it has not seen, in the order time gives, and never on how closely it fits its own history. (Meridian's view)

Prediction intervals usually grow wider the further ahead a forecast reaches, because the further ahead the forecast, the more uncertainty is associated with it.^[3] Some forecasting methods are extremely simple and surprisingly effective.^[3] In RapidMiner, the Default Forecast operator predicts the same value for every future step, such as the last value or a recent mean, as a baseline to test methods such as ARIMA against.^[4] Fit a simple benchmark beside any elaborate model, and keep the model only if it forecasts the test data better than the benchmark does. (Meridian's view)

The method

The mean method forecasts every future value as the average of the historical data.^[3] The naïve method sets every forecast to the value of the last observation, and works remarkably well for many economic and financial time series.^[3] The seasonal naïve method sets each forecast to the last observed value from the same season, such as the same month of the previous year.^[3] The drift method lets forecasts rise or fall by the average change seen in the historical data, which is equivalent to extending a line drawn between the first and last observations.^[3]

Exponential smoothing was proposed in the late 1950s, in work by Brown, Holt and Winters.^[5] Forecasts from exponential smoothing are weighted averages of past observations, with weights that decay exponentially as the observations get older.^[5] Simple exponential smoothing suits data with no clear trend or seasonal pattern, and its forecasts are flat: every future value equals the last estimated level.^[5] Holt extended simple exponential smoothing in 1957 to forecast data with a trend, using one smoothing equation for the level and one for the trend.^[5] Holt's linear trend method carries a constant trend indefinitely into the future, and empirical evidence indicates that it tends to over-forecast, especially for forecasts further ahead.^[5] Gardner and McKenzie introduced a parameter in 1985 that dampens the trend to a flat line some time in the future, and damped trend methods are arguably the most popular individual methods when many series must be forecast automatically.^[5] The Holt-Winters seasonal method, from Holt and Winters, adds a third smoothing equation, for the seasonal component, to the level and trend equations.^[5] The additive form of the Holt-Winters method is preferred when seasonal variations are roughly constant through the series, and the multiplicative form when they change in proportion to the level of the series.^[5]

Underlying the exponential smoothing methods are state space models, which generate the same point forecasts but can also generate prediction intervals.^[5] Each of the state space models is labelled ETS for its Error, Trend and Seasonal components, a label that can also be read as ExponenTial Smoothing.^[5] In the ETS framework, information criteria such as Akaike's Information Criterion (AIC) and the AIC corrected for small sample bias (AICc) can be used to choose the most appropriate model for a series.^[5] ETS models with multiplicative errors are not numerically stable when the data contain zeros or negative values, so they are considered only for strictly positive series.^[5]

Where exponential smoothing models are based on a description of the trend and seasonality in the data, ARIMA models aim to describe the autocorrelations in the data.^[2] A stationary time series is one whose statistical properties do not depend on the time at which it is observed, so a series with a trend or seasonality is not stationary.^[2] Differencing can help stabilise the mean of a time series by removing changes in its level, eliminating or reducing trend and seasonality.^[2] An ARIMA model combines differencing with autoregression and a moving average model, and the integration in its name is the reverse of differencing.^[2] An ARIMA(p, d, q) model is defined by p, the order of the autoregressive part, d, the degree of first differencing involved, and q, the order of the moving average part.^[2] The classic text that popularised ARIMA modelling was published by Box and Jenkins in 1970.^[2] ARIMA point forecasts are calculated one step at a time, replacing future observations with their forecasts and future errors with zero, and repeating for each further step.^[2]

Compared with the alternatives

A commonly held myth is that ARIMA models are more general than exponential smoothing; linear exponential smoothing models are all special cases of ARIMA models, but the non-linear ones have no ARIMA counterparts.^[2] Many ARIMA models have no exponential smoothing counterpart, and all ETS models are non-stationary, while some ARIMA models are stationary.^[2] The AICc can select between models of the same class, but cannot compare ETS with ARIMA models, because the two classes compute the likelihood in different ways.^[2] When forecasting models are compared on a test set, the comparison is valid however the forecasts were produced.^[2] To choose between an ETS model and an ARIMA model, compare their forecasts on held-out test data rather than their information criteria. (Meridian's view)

In RapidMiner, a forecast model already contains the series it was fitted to, so Apply Forecast takes only the model as input, unlike Apply Model, which needs a data table compatible with the model.^[6]

Sources

1. [1]1 Getting started – Forecasting: Principles and Practicesecondary source
2. [2]9 ARIMA models – Forecasting: Principles and Practicesecondary source
3. [3]5 The forecaster’s toolbox – Forecasting: Principles and Practicesecondary source
4. [4]Default Forecast - Rapidminer Documentationdocumentation
5. [5]8 Exponential smoothing – Forecasting: Principles and Practicesecondary source
6. [6]Apply Forecast - Rapidminer Documentationdocumentation