Editing Forecasting (section)

==Categories of forecasting methods==

===Qualitative vs. quantitative methods===

Qualitative forecasting techniques are subjective, based on the opinion and judgment of consumers and experts; they are appropriate when past data are not available.
They are usually applied to intermediate- or long-range decisions. Examples of qualitative forecasting methods are{{citation needed|date=May 2012}} informed opinion and judgment, the [[Delphi method]], [[market research]], and historical life-cycle analogy.

Quantitative forecasting [[mathematical model|models]] are used to forecast future data as a function of past data. They are appropriate to use when past numerical data is available and when it is reasonable to assume that some of the patterns in the data are expected to continue into the future.
These methods are usually applied to short- or intermediate-range decisions. Examples of quantitative forecasting methods are{{citation needed|date=May 2012}} last period demand, simple and weighted N-Period [[moving average]]s, simple [[exponential smoothing]], Poisson process model based forecasting<ref>{{cite conference |title= A poisson process model for activity forecasting |last1= Mahmud |first1= Tahmida |last2= Hasan |first2= Mahmudul |last3= Chakraborty |first3= Anirban |last4= Roy-Chowdhury |first4= Amit |date= 19 August 2016 |publisher= IEEE |conference= 2016 IEEE International Conference on Image Processing (ICIP) |doi= 10.1109/ICIP.2016.7532978 }}</ref> and multiplicative seasonal indexes. Previous research shows that different methods may lead to different level of forecasting accuracy. For example, [[Group method of data handling|GMDH]] neural network was found to have  better forecasting performance than the classical forecasting algorithms such as Single Exponential Smooth, Double Exponential Smooth, ARIMA and back-propagation neural network.<ref>{{Cite journal | doi=10.1080/14445921.2016.1225149|title = Forecasting the REITs and stock indices: Group Method of Data Handling Neural Network approach| journal=Pacific Rim Property Research Journal| volume=23| issue=2| pages=123–160|year = 2017|last1 = Li|first1 = Rita Yi Man| last2=Fong| first2=Simon| last3=Chong| first3=Kyle Weng Sang|s2cid = 157150897}}</ref>

===Average approach===

In this approach, the predictions of all future values are equal to the mean of the past data. This approach can be used with any sort of data where past data is available. In time series notation:

:<math>\hat{y}_{T+h|T} = \bar{y} = ( y_1 + ... + y_T ) / T </math> <ref name="otexts.org">{{cite book|chapter-url=https://www.otexts.org/fpp/2/3|chapter=2.3 Some simple forecasting methods |publisher=OTexts |title=Forecasting: Principles and Practice |first1=Rob J |last1=Hyndman |first2=George |last2=Athanasopoulos |access-date=16 March 2018 |url-status=live |archive-url= https://web.archive.org/web/20170813151321/https://www.otexts.org/fpp/2/3 |archive-date= Aug 13, 2017 }}</ref>

where <math>y_1, ... , y_T</math> is the past data.

Although the time series notation has been used here, the average approach can also be used for cross-sectional data (when we are predicting unobserved values; values that are not included in the data set). Then, the prediction for unobserved values is the average of the observed values.

===Naïve approach===

Naïve forecasts are the most cost-effective forecasting model, and provide a benchmark against which more sophisticated models can be compared. This forecasting method is only suitable for time series [[data]].<ref name="otexts.org" /> Using the naïve approach, forecasts are produced that are equal to the last observed value. This method works quite well for economic and financial time series, which often have patterns that are difficult to reliably and accurately predict.<ref name="otexts.org"/> If the time series is believed to have seasonality, the seasonal naïve approach may be more appropriate where the forecasts are equal to the value from last season. In time series notation:

:<math>\hat{y}_{T+h|T} = y_T </math>

===Drift method===
A variation on the naïve method is to allow the forecasts to increase or decrease over time, where the amount of change over time (called the [[Stochastic drift|drift]]) is set to be the average change seen in the historical data. So the forecast for time <math>T+h</math> is given by

:<math>\hat{y}_{T+h|T} = y_T + \frac{h}{T-1}\sum_{t=2}^T (y_{t}-y_{t-1}) = y_{T}+h\left(\frac{y_{T}-y_{1}}{T-1}\right).</math>  <ref name="otexts.org"/>

This is equivalent to drawing a line between the first and last observation, and extrapolating it into the future.

===Seasonal naïve approach===

The seasonal naïve method accounts for seasonality by setting each prediction to be equal to the last observed value of the same season. For example, the prediction value for all subsequent months of April will be equal to the previous value observed for April. The forecast for time <math>T + h</math> is<ref name="otexts.org"/>

:<math>\hat{y}_{T+h|T} = y_{T+h-m(k+1)}</math>
where <math>m</math>=seasonal period and <math>k</math> is the smallest integer greater than <math>(h-1) / m</math>.

The seasonal naïve method is particularly useful for data that has a very high level of seasonality.

=== Deterministic approach ===
A deterministic approach is when there is no stochastic variable involved and the forecasts depend on the selected functions and parameters.<ref name=":1">{{Cite journal |last1=Stoop |first1=Ruedi |last2=Orlando |first2=Giuseppe |last3=Bufalo |first3=Michele |last4=Della Rossa |first4=Fabio |date=2022-11-18 |title=Exploiting deterministic features in apparently stochastic data |journal=Scientific Reports |language=en |volume=12 |issue=1 |pages=19843 |doi=10.1038/s41598-022-23212-x |issn=2045-2322|doi-access=free |pmid=36400910 |pmc=9674651 |bibcode=2022NatSR..1219843S |hdl=11311/1233353 |hdl-access=free }}</ref><ref>{{Cite journal |last1=Orlando |first1=Giuseppe |last2=Bufalo |first2=Michele |last3=Stoop |first3=Ruedi |date=2022-02-01 |title=Financial markets' deterministic aspects modeled by a low-dimensional equation |journal=Scientific Reports |language=en |volume=12 |issue=1 |pages=1693 |doi=10.1038/s41598-022-05765-z |issn=2045-2322|doi-access=free |pmid=35105929 |pmc=8807815 |bibcode=2022NatSR..12.1693O |hdl=20.500.11850/531723 |hdl-access=free }}</ref> For example, given the function    

:<math>\begin{aligned} f_n(x_t) = \dfrac{1}{(1+x_t^n)} \, , \qquad n \in {\mathbb {N}},\;x\in {\mathbb {R}}. \end{aligned} </math>


The  short term behaviour <math>x_t</math>  and the is the medium-long term trend <math>y_t</math> are 
:<math>
\begin{aligned} {\left\{ \begin{array}{ll} x_{t+1} = \alpha f_n(x_t) + \gamma \, y_t + \delta \\ y_{t+1} = \beta \, y_t - \mu \, x_{t} + \eta  \end{array}\right. } \end{aligned}
</math>

where <math> \alpha, \gamma, \beta, \mu, \eta </math> are some parameters.

This approach has been proposed to simulate bursts of seemingly stochastic activity, interrupted by quieter periods. The assumption is that the presence of a strong deterministic ingredient is hidden by noise. The deterministic approach is noteworthy as it can reveal the underlying dynamical systems structure, which can be exploited for steering the dynamics into a desired regime.<ref name=":1" />

===Time series methods===
[[Time series]] methods use historical data as the basis of estimating future outcomes. They are based on the assumption that past demand history is a good indicator of future demand.

*[[Moving average]]
*[[Weighted moving average]]
*[[Exponential smoothing]]
*[[Autoregressive–moving-average model|Autoregressive moving average (ARMA)]] (forecasts depend on past values of the variable being forecast and on past prediction errors)
*[[Autoregressive integrated moving average|Autoregressive integrated moving average (ARIMA)]] (ARMA on the period-to-period change in the forecast variable)
:e.g. [[Box–Jenkins]] 
:Seasonal ARIMA or SARIMA or ARIMARCH,
*[[Extrapolation]]
*[[Linear prediction]]
*[[Trend estimation]] (predicting the variable as a linear or polynomial function of time)
*[[Growth curve (statistics)]]
*[[Recurrent neural network]]

===Relational methods===

Some forecasting methods try to identify the underlying factors that might influence the variable that is being forecast. For example, including information about climate patterns might improve the ability of a model to predict umbrella sales. Forecasting models often take account of regular seasonal variations. In addition to climate, such variations can also be due to holidays and customs: for example, one might predict that sales of college football apparel will be higher during the football season than during the off season.<ref name="NahmiasOlsen2015">{{cite book|author1=Steven Nahmias|author2=Tava Lennon Olsen|title=Production and Operations Analysis: Seventh Edition|url=https://books.google.com/books?id=SIsoBgAAQBAJ&q=forecasting|date=15 January 2015|publisher=Waveland Press|isbn=978-1-4786-2824-8}}</ref>

Several informal methods used in causal forecasting do not rely solely on the output of mathematical [[algorithm]]s, but instead use the judgment of the forecaster. Some forecasts take account of past relationships between variables: if one variable has, for example, been approximately linearly related to another for a long period of time, it may be appropriate to extrapolate such a relationship into the future, without necessarily understanding the reasons for the relationship.

Causal methods include:

*[[Regression analysis]] includes a large group of methods for predicting future values of a variable using information about other variables. These methods include both [[parametric statistics|parametric]] (linear or non-linear) and [[Nonparametric regression|non-parametric]] techniques.
*[[ARMAX|Autoregressive moving average with exogenous inputs (ARMAX)]]<ref>{{cite book|last=Ellis|first=Kimberly|title=Production Planning and Inventory Control Virginia Tech|year=2008|publisher=McGraw Hill|isbn=978-0-390-87106-0}}</ref>

Quantitative forecasting models are often judged against each other by comparing their in-sample or out-of-sample [[mean square error]], although some researchers have advised against this.<ref>{{cite journal | url = http://marketing.wharton.upenn.edu/ideas/pdf/armstrong2/armstrong-errormeasures-empirical.pdf | title = Error Measures For Generalizing About Forecasting Methods: Empirical Comparisons | author = [[J. Scott Armstrong]] and Fred Collopy | journal = [[International Journal of Forecasting]] | volume = 8 | pages = 69–80 | year = 1992 | doi = 10.1016/0169-2070(92)90008-w | url-status = dead | archive-url = https://web.archive.org/web/20120206182744/http://marketing.wharton.upenn.edu/ideas/pdf/armstrong2/armstrong-errormeasures-empirical.pdf | archive-date = 2012-02-06 | citeseerx = 10.1.1.423.508 }}</ref> Different forecasting approaches have different levels of accuracy. For example, it was found in one context that [[GMDH]] has higher forecasting accuracy than traditional ARIMA.<ref>16. Li, Rita Yi Man, Fong, S., Chong, W.S. (2017) [http://www.tandfonline.com/eprint/pgjIcAMrJBP4WcHIZH7F/full Forecasting the REITs and stock indices: Group Method of Data Handling Neural Network approach], Pacific Rim Property Research Journal, 23(2), 1-38</ref>

===Judgmental methods===

Judgmental forecasting methods incorporate intuitive judgement, opinions and subjective [[probability]] estimates. Judgmental forecasting is used in cases where there is a lack of historical data or during completely new and unique market conditions.<ref>{{cite book|chapter-url=https://www.otexts.org/fpp/3/1|chapter=3.1 Introduction |publisher=OTexts |title=Forecasting: Principles and Practice |first1=Rob J |last1=Hyndman |first2=George |last2=Athanasopoulos |access-date=16 March 2018 }}</ref>

Judgmental methods include:

*Composite forecasts{{citation needed|date=January 2019}}
*Cooke's method{{citation needed|date=January 2019}}
*[[Delphi method]]
*[[Forecast by analogy]]
*[[Scenario planning|Scenario building]]
*[[Statistical survey]]s
*[[Technology forecasting]]

===Artificial intelligence methods===
*[[Artificial neural networks]]
*[[Group method of data handling]]
*[[Support vector machine]]s

Often these are done today by specialized programs loosely labeled
*[[Data mining]]
*[[Machine learning]]
*[[Pattern recognition]]

===Geometric extrapolation with error prediction===

Can be created with 3 points of a sequence and the "moment" or "index". This type of extrapolation has 100% accuracy in predictions in a big percentage of known series database (OEIS).<ref>{{cite web |title=Probnet: Geometric Extrapolation of Integer Sequences with error prediction |author=V. Nos |date=2021-06-02 |url=https://hackage.haskell.org/package/Probnet |website=Hackage |publisher=Haskell |access-date=2022-12-06 |url-status=live |archive-url=https://web.archive.org/web/20220814084157/https://hackage.haskell.org/package/Probnet |archive-date=2022-08-14}}</ref>

===Other methods===
*[[Granger causality]]
*[[Simulation]]
*[[Demand forecasting]]
*[[Probabilistic forecasting]] and [[Ensemble forecasting]]