# 1 Introduction^{1}

Forecasting has come a long way since early humans looked at the sky to see if the weather would be suitable for hunting, and even since hunters could get a forecast such as “a high of 40 with a chance of rain”. Now a hunter can look at a smartphone to instantly get hour-by-hour forecasts of temperatures and probabilities of rain at multiple locations as well as videos of maps showing forecasted weather patterns over the coming hours. Tailored forecasts of increasing sophistication can be generated to inform important decisions of many different types by managers, public officials, investors, and other decision makers.

In the 15 years since the excellent review paper by De Gooijer & Hyndman (2006), the field of forecasting has seen amazing growth in both theory and practice. Thus, this review is both timely and broad, ranging from the highly theoretical to the very applied.

Rapid advances in computing have enabled the analysis of larger and more complex data sets and stimulated interest in analytics and data science. As a result, the forecaster’s toolbox of methods has grown in size and sophistication. Computer science has led the way with methods such as neural networks and other types of machine learning, which are getting a great deal of attention from forecasters and decision makers. Other methods, including statistical methods such as Bayesian forecasting and complex regression models, have also benefited from advances in computing. And improvements have not been limited to those based on computing advances. For example, the literature on judgmental forecasting has expanded considerably, driven largely by the “wisdom of crowds” notion.

The combining, or aggregation, of forecasts, which is not a new idea, has received increased attention in the forecasting community recently and has been shown to perform well. For example, the top-performing entries in the M4 Competition run by Spyros Makridakis combined forecasts from multiple methods. Many models have been developed to forecast the number of deaths that will be caused by COVID-19, and combining the forecasts makes sense because it is hard to know which one will be the most accurate. It is consistent with Bayesian ideas since it can be viewed as updating, with each individual forecast added to the combined forecast (also called an ensemble) contributing some new information.

Despite the excitement surrounding these new developments, older methods such as ARIMA and exponential smoothing are still valuable too. Exponential smoothing, along with other simple approaches, are quite robust and not as prone to overfitting as more complex methods. In that sense, they are useful not only on their own merits, but as part of an ensemble that also includes more sophisticated methods. Combined forecasts are more valuable if the forecasts come from methods that are diverse so that their forecast errors are not highly correlated.

The conditions leading to larger, more sophisticated toolboxes for forecasters have also led to larger data sets with denser grids and improved models in areas of application. This has happened with models of the atmosphere, which are important in formulating improved weather forecasts. More detailed information about customers and their preferences allows the development of improved models of customer behaviour for managers. In turn, forecasting methods that can handle all of that information quickly are valuable for decision-making purposes. This process has spurred an explosion in trying to gather information on the internet.

Risk is an important consideration in decision making, and probability forecasts can quantify such risks. Theoretical work in probability forecasting has been active for some time, and decision makers in many areas of practice have embraced the use of probability forecasts. In the Bayesian approach, inferences and forecasts are probabilistic in nature, and probability forecasts can be generated in many other ways too.

The U.S. National Weather Service began issuing probabilities of precipitation to the public in the 1960s. Yet extensive widespread use and dissemination of probabilities has only developed since the turn of the century. Now probability forecasts are increasingly communicated to the public and used as inputs in decision making. Nate Silver’s FiveThirtyEight.com report gives probability forecasts for elections, medicine and science, sporting events, economic measures, and many other areas, often looking at multiple forecasting models individually and also combining them.

It is natural for people to desire certainty. When probability forecasts of precipitation were first disseminated widely, many were very sceptical about them, with some accusing the forecasters of hedging and saying “Don’t give me a probability. I want to know if it’s going to rain or not”. Of course, point forecasts often are given along with probability forecasts. The current frequent exposure to probabilities helps the general public better understand, appreciate, and feel more comfortable with them. And the current situation in the world with COVID-19, increases in huge fires, big storms, political polarisation, international conflicts, etc., should help them realise that we are living in an age with huge uncertainties, and forecasts that quantify these uncertainties can be important. Where possible, visualisation can help, as indicated by the saying that a picture is worth a thousand words. Examples are the cones of uncertainty on maps in forecasts of the speed, severity, and future path of hurricanes, and the time line of the probability of a team winning a game, updated quickly after each play.

Put simply, this is an exciting time for the field of forecasting with all of the new theoretical developments and forecasting applications in practice. Forecasting is so ubiquitous that it’s not possible to cover all of these developments in a single article. This article manages to cover quite a few, and a good variety. Using short presentations for each one from an expert “close to the ground” on that theoretical topic or field of practice works well to provide a picture of the current state of the art in forecasting theory and practice.

### Bibliography

De Gooijer, J. G., & Hyndman, R. J. (2006). 25 years of time series forecasting. *International Journal of Forecasting*, *22*, 443–473.

This subsection was written by Robert L. Winkler (last update on 22-Oct-2021).↩︎