1. INTRODUCTION
Climate change around the world has recently intensified. For example, in Japan, in the hot summer of 2010, the seasonal mean temperature was the highest in the historical record, collected by the Japan Meteorological Agency (JMA) since 1898 (JMA, 2010). Meanwhile, in the cold winter of 2012, two years later, the monthly mean temperature in Eastern Japan and Western Japan fell below normal (mean) for three consecutive months for the first time in 26 years (JMA, 2012). Such severe fluctuations in climate have a great influence on products whose demand fluctuates according to the season (seasonally dependent products: Kurihara, 2012). That is, the demand for seasonally dependent products such as home air conditioners, heating appliances, and beer is affected not only by the usual seasonal change but also by extreme seasonal changes influenced by climate change.
The manufacturing industry has experienced major changes in the market environment in recent years, and to manufacture and supply at low cost with consideration of changing demand, without missing sales opportunities, has become a significant issue. Therefore, production management has to solve a difficult problem: balancing “agile production,” which supplies products in a timely manner to the market for inventory reduction, and “stable production,” which is aimed at reducing manufacturing costs. We regard this as a balancing problem between inventory reduction and production leveling in production planning, especially aggregate production planning; based on this conceptualization of the problem, we have proposed harmonized models and their solutions (Kurihara and Yamashita, 2013;Kurihara and Yamashita, 2015;Kurihara et al., 2016) based on the “maximum entropy principle” (Klir and Folger, 1988).
If highly accurate demand analysis/forecast is possible, it will be easier to realize production that harmonizes agility and stability by being sufficiently prepared for production and procurement. In particular, for seasonally dependent products, the effects of extracting the characteristics from the aspect of demand and analyzing fluctuating demand based on these characteristics seem to be significant. If the whole demand can be structurally and concisely analyzed in relation to factors and regions, it can support timely and effective coordination between business planning (corporate planning and finance department), sales planning (sales department), and manufacturing/ procurement resource planning (manufacturing department), focusing on aggregate production planning based on the whole demand, in a companywide sales and operation planning (S&OP) process (Bowersox et al., 2010).
Therefore, with respect to seasonally dependent products, this study aims to explore a demand analysis model that can structurally and concisely express the relationship between the whole demand, factors, and regions, to provide a basis for whole demand forecasting.
Demand analysis models are classified into “regression models,” which analyze demand based on causal relationships with explanatory variables, and “time series models,” which analyze demand based on past demand fluctuation patterns (Makridakis et al., 1998;Sanders, 2000;Honda, 2000). Demand for seasonally dependent products is said to be greatly affected by climate and weather, and the climate has seen dynamic changes beyond past levels in recent years. Consequently, when analyzing demand for seasonally dependent products, regression models that can reflect the changes in climate may be more accurate than time series models that rely solely on past demand fluctuation patterns. However, it should be noted that the characteristics of climate vary depending on the region.
Therefore, this study takes the viewpoint that demand for seasonally dependent products is strongly influenced by climate/weather (e.g., air temperature, precipitation) at the time at which they are sold, rather than the fluctuation patterns of past demand (referred to as “seasonal dependency”). Meanwhile, we need to consider that climate/weather varies depending on the region (referred to as “regional characteristics of climate”), but the influence on demand does not change in different regions if their climate/weather conditions are the same, given the relatively homogeneous Japanese character (referred to as “regional homogeneity of demand”) in the demand analysis of seasonally dependent products in Japan.
Therefore, based on our previous study (Kurihara and Yamashita, 2013), we attempt to express not only the three features of the seasonally dependent product mentioned above (“seasonal dependency,” “regional characteristics of climate,” “regional homogeneity of demand”) as an inseparable product of mutually independent meteorological and regional factors, but also social custom factors, policy factors, and trend factors as a new sum. Then, we attempt to apply the alternating least squares method to estimate the parameters, which are mixed forms of an inseparable product and a sum.
This study focuses on Japanese beer as a typical example of a seasonally dependent product and attempts to build a new weighted additive model for demand analysis using meteorological and regional data with nationwide demand as the whole demand employing a twostage approach.
Through empirical analysis based on beer demand and meteorological data, this study confirms the validity and effectiveness of the proposed model. For that reason, this study uses monthly data from the Family Income and Expenditure Survey (Statistics Bureau, Japan, 20102014), as well as air temperature and precipitation (JMA, 2010 2014), which are open data.
2. LITERATURE OVERVIEW
In general, demand analysis models are roughly categorized into regression models and time series models, as described in the previous section. The former regression models include single regression analysis and multiple regression analysis (Draper and Smith, 1981). The latter time series models include the TCSI decomposition method (moving average, CENSUS method (Shiskin, 1961), etc.), exponential smoothing (Holtz method, HoltzWinter method (Winters, 1960), etc.), and the autoregressive model (ARMA model (Box and Jenkins, 1976), ARIMA model, etc.).
Industries in which climate and weather affect demand include manufacturing industries, such as apparel, food, and beverage; energy industries, such as electric power and gas; and service industries, such as tourism and leisure.
A number of previous studies on demand analysis/ forecast in these industries has been conducted based on the aforementioned demand analysis methods. For example, in the field of electric power, there are studies by Apadula et al. (2012) and Mirasgedis et al. (2006). In the field of tourism, there are many studies, including those by Liang (2014) and ÁlvarezDíaz and Rosselló Nadal (2010). Here, we will focus on previous studies on beer, which is the target for this study, and food and beverage including beer. We will also compare these previous studies with this study.
First, in the field of study of demand analysis for food and beverage, Arunraj and Ahrenz (2015) proposed a mixed model of time series analysis and regression analysis, and Mirasgedis et al. (2014) proposed a multiple regression analysis model. However, neither of these models consider regional characteristics of climate because they employ meteorological factors from only one region to indicate the climate characteristics of multiple regions. In contrast, our proposed model uses multiple regression analysis, which can reflect the climate characteristics in multiple regions, based on the regional homogeneity of demand.
Furthermore, in the field of beer, Kros and Keller (2010) and Koksalan et al. (1999) proposed models of type I quantification analysis (Hayashi, 1952) that analyze seasonal fluctuations using monthly dummy variables and a kind of regression model. Nojyu of Kirin Brewery (1998) proposed a TCSI decomposition method model using a monthly composition ratio (= demand for a month / demand for the year) as a seasonal index, and Lenten and Moosa (1999) proposed an autoregressive model, both of which are time series models. In other words, all of the above models are developed for analysis based on historical demand fluctuation patterns. In contrast, the proposed model is a multiple regression analysis based on meteorological factors considered to greatly influence demand fluctuation as explanatory variables. Therefore, even if the demand fluctuation greatly deviates from the past fluctuation pattern, there is still a possibility that it may become a more accurate demand analysis, provided it can capture the relationship between demand and meteorological factors.
Bratina and Faganel (2008) proposed a mixed model of time series analysis and regression analysis that also took into account meteorological factors (temperature); however, this model cannot consider regional climate differences, such as in Mirasgedis’s et al. (2014). In addition, while the Beer Industry Electronic Commerce Coalition (BIECC) (2009) mostly focused on the analysis of demand influencing factors and the approach to demand analysis, this study primarily focuses on the analysis of those factors and the development of a demand analysis model based on them.
We have conducted previous studies on the demand analysis of seasonally dependent products as follows. First, in the previous study (Kurihara, 2012), to extract the product characteristics from the aspect of demand analysis, we classify the products using two axes: degree of seasonal fluctuation (seasonality/nonseasonality) and degree of product durability (durables/nondurables), as shown in Figure 1. In that study, the demand characteristics of each product are observed from the viewpoint of purchasing frequency, and seasonally dependent products are divided into “seasonal durables” and “seasonal nondurables.”
That is, in the former, demand in one period interferes with demand in the following periods because the product purchase cycle is longer, whereas in the latter, demand in one period does not interfere with demand in the following periods because the product is consumed quickly after it is purchased.
The previous study (Kurihara and Yamashita, 2012) chose home air conditioners as a typical example of seasonal durables and focused on meteorological factors such as monthly mean temperature and monthly total precipitation as demand influencing factors. Furthermore, based on data from the Japan Meteorological Agency (20102014), the value of each meteorological element was decomposed into normal (mean over the past 30 years) and deviation from the normal (anomaly = the actual value for the corresponding month – the normal). The influence on the demand fluctuation caused by the change in meteorological elements was analyzed by dividing it into the normal seasonal change and the abnormal fluctuation of the year (such as a hotter or colder summer than usual).
In addition as indicated by our previous study and described in the previous section, the demand for a seasonally dependent product has characteristics that include “seasonal dependency,” “regional characteristics of climate,” and “regional homogeneity of demand,” that is, the climate/weather, which has a strong effect on the demand, varies depending on the region, but if the climate/ weather is the same, the influence on the demand is the same in different regions. This means that although the climate/weather itself may vary depending on the region, the meteorological factors are independent factors, different from the regional factors (regional weights) that affect the nationwide demand. Therefore, based on these considerations, that study proposed “a weighted additive model for the whole demand analysis” by combining meteorological factors and regional weights that are mutually independent. Then, the alternating least squares method was used to estimate the parameters, taking the form of an inseparable product. This alternating least squares method was proposed as a parameter estimation method for the individual differences additive model in the field of psychology by Takane et al. (1980) and was applied to parameter estimation for the personnel rating model in the field of personnel management by Yamashita (2000). However, it has rarely been applied in the field of demand analysis targeted by this study.
As a method to analyze the whole (nationwide) demand, focusing only on regional characteristics of climate, a method of accumulating the estimated regional demand after analyzing the relationship between regional demand and meteorological factors by region, using normal multiple regression analysis, is often used. However, in such a method, different regions will have different impacts on the demand, even if the conditions of climate/weather are the same. This method thus cannot consider the “regional homogeneity of demand,” defined as the case where “the influence on the demand is the same in different regions under the same climate/weather conditions.”
Based on the discussion above, in this study, we change the target from seasonal durables to seasonal nondurables, choose beer as a typical example of seasonal nondurables, and attempt to develop an analysis model with the nationwide demand as the whole demand, employing a twostage approach. Then, in addition to considering the characteristics of seasonal nondurables, regional characteristics of climate, and regional homogeneity of demand based on our previous studies (Kurihara, 2012;Kurihara and Yamashita, 2012), this study incorporates social customs factors (e.g., a yearend gift called “Seibo”), policy factors (e.g., a tax increase), and a trend in demand fluctuations to further improve the accuracy. Then, the parameters in the mixed form of an inseparable product and sum are estimated using the alternating least squares method.
3. BEER DEMAND AND ITS FACTORS IN JAPAN
3.1 Climate of Japan
From the perspective of climate as a habitable environment, the regional climate is categorized based on the annual mean air temperature, its monthly mean, the annual mean precipitation, and its monthly mean (Tamiya, 1995), as shown in Köppen’s climate division (Kottek et al., 2006). Among them, the climate of Japan is classified as a “humid subtropical climate,” where the four seasons are clear and the summer is hot and humid, except for Hokkaido and some of Okinawa. Areas with humid subtropical climates are primarily distributed in the middle latitudes, especially on the east coast of continents, such as the southeastern United States of America and eastern Australia.
However, because Japan is slender in the north and south, the climate can be quite different, depending on the region. Therefore, to grasp the difference in climate per region, the Japan Meteorological Agency broadly divides Japan into three regions: northern Japan, eastern Japan, and western Japan. Figure 2, based on a map of East Asia (TeikokuShoin, 2016), shows the location of Japan on the east coast of the Eurasian Continent, and the locations of northern Japan, eastern Japan, and western Japan as denoted by the Japan Meteorological Agency.
3.2 Alcoholic Beverage Consumption and Related Social Customs and Policies in Japan
The demand for many beverages, including alcoholic beverages，is affected by climate. Among them, the alcoholic beverage most familiar in Japanese families is beer. Based on the Liquor Tax Act, beertype beverages are divided as follows, according to the malt usage ratio and material type (Sapporo breweries, 2016).

beer (malt ratio ≧ 2/3)

lowmalt beerlike beverage called “Happoshu” (2/3 > malt ratio> 0)

Other effervescent brewed alcoholic beverages/ liqueurs (socalled “third beer” using cereals, saccharides, and other ingredients instead of malt)
The reason why “Happoshu” and “third beer,” as alternatives to beer, exist is that the beer tax in Japan is high (4th highest among 30 OECD member countries as of 2014), and market competition can be intense. As a result, beer manufacturers have devised a malt ratio with certain raw materials, developing “Happoshu” and “third beer” as beerlike beverages to avoid paying a high tax (Nihon keizai shinbun, 2016a).
Table 1 shows the alcoholic beverage consumption per month per household and its ratio in 2014, calculated based on Family Income and Expenditure Survey Data (Statistics Bureau, Japan, 20102014).
As shown in Table 1, beertype beverages, including beer, “Happoshu,” and “third beer,” account for approximately half (49.8%) of the total alcoholic beverage consumption in Japan
Factors influencing the demand for these beertype beverages include social customs factors (determining when they are purchased), policy factors that affect product prices, trends in demand fluctuations, and economic factors that may be considered, in addition to the meteorological factors mentioned above. Regarding the social customs in Japan, there are “Chugen (midyear gift)” in July and “Seibo (yearend gift)” in December, both of which are given to those who have helped an individual, either personally or professionally. In addition, there are New Year’s preparation and Christmas in December as well. The recent policy that affected product prices is the tax system change in April 2014, when the consumption tax was increased from 5% to 8%.
As a typical example of seasonal nondurables, in this study, we choose beer, which is the most frequently consumed among beertype beverages, focus on its household demand, and then examine factors that affect the demand.
3.3 Beer Consumption in Japan and its Factors
First, this study focuses on the largest consumption area, the Kanto district, and looks at the trend of consumption per household per month from January 2010 to December 2014, together with Tokyo’s monthly mean temperature, as shown in Figure 3. The Kanto district is located in the eastern part of eastern Japan, and Tokyo is located in the southern part of the Kanto district.
In Figure 3, when looking at the overall fluctuation pattern, it is clear that the fluctuation pattern of demand is similar to that of the temperature, except in December, during the lowtemperature period. This suggests that meteorological factors have a large influence on beer demand. In addition, as an overall trend, beer demand has been decreasing.
Then, when looking at the demand peaks individually, we can see that there are peaks in July to August and December every year, and a slightly smaller peak in March 2014. First, July to August are hot summer months, and the climate can be considered the cause of the demand peak. More precisely, it turns out that despite the fact that the peak temperature is in August, the demand peaks for the summers of 2010, 2011, and 2013 are in July. This suggests that there are factors other than meteorological factors, such as “Chugen,” as a demand influencing factor in July. Next, in December, we can see that the demand peaks, despite the low temperature. This suggests that December is the month for not only “Seibo” but also New Year’s preparation and Christmas, and these social customs have contributed to a demand peak. Furthermore, we can consider that the single peak in March 2014 is a temporary rush demand that occurred before the consumption tax increase in the following month, and the policy factor associated with the tax system change led to this peak.
Based on the above results, this study examines an analysis model of beer demand that includes social customs factors, policy factors, demand trends, and economic factors, in addition to meteorological factors. However, because beer is a seasonal nondurable, it is important to note that meteorological factors affecting monthly demand are only for the month, unlike home air conditioners, which are seasonal durables.
3.4 Comparison between the United States and Japan Regarding the Demand Influencing Factors for Beer
Are the demand influencing factors for beer that this study has examined so far unique to Japan? Let us compare the demand influencing factors for Japan with those of the United States, the world’s secondlargest beerconsuming country as of 2012 (Kirin holdings, 2012). Table 2 compares the demand influencing factors for beer between the United States and Japan, based on BIECC analysis of the United States (2009).
There are differences in companyspecific parts (existence of event factors, contents of policy factors, etc.) because of different levels of beer demand (company or nationwide), and differences in the contents of social customs factors for the different countries. However, from Table 2, it can be seen that although the contents of the demand influencing factors are not the same, the existence of demand influencing factors such as meteorological and social customs factors is the same. This seems to suggest that even in countries other than Japan, this study can be applied to beer demand analysis, taking into consideration the similar demand influencing factors. In parThere are differences in companyspecific parts (existence of event factors, contents of policy factors, etc.) because of different levels of beer demand (company or nationwide), and differences in the contents of social customs factors for the different countries. However, from Table 2, it can be seen that although the contents of the demand influencing factors are not the same, the existence of demand influencing factors such as meteorological and social customs factors is the same. This seems to suggest that even in countries other than Japan, this study can be applied to beer demand analysis, taking into consideration the similar demand influencing factors. In particular, other countries with high beer consumption rates among alcoholic beverages in midlatitude zones (with similar climates to Japan), such as Australia, New Zealand, the United Kingdom, and Germany (Fogarty, 2008), can be candidates for applying this study.
Furthermore, even when expanding from beer to food and beverage, Agnew and Thornes (1995) pointed out the demand influencing factors in the UK retail and distribution industry included economic factors, political/ legal factors (policy factors), technical development, social trends, seasonal fluctuations, the holiday period (social customs factors), and weather changes. When comparing these factors with the demand influencing factors for beer, it is clear that the similarity is high, so it appears that this study is also applicable to foods and beverages with high seasonal dependency.
With respect to the nationwide household demand for beer in Japan, this study will investigate the influence of each factor (regional climate differences, social customs, policy, and so on) and the size of the fundamental demand, based on monthly data.
4. STUDY METHOD
4.1 Approach and Prerequisites
Taking nationwide demand as the whole demand for Japanese beer, which is a seasonal nondurable item, this study proposes the following twostage approach to develop a new weighted additive model for nationwide demand analysis with meteorological and regional data, taking into account social customs factors, policy factors, and so on.
(First stage) To build a base model for regional demand based on candidate factors of nationwide demand, to validate the factors based on the results of multiple regression analysis, and to determine factors to be used for analysis of nationwide demand.
(Second stage) To build a weighted additive model that considers meteorological factors and regional factors based on “seasonal dependency,” “regional characteristics of climate,” and “regional homogeneity of demand,” and newly consider social customs factors, policy factors, and so on for the nationwide demand, based on the factors extracted and validated in the first stage, and to estimate parameters in the mixed form of an inseparable product and a sum using the alternating least squares method.
The main prerequisites for formulating the problem in this study are as follows.

(1) The climate of the region is represented by the climate of the largest city in the region.

(2) The optimal value of the regional weight is in the vicinity of the ratio of household consumption in that region to the whole country.

(3) Social customs factors, policy factors, economic factors, and trends are nationwide and do not have regional characteristics.
4.2 Problem Formulation
In this study, we aim to develop a model that can structurally and concisely reproduce the relationship between the whole (nationwide) demand and demand factors. Therefore, in the first stage, focusing on the Kanto district, which is the largest consumer area as a microcosm of nationwide demand, to extract demand factors that can become candidates for nationwide demand, we build a base model based on our previous study (Kurihara and Yamashita, 2012) and examine the validity of candidate factors based on the results of multiple regression analysis. As an economic factor, we consider the “coincident indicator of composite indexes (CI),” which is a monthly indicator and can combine nine indicators of monthly production, shipment, sales, and labor (overtime hours and active job opening ratio) to comprehensively represent economic trends with one indicator.
For this case, the social customs factors for demand in July and December and policy factors for demand in March 2014 are expressed using dummy variables. Therefore, a base model with household beer consumption in the Kanto district as the objective variable, with meteorological factors, social customs factors, a policy factor, an economic factor, and trends as explanatory variables, can be expressed as a multiple regression model, as shown in Eq. (1).
where

s_{t}: household beer consumption per month per household in the Kanto district (yen),

t: month index,

i: factor index,

i = 1: normal of monthly mean air temperature (normal temperature),

i = 2: deviation from normal of monthly mean air temperature (temperature deviation),

i = 3: normal of monthly total precipitation (normal precipitation),

i = 4: deviation from normal of monthly total precipitation (precipitation deviation), (Above, i = 1~4 represents meteorological factors)

i = 5: social customs factor 1 (demand in December),

i = 6: social customs factor 2 (demand in July),

i = 7: policy factor (demand in March 2014),

i = 8: trend factor,

i = 9: economic factor,

a_{i}: parameter (partial regression coefficient),

a_{0}: constant term,

x_{1t}: normal temperature for month t (°C),

x_{2t}: temperature deviation for month t (°C) = monthly mean air temperature for month t–normal temperature for month t,

x_{3t}: normal precipitation for month t (mm),

x_{4t}: precipitation deviation for month t (mm) = monthly total precipitation for month t – normal precipitation for month t, (Here, observation values in Tokyo are used for x_{1t} to x_{4t}),

x_{5t}: dummy variable corresponding to demand in December (the value for December is 1; the values for other months are 0)

x_{6t}: dummy variable corresponding to demand in July (the value for July is 1; the values for other months are 0)

x_{7t}: dummy variable corresponding to demand in March 2014 (the value for March 2014 is 1; the values for other years and months are 0)

x_{8t}: Monthly number indicating the passage of years and months (x_{8t} = t),

x_{9t}: coincident indicator of composite indexes (CI) representing current economic situation (this is an index to grasp the magnitude and tempo of the economic fluctuation announced by the Cabinet Office every month, and it moves nearly in line with the economy)

e_{t}: residual term
To verify the validity of the model formula and factors based on Eq. (1), we conducted an empirical analysis with the following data from the previous five years (January 2010–December 2014).

(a) Household demand for beer data in the Kanto district: Household consumption data per month per household in the Kanto district (Statistics Bureau, Japan 20102014)

(b) Meteorological data in Tokyo: actual values of monthly mean air temperature and their normal values; actual values of monthly total precipitation and their normal values (JMA, 20102014)

(c) Economic data: coincident indicator of CI (Cabinet Office, Japan, 20102014)
The results of the analysis are shown in Table 3, where we can see that the model of Eq. (1) has a high multiple correlation coefficient of 0.956, which replicates regional demand in the Kanto district well overall. However, Table 3 also indicates that the Pvalue of the economic factor is as high as 0.379 compared with the Pvalues of other factors, and the economic factor cannot be said to be statistically significant. Therefore, in the second stage, this study excludes the economic factor from the demand influencing factors for beer from the viewpoint of certainty and simplicity. The selected factors are verified again from the viewpoint of the validity of the model in “5. Empirical Analysis.”
Next, in the second stage, to structurally analyze the beer demand at the nationwide level, it is necessary to quantitatively comprehend the relationship between the nationwide demand and factors while considering “regional characteristics of climate” and “regional homogeneity of demand” together.
Therefore, based on the demand analysis model for home air conditioners (Kurihara and Yamashita, 2012), this study introduces a regional weight indicating the magnitude of the influence of a region on the demand.
Then, paying attention to the second term in Eq. (1), by expressing the relationship between the regional weight (w_{k}) and the meteorological factors in the form of a mutually independent product, adding new nationwide factors (social customs factors, a policy factor, and trends) in the form of a sum, and formulating the relationship between the nationwide demand and factors, an analysis model of the nationwide demand for beer can be expressed as Eq. (2).
where

y_{t}: nationwide household beer consumption per month per household (yen),

k: region code (K : number of region codes),

w_{k}: regional weight,

x_{1tk}: normal temperature in region k for month t (°C),

x_{2tk}:temperature deviation in region k for month t (°C),

x_{3tk}: normal precipitation in region k for month t (mm),

x_{4tk}: precipitation deviation in region k for month t (mm)
The symbols a_{0} to a_{8} and x_{5t} to x_{8t}, other than those above, are the same as in Eq. (1).
Eq. (2) has two kinds of parameter groups: a meteorological parameter, a_{i}, and a regional weight, w_{k}, so that the nationwide factors and the regional factors can be considered together. This makes it possible to quantitatively comprehend not only the effects of climate/weather, social customs, policies, and trends (parameter a_{i}) on the nationwide household beer consumption (y_{t}), but also its regional influences (regional weight w_{k}). Based on the discussion above, Table 4 summarizes the model proposed in this study from the viewpoint of demand influencing factors and compares it with the model used in the previous study (Kurihara and Yamashita, 2012).
4.3 Estimation Method for Parameters
The proposed model of this study has “double degrees of freedom” (Takane, 1976) because it has the above two types of parameter groups (parameter, a_{i}, of meteorological factors that does not have the subscript of region code k and region weight, w_{k} that does not have the subscript of factor index i). Therefore, the question here is how to eliminate the double degrees of freedom of the model and estimate the least square solutions. Two types of parameter groups can be expressed as a “product,” which makes them “inseparable” (Takane, 1976), and cannot be estimated at one time by the least squares solutions. Therefore, we introduce the algorithm of the alternating least squares method (Takane et al., 1980) for such an inseparable parameter group and estimate the parameter a_{i} and the regional weight w_{k} using the following procedure.
Unlike the demand analysis model (Kurihara and Yamashita, 2012), in the model proposed in this study, the relationship between the whole demand and the factors is expressed in the mixed form of a product (a meteorological parameter and a regional weight) and a sum (social customs factors, a policy factor, and trends). Therefore, the feature of this study is that it corresponds to a mixed form by estimation of the regional weight, w_{k}, in step 3.
[Step 1] Initial value setting for regional weight, w_{k}
First, the household consumption ratio for each region relative to the whole country, which seems to be close to the optimum value, is calculated based on the Family Income and Expenditure Survey data (Statistic Bureau, Japan, 20102014) from January 2010 to December 2014. Then, the value is set as the initial value of the region weight w_{k} (${\sum}_{k=1}^{K}{w}_{k}=1$).
[Step 2] Estimation of parameter, a_{i}
For i = 1 to 4, u_{it} is calculated by Eq. (3), using the regional weight, w_{k} (the initial value of Step 1 or the estimated value normalized in Step 3).
Furthermore, with u_{it} = x_{it} for i = 5 to 8, Eq. (2) is transformed as follows.
Here, if the parameter vector with a_{i} as its element is a, and the objective variable vector with y_{t} (nationwide household beer consumption) as its element is y, and the explanatory variable matrix with u_{it} as its element is U, and the residual vector with e_{t} (residual term) as its element is e, Eq. (4) can be rewritten as follows
In this case, the least squares solution of the parameter vector a is given by the following normal equation.
[Step 3] Estimation of regional weight, w_{k}
For i = 1 to 4, v_{tk} is calculated by Eq. (7) using the parameter a_{i} for the meteorological factors estimated in Step 2.
Here, if the residual term is ε_{t}, Eq. (2) is transformed as follows.
Here, if ${z}_{t}={y}_{t}{a}_{0}{\displaystyle {\sum}_{i=5}^{8}{a}_{i}{x}_{it}}$, Eq. (8) can be rewritten as shown in Eq. (9).
Similar to Step 2, if the parameter vector of w_{k} as its element is w, and the objective variable vector with z_{t} as its element is z, and the explanatory variable matrix with v_{tk} as its element is V, and the residual vector with ε_{t} (residual term) as its element is ε, Eq. (9) can be rewritten as follows.
In this case, the least squares solution of the parameter vector w is given by the following normal equation.
The parameter vector w can be estimated using the normal equation of Eq. (11), and the estimated w_{k} is standardized so that ${\sum}_{k=1}^{K}{w}_{k}=1$.
[Step 4] Confirmation of the convergence condition
If the parameter vectors a and w respectively estimated in Step 2 and Step 3 satisfy the predetermined convergence condition, the process proceeds to Step 5; if not, the process returns to Step 2. Here, the predetermined convergence condition means that the value of the multiple correlation coefficient R is barely improved, that is, the difference between the previously calculated R and the currently calculated R is a minute value (not greater than 0.00001).
[Step 5] Calculation of the multiple correlation coefficient
After regression statistics for the model are calculated using a_{i} and w_{k}, estimated by repeating Step 2 and Step 3, the procedure is terminated.
5. EMPIRICAL ANALYSIS
5.1 Specification of the Models to be Compared with the Proposed Model
We compared the proposed model with the following two models to verify the validity and effectiveness of the proposed model based on multiple regression analysis.

(a) Meteorological factor model (weighted additive model using only meteorological and regional data): As with our previous study (Kurihara and Yamashita, 2012), this model is a weighted additive model by which meteorological and regional data take into account regional characteristics of climate and regional homogeneity of demand but do not consider factors other than meteorological and regional ones. By comparison with this model, the effectiveness of the added factors (social customs factors, policy factors, and trends) in the proposed model is verified. This model is based on Eq. (2) without its third term.

(b) 1region representative model (traditional multiple regression model): This model is a traditional multiple regression analysis model that employs meteorological factors from one region as a climate characteristic for multiple regions and takes into account the social customs factors, policy factors, and trends, as well as the proposed model. By comparison with this model, the effectiveness of considering regional characteristics of climate and regional homogeneity of demand is verified. This model is based on Eq. (1), in which the objective variable is changed to nationwide demand, the climate of one region (Tokyo) is represented as a meteorological factor, and the economic factor is removed.
The results of a comparison of the specifications of the above three models are shown in Table 5.
Here, based on the classification of the Japan Meteorological Agency, as regions that take into consideration the climate characteristics, the three regions of northern Japan, eastern Japan, and western Japan are selected, as shown in Figure 2. In addition, this study selects the largest city (Sapporo, Tokyo, Osaka) in the region and uses the climate of that city as the climate representing that region. Thus, the region k is as follows.
5.2 Description of Data
In the empirical analysis using the proposed model, the following data for five years (January 2010 to December 2014) is used.

(a) Nationwide demand data for beer: Nationwide household consumption data for beer per month per household from the Family Income and Expenditure Survey (Statistics Bureau, Japan, 2010 2014)

(b) Weather data in Tokyo, Sapporo, and Osaka: Actual values of monthly mean temperature and their normal values, and actual values of monthly total precipitation and their normal values (JMA, 20102014)
As an initial value of regional weight, w_{k}, based on the Family Income and Expenditure Survey data (Statistics Bureau, Japan, 20102014) from January 2010 to December 2014, the household consumption ratio for each region relative to the whole country is calculated as follows.
5.3 Results and Discussion
Based on the above data, an outline of the procedures for estimating parameters is shown in Figure 4. The results of the analysis are shown in Tables 6 and 7.
5.3.1 Comparison among the Proposed Model, the Meteorological Factor Model, and the 1Region Representative Model
Regarding the regression statistics, Table 6 shows the results of a comparison of the proposed model, the meteorological factor model, and the 1region representative model. In Table 6, it can be seen that both multiple correlation coefficients were improved with the proposed model, indicating accuracy, and AIC, indicating the degree of good balance between accuracy and simplicity, was also improved compared with the meteorological factor model and the 1region representative model. Furthermore, the value of the multiple correlation coefficient itself is 0.965, which indicates that the proposed model is a highly accurate demand analysis model.
Furthermore, the variance ratio (F) calculated based on the variance of the residual and the variance of regression in Table 6, the number of explanatory variables (11), and the sample size (60), is 58.5, which is much higher than the F value (2.64) of the F distribution at the significance level of 1%. Furthermore, the null hypothesis (the model equation is not effective for estimating the objective variable) can be rejected at the significance level of 1%, suggesting the significance of the proposed model equation from the viewpoint of analysis of variance.
Based on these facts, it is suggested that this weighted additive model using meteorological and regional data has not only validity but also higher analysis accuracy, accomplished by considering “regional characteristics of climate” and “regional homogeneity of demand” together and newly considering factors (social customs factors, policy factors, and trends) selected in “4.2 Problem formulation.”
5.3.2 Fundamental Demand for Beer and its Trend
Based on the parameter values shown in Table 7, the following findings were obtained. First, the fundamental demand for beer excluding the influence of all factors was 897 yen (nationwide consumption per month per household) as of January 2010, but decreased at a pace of approximately 5 yen/month. This confirms the declining trend in beer demand, which is said to be caused by the youth’s shift away from beer, declining birthrate, and an aging population (Nihon Keizai Shinbun, 2016b). Although beer is still the alcoholic beverage that is most consumed at home, this result suggests that beer has already moved from the maturity stage to the decline stage in the product life cycle.
5.3.3 Meteorological Factors
Next, when comparing the standard partial regression coefficients, showing the influence degree of demand factors as absolute values, it is understood that normal temperature and normal precipitation are larger, in this order. This confirms that climate is the biggest factor in beer demand, and beer is a typical “seasonally dependent product.” On the contrary, temperature deviation and precipitation deviation are the smallest factors, in that order. These findings suggest an interesting result: beer demand is greatly influenced by seasonal change expressed by “normal,” while the influence of weather fluctuation within the season represented by “anomaly” (deviation from normal) is small. This also seems to suggest the possibility of analyzing beer demand easily without lowering accuracy considerably, even if only normal values (normal temperature and normal precipitation) are used as meteorological data.
5.3.4 Social Customs Factors and Policy Factor
It is understood that the custom factor in December, including “Seibo,” New Year's preparations, and Christmas, has the second greatest influence, after the normal values of meteorological factors, according to the standard partial regression coefficients. This suggests that beer is one of the nondurables representing yearend sales warfare in Japan and is a typical alcoholic beverages consumed at home, as it is warm in the room even in winter. In contrast with this, the standard partial regression coefficient of the custom factor in July (“Chugen”) is 0.148, approximately 1/5 of that in December (“Seibo”). Certainly, beer demand in December includes the demand for not only “Seibo,” but also New Year's preparations and Christmas. This also seems to suggest that “Seibo” is more important than “Chugen” as a giftgiving custom (Daimaru Matsuzakaya Department Stores, 2016).
It is also interesting to see that the standard partial regression coefficient for the policy factor related to the temporary rush demand in March 2014 (before the tax increase) is not as large as that for the custom factor in December. This seems to be a result of the fact that the nondurables consumed on a daily basis were purchased immediately before the tax increase according to the magnitude of the tax increase because of the characteristics of the product.
Furthermore, it is also a feature of this model that the impact of an increase in the consumption tax scheduled in the future (October 2019) on the household demand for beer can be roughly estimated from the parameter of the current policy factor.
5.3.5 Regional Weights
When comparing the estimated values for the regional weights with their initial values, it is clear that the value in western Japan is slightly larger (+0.017), whereas the value in eastern Japan is slightly smaller (0.019), than the initial value. This also suggests that the demand for beer is more affected by a hotter region (western Japan) in regions with high consumption because the demand is greatly influenced by climate.
6. CONCLUSION
In this study, we attempted to develop a demand analysis model that can structurally and concisely reproduce the relationship between the whole demand and factors for a seasonally dependent product, based on the characteristics of “seasonal dependency,” “regional characteristics of climate,” and “regional homogeneity of demand.” Taking Japanese beer as an example of seasonal nondurable goods, we proposed a new weighted additive model for beer demand analysis using meteorological and regional data, considering social customs factors, a policy factor, and the trend, taking nationwide demand as the whole demand. We also proposed a twostage approach to build the model and a procedure for deriving a solution of the model employing the alternating least squares method. The validity of the proposed model can be confirmed by the results of the empirical analysis. In addition, from the estimated parameter and weight values, suggestions for a simple analysis model for beer demand and findings related to the impact on beer demand of social customs, etc., were obtained.
In the future, if demand forecasting models can be developed based on the model proposed in this study, and forecasting of demand with higher accuracy becomes possible, we can expect to not only make timely and effective sales and operation planning (S&OP) but also prepare for production and procurement, sufficiently and efficiently, even with a large demand fluctuation in seasonally dependent products.
However, in such demand forecasting models, it is necessary to reproduce the relationship between demand and factors more accurately. To that end, it will be a challenge to review factors (including economic factors), review indicators corresponding to factors, and improve the models using the reviewed factors, in training–validation– testing. In this study, we proposed a model that can serve as a basis for such demand forecasting models.
Furthermore, it seems that the approach used in this study can be applied not only to Japanese beer, but also to beer consumed in other countries, and even to other seasonal nondurables.
In the case of Japanese beer, this study suggests that it is possible to analyze demand concisely without lowering accuracy considerably, even if only normal values (past mean values) are used. However, in general, to forecast the demand for seasonally dependent products with higher accuracy using a regression model, highly accurate meteorological forecast data, including not only the “normal” but also “anomaly,” would be required. Therefore, formulating how to incorporate weather forecast data into the model will be one of the important issues to consider when applying this study to other seasonally dependent products in the future.