Chapter 1 Getting started

What can be forecast?

The predictability of an event or a quantity depends on several factors including:

  1. how well we understand the factors that contribute to it;
  2. how much data is available;
  3. whether the forecasts can affect the thing we are trying to forecast.

It is important to know when something could be forcast with precision:

Good forecasts capture the genuine patterns and relationships which exist in the historical data, but do not replicate past events that will not occur again.

Usual assumption of forecasting:

What is normally assumed is that the way in which the environment is changing will continue into the future

Two forecasting methods:

  1. quantitative forecasting: If there are no data available, or if the data available are not relevant to the forecasts

  2. qualitative forecasting

  • numerical information about the past is available
  • it is reasonable to assume that some aspects of the past patterns will continue into the future

There is a wide range of quantitative forecasting methods, often developed within specific disciplines for specific purposes. Each method has its own properties, accuracies, and costs that must be considered when choosing a specific method.

Most quantitative prediction problems use either time series data (collected at regular intervals over time) or cross-sectional data (collected at a single point in time). Here we only focus on time series data.

Anything that is observed sequentially over time is a time series. In this book, we will only consider time series that are observed at regular intervals of time (e.g., hourly, daily, weekly, monthly, quarterly, annually). When forecasting time series data, the aim is to estimate how the sequence of observations will continue into the future.

3 types of models:

  • explanatory model (it helps explain what causes the variation in outcome.)

\[ ED = f(current\;temperature,\, strength\;of\;economy,\;population,time\;of\;day, \;error) \]

  • time series model (based on past values of a variable, but not on external variables which may affect the system):

\[ ED_{t+1} = f(ED_t,\,ED_{t−1},\,ED_{t−2},\,ED_{t−3},\,...,\, error) \]

  • dynamic regression models, panel data models, longitudinal models, transfer function models, and linear system models (assuming that \(f\) is linear)

\[ ED_{t+1} = f(ED_t,\,current \; temperature, time \; of \; day,\,day\;of\;week,\,error) \]

An explanatory model is useful because it incorporates information about other variables, rather than only historical values of the variable to be forecast. However, there are several reasons a forecaster might select a time series model rather than an explanatory or mixed model. First, the system may not be understood, and even if it was understood it may be extremely difficult to measure the relationships that are assumed to govern its behaviour. Second, it is necessary to know or forecast the future values of the various predictors in order to be able to forecast the variable of interest, and this may be too difficult. Third, the main concern may be only to predict what will happen, not to know why it happens. Finally, the time series model may give more accurate forecasts than an explanatory or mixed model.

The model to be used in forecasting depends on the resources and data available, the accuracy of the competing models, and the way in which the forecasting model is to be used.

5 steps in a forecasting task:
1. problem definition
2. gathering information
3. exploratory data analysis
4. chossing and fitting models
5. using and evaluating the model

When we obtain a forecast, we are estimating the middle of the range of possible values the random variable could take. Often, a forecast is accompanied by a prediction interval giving a range of values the random variable could take with relatively high probability. For example, a 95% prediction interval contains a range of values which should include the actual future value with probability \(95\%\).

Forecast distribution is the probability distribution of the outcome of interest at time \(t\), \(y_t\), given all information related, \(I\).

When we talk about the “forecast”, we usually mean the average value of the forecast distribution, and we put a “hat” over \(y\) to show this. Thus, we write the forecast of \(y_t\) as \(\hat{y}_t\), meaning the average of the possible values that \(y_t\) could take given everything we know. Occasionally, we will use \(\hat{y}_t\) to refer to the median (or middle value) of the forecast distribution instead.

\(y_{T+h|T}\) means the forecast of \(y_{T+h}\) taking account of \(y_1,\,...,y_T\) (i.e., an h-step forecast taking account of all observations up to time \(T\)).