Seasonal multiplicative model

The technique of seasonal decomposition uses the same additive and multiplicative models as in exponential smoothing, but without the smoothing procedure. 

The patterns of both ACF (Fig. 6-27) and PACF (Fig. 6-28) of the errors from the previous model are used to find the complete multiplicative trend and the seasonal ARIMA model. The following conclusions can be drawn ... 

ABSTRACT This paper is to research an application of the multiplicative seasonal model to forecast the total number of China s coal mine accidents. By the empirical analysis on the data of coal mine accidents from January 2006 to December 2010, an accepted multiplicative seasonal forecasting model ARIMA(4,1,1)(1,1,1) is built up after differing the series to be stationary and estimating the order and parameters of the model. Furthermore, the test of this multiplicative seasonal model shows that it has a desirable fitting effect on the data of coal mine accidents. At last, this model is applied to forecast the number of national coal mine accidents from January 2010 to December 2010, and the forecasted values have a high accuracy when compared to the actual data. 

Models for multiplicative effects are also available. Multiplicative seasonal factors are suitable if the variance, v2[a (/)J increases as the mean increases ... 

These were the general conclusions from ACF and PACF of the time series for modeling ARIMA (p,d,q) sp,sd,sq). Now the second term of the multiplicative ARIMA model - the seasonal ARIMA component, ARIMA (()S)S))(sp,sd,sq) - must be estimated. 

PACF of the errors indicates a first-order autoregression component, therefore the parameter p is set to unity. The spikes at lags 13 and 25 are a consequence of the multiplicative seasonal model. 

Table 4.1. models (using backwards elimination stepwise multiple regression) for the study lakes based on suites of parameters. The global indices based on annual and seasonal data, with and without lags (1 and 2-year) and deposition provided the best models. Local and regional-scale climate parameters (e.g., precipitation, temperature, etc.) did not provide significant models... 

The empirical analysis of ARIMA multiplicative seasonal model to forecast the total number of coal mine accidents... 

Also, a time series changing seasonally couldn t be modeled by ARMA immediately. Firstly, it should be differed to remove the seasonality of the series, and the length of one difference step is one seasonal cycle. Generally, if the series is processed by a D-th order difference the cycle length of which is s, the seasonality will almost be removed. Furthermore, if the series needs to be turned stationary by a d-th order difference before it s impacted by the seasonal difference, a model called the Multiplicative Seasonal Mod-el ARIMA(p, d, q) (P, D, Q) could analyze the original series, and the model is described as below ... 

Compared to ordinary time series models, the Multiplicative Seasonal Model needs more historical data, and the Multiplicative Seasonal Model can be applied to a wider field because data in daily life always have an obvious trend and seasonal features. Therefore, the Multiplicative Seasonal Model can well solve such problems that involve some issues about forecasting, and as well as reach a high precision. The model in this paper, ARIMA (4,1,1)(1,1,1) well matches the monthly changing number of national coal mine accidents. Moreover, the more historical data, the more accurate the forecasted result is. AH above, the Multiplicative Seasonal Model is a practical tool for us to forecast or to apply in many other fields. 

Guess how many different models of an automotive brand are introduced per year Will that be a couple of two- and four-door sedan models, a few cross-over models, a few SUV models, a few convertible models, and a couple of sports models with a few colors Now compare this with an apparel brand, which produces many product lines each with many styles in multiple colors/prints, in multiple sizes, for a number of seasons per year. In most situations the brand may also have different product lines for different markets with different price points. The number of stock-keeping units (SKUs) produced by an apparel brand per year cannot be meaningfiiUy compared to the SKUs developed by an automotive brand. Now if you think of the product-development (PD) efforts for these two industries, it is quite clear that the PD for an apparel brand is quite complex. The PD gets further complicated with the time pressure to meet market demand for fashion items. Therefore, it is important and beneficial to understand the apparel PD process for anyone interested in careers in the apparel industry. 

The aim of this probabilistic Monte Carlo framework is to describe the huge variation that undoubtedly occurs in real water supply zones. If we can mimic this real-world variation then the model can be used for predictive purposes, as has been demonstrated by case studies (Hayes et al, 2006, 2008). It should be appreciated that the average lead concentrations predicted by the model relate to a single plumbosolvency condition occurring in time, whether it is applied as a constant throughout an area or as a range. In consequence, the predicted results relate to an average condition over time. This is reasonable if the periods of time under consideration extend to multiples of a year, such that seasonal variation is accommodated. 

Having evaluated our method on 1993 sales data, we also wanted to determine how well it would perform across multiple years, since in actual use we would be fitting the model on sales that had occurred 1-year prior to actual introduction of the styles being tested. We obtained sales data for 30 style/colors Ifom the 1994 fall season in the Knit Tops division that had been tested in 25 stores chosen by the planners. These were all new products that had not been on sale during 1993. We fit our clustering model as fit on the 1993 data to these 30, 1994 products. This exactly replicates how the model would be used in practice and hence is an accurate measure of its effectiveness. 

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