Forecast error). Property 2 and the continuous functional calculus ensure that Φ preserves the *-operation. Finally, the semicommutator property shows that Φ is multiplicative. Therefore the theorem holds. Wold's decomposition In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator. In time series analysis, the theorem implies that any stationary discrete-time stochastic process can be decomposed into 	Wold's decomposition
24030949	Watkins (surname) Watkins (surname) Watkins is an English and Welsh surname derived as a patronymic from Watkin, in turn a diminutive of the name Watt (also Wat), a popular Middle English given name itself derived as a pet form of the name Walter. Notable people with the surname include: Watkins (surname) Watkins is an English and Welsh surname derived as a patronymic from Watkin, in turn a diminutive of the name Watt (also Wat), a popular Middle English given name itself derived as a pet form of the name Walter. Notable people with the surname include: 	Watkins (surname)
24030950	Forecast error Forecast error In statistics, a forecast error is the difference between the actual or real and the predicted or forecast value of a time series or any other phenomenon of interest. Since the forecast error is derived from the same scale of data, comparisons between the forecast errors of different series can only be made when the series are on the same scale. In simple cases, a forecast is compared with an outcome at a single time-point and a summary of forecast errors is constructed over a collection of such time-points. Here the forecast may be assessed using 	Forecast error
24030951	the difference or using a proportional error. By convention, the error is defined using the value of the outcome minus" the value of the forecast. In other cases, a forecast may consist of predicted values over a number of lead-times; in this case an assessment of forecast error may need to consider more general ways of assessing the match between the time-profiles of the forecast and the outcome. If a main application of the forecast is to predict when certain thresholds will be crossed, one possible way of assessing the forecast is to use the timing-error—the  difference in time between when the outcome crosses the threshold and when the forecast does so. When there is interest in the maximum value being reached, assessment of forecasts can be done using any of: Forecast error can be a calendar forecast error or a cross-sectional forecast error, when we want to summarize the forecast error over a group of units. If we observe the average forecast error for a time-series of forecasts for the same product or phenomenon, then we call this a calendar forecast error or time-series forecast error. If we observe this for multiple  products for the same period, then this is a cross-sectional performance error. Reference class forecasting has been developed to reduce forecast error. Combining forecasts has also been shown to reduce forecast error. Calculating forecast error. The forecast error is the difference between the observed value and its forecast based on all previous observations. If the error is denoted as formula_1 then the forecast error can be written as; formula_2 where, formula_3 = observation formula_4 = denote the forecast of formula_3 based on all previous observations Forecast errors can be evaluated using a variety of methods namely mean percentage error , root mean squared error, mean absolute percentage error, mean squared error. Other methods include tracking signal and forecast bias. For forecast errors on training data formula_3 denotes the observation and formula_4 is the forecast For forecast errors on test data formula_8 denotes the actual value of the h-step observation and the forecast is denoted as formula_9 Academic literature. Dreman and Berry in 1995 "Financial Analysts Journal", argued that securities analysts' forecasts are too optimistic, and that the investment community relies too heavily on their forecasts. However, this was countered by Lawrence D. Brown in 1996 and then again in  1997 who argued that the analysts are generally more accurate than those of "naive or sophisticated time-series models" nor have the errors been increasing over time. Hiromichi Tamura in 2002 argued that herd-to-consensus analysts not only submit their earnings estimates that end up being close to the consensus but that their personalities strongly affect these estimates. Examples of forecasting errors. Michael Fish - A few hours before the Great Storm of 1987 broke, on 15 October 1987, he said during a forecast: "Earlier on today, apparently, a woman rang the BBC and said she heard there was a  hurricane on the way. Well, if you're watching, don't worry, there isn't!". The storm was the worst to hit South East England for three centuries, causing record damage and killing 19 people. Great Recession - The financial and economic crisis that erupted in 2007—arguably the worst since the Great Depression of the 1930s—was not foreseen by most of the forecasters, even if a few lone analysts had been predicting it for some time (for example, Nouriel Roubini and Robert Shiller). The failure to forecast the "Great Recession" has caused a lot of soul searching in the  profession. The UK's Queen Elizabeth herself asked why had nobody noticed that the credit crunch was on its way, and a group of economists—experts from business, the City, its regulators, academia, and government—tried to explain in a letter. It was not just forecasting the Great Recession, but also its impact where it was clear that economists struggled. For example, in Singapore, Citi argued the country would experience "the most severe recession in Singapore’s history". The economy grew in 2009 by 3.1% and in 2010, the nation saw a 15.2% growth rate. At the end of 2019  the International Monetary Fund estimated global growth in 2020 to reach 3.4%, but as a result of the coronavirus pandemic, the IMF have revised its estimate in November 2020 to expect the global economy to shrink by 4.4%. Forecast error In statistics, a forecast error is the difference between the actual or real and the predicted or forecast value of a time series or any other phenomenon of interest. Since the forecast error is derived from the same scale of data, comparisons between the forecast errors of different series can only be made when the series are on the same