Purchase Preference Analaysis and Trend Forecast
-- A Fuzzy Utility Function Approach to Data Mining

I.    Brief Background on Consumption Theory

• Consumer Behaviors and Preference
One consumer would in general have different consumption behaviors or preferences from another.  He may spend money on computers and technical books, while the other may spend on clothing and food. Availability of this information on consumer preference will be of great value to a marketing company, a bank, or a credit card company that can use this information to target different groups of consumer for improved response rate or profit.  By the same token, information on consumption preference of the residents in one specific region can help businesses in planning their operations in this region for improved profit.   Therefore, it is very important to have a tool that can help analyze consumers behaviors and forecast the changes in purhcase patterns and changes in purchase trend.

• Fuzzy Consumption Utility Functions-based Utility Theory
In studying advanced methodology for consumption behaviors, AI researchers at Zaptron Systems have developed the so called fuzzy utility functions that can model and describe the consumption behaviors of a target consumer group.

• Consumption Utility - it is a criterion (or index) used to evaluate the effectiveness of customers consumption. A low value of consumption utility, say 0.15,  indicates that a customer is not satisfied with the consumption of a certain commodity; while high value, say 0.96, indicates that the customer is very satisfied.  There are formal theories on utility, including ordinal utility, cardinal utility and marginal utility.

• Consumption utility function - The behavioral characteristics of human beings can be represented by the concept of consumption utility, and consumption utility function is the mathematical description of this concept.  In addition, human consumption behaviors are determined by the following two types of factors:

• objective factors - the physical, chemical, biological and artistic properties of goods;

• subjective factors - consumer's interest, preference and psychological state.

II.    Fuzzy Utility Function for Consumption

• Fuzzy Set Theoretical Approach -- In fact, consumption utility is a fuzzy concept. To model the above subjective factors, fuzzy set theory is used to describe different levels of consumers satisfaction with respect to various consumption plans (spending patterns), such as "not satisfied," "somehow satisfied," "very satisfied," and etc.  Mathematically, the fuzzy utility function is a more  accurate measure on the consumption utility.  It can describe the relationships among spending, price, consumption composition (decomposition), preference and subjective measure on commodity or service values.

• Mathematical model of fuzzy consumption functions - to model the consumption of one commodity, a fuzzy function Ui must satisfy the following three conditions (Xi is the spending on i-th commodity):

1. Ui:      Xi --> Ui

2. Ui --> 1 if Xi is infinity,  Ui --> 0 if Xi is 0

3. The first derivative of Ui w.r.t. Xi must be positive,and the second derivative negative.

One commonly used Ui has the following Fi(.) form (subscript i is for the i-th commodity or service):

Ui = Fi(Di, Pi, Si, C)

For N commodities or services, the total fuzzy utility function U is expressed as a weighted sum of utility function Ui for individual category, i.e.,

U = W1*U1 + W2*U2 + W3*U3 + ..... + WN*UN
= W1*F1 + W2*F2  + W3*F3 + ..... + WN*FN

where

• U - Total utility function for all possible consumption categories. It is an indicator of consumer's level of satisfaction, with U =1 for maximum satisfaction and U = 0 for non satisfaction. 0.0 < U < 1.0.

• Ui or Fi - The utility function for i-th consumption category, with Ui =1 for maximum satisfaction and Ui = 0 for non satisfaction. 0.0 < Ui < 1.0.

• Di - a parameter related to customer's subjective evaluation on consumption (depreciation)

• Qi  = Di*Pi, appreciation (not "satisfaction"), consumer's subjective measure on the value of a consumption

• Pi - price of the i-th commodity or service

• Si - percentage of spending on i-th commodity, 0 < Si < 1 and Si = Ai + Bi/C

• Ai - Limit spending percentage of the i-th commodity in total spending when average personal income level  increases so greatly that one could afford to spend to "limit" (as much as he or she desires) with unlimited financial resources. We always have 0.0 < Ai < 1.0.

• Bi - The trend of change in Ai due to change in average personal income level. Note the relationship between Ai an Bi as follows:
Bi > 0 if Ai decreases, and
Bi < 0 if Ai increases.

• C - total spending amount, in dollars, on all N commodities or services

• Wi - a weighting factor representing a consumer's preference to the i-th commodity (0<Wi<1)

• Computation of Parameters {Di, Qi, Si, Ai, Bi, Wi} - Based on the Maximum Utility Principle, they can be computed by solving a set of complex mathematical equations.  Zaptron's DataX™ software suite has special modules that compute them.

III.    A Case Study using DataX™

• Problem Background - The annual data of the average spending amount (in dollars) on 5 categories by customers in a remote rural area (with severe electric power shortage, indicated by the smaller spending amount in the "Energy" column of Table-1) are available (see Table-1 below).  Table-2 shows the parameters computed using the Zaptron model.

Table-1 Annual average consumption data for 5 categories (unit: 100 US\$):

 Year Food Clothing Energy Housing Supplies Entertainment 1980 171.9 37.2 26.2 19.6 29.8 5 1981 175 39.3 26.8 100 35 5.2 1982 155.7 36.3 24.1 49.7 37.9 4.2 1983 188.1 43.4 28.4 43.2 44.9 6.6 1984 220.3 40.2 27.5 70.8 56.9 7.2

Table-2 Parameters computed using fuzzy consumption function:

 Consumption Category Limit Spending Ai (%) Spending Trend Bi Preference Wi Appreciation Qi Food 36.8 52.081 0.149 1086.84 Clothing 2.90 29.169 0.385 84.94 Energy 1.80 20.329 0.383 52.68 Housing 41.4 -88.76 0.041 1221.63 Supplies 15.4 -13.09 0.026 453.45 Entertainment 1.70 -0.454 0.017 51.20
• Explanation of the Result -
By studying Table-1 and Table-2, we can make the following observations:

(1) Limit Spending Percentage (parameter Ai in %) has the largest value for "housing" consumption, and smallest value for "energy" consumption.  These Ai values are correct because people in general would always like to live in a better house (or to continue spending on home improvement), and do not spend extra money on energy, when their incomes permit (with unlimited income).

(2) Consumption Preference (parameter Wi) for consumers in this rural area (with severe power shortage) is from high to low for clothing (Wi = 0.385), energy (Wi = 0.383), food (Wi = 0.149), housing (Wi = 0.041), supplies (Wi = 0.026) and entertainment (Wi = 0.017). These numbers show that people most prefer buying new clothing (Wi = 0.385). They also show a high Wi value (Wi = 0.383) for energy, This means that people in this rural area have to pay high attention (strong preference) to energy consumption to ensure the basic human needs (for instance, they need to use homemade firewood to cook food daily), given the natural condition of severe power shortage in the area.  Naturally, preference for food (0.149) and housing (0.041) is higher than that for supplies (0.026) and entertainment (0.017). The preference for entertainment is the lowest with Wi = 0.017.

(3) Spending Trend (parameter Bi) - Parameter values for Bi show the positive or negative changes (spending trend) in Limit Spending Percentage (parameter Ai). Positive Bi values, such as those for food, clothing and energy of Table-2, indicate a decreasing trend in limit spending percentage on those categories.  Negative values, such as those for housing, supplies and entertainment of Table-2, indicate an increasing trend in limit spending percentage on those items.  In other words, with unlimited personal incomes, people would spend more (to limit) on housing, supplies and entertainment, and less on food, clothing and energy.  This makes sense to most people in most areas of the world.

(4) Appreciation (parameter Qi) -   these values show how the people evaluate these 5 types of  consumption, and large numbers indicate higher value. Table-1 show that Appreciation (Qi) values are in the same order as Limit Spending Percentage (Ai) values, and this result makes sense and is expected.

IV.  Consumption Trend Forecast

The method can be used in combination with the time series analyais method to predict the purchase trend and customer's satisfaction level. Here tTwo examples are given to show the forecast result of (1) purchase trend and (2) consumer's satisfaction level.

• 4.1 Forecast on Future Consumption Trend
We used the data for the years from 1980 to 1984 given in Table-2 above to forecast the consumption trend using the Time Series Analysis Tool of DataX.  In forecasting, it is assumed that the annual spending rate increases by 8% and population increases by 0.9%. The forecast results so obtained for the years 1985, 1990, 1995 and 2000 are listed in Table-3. The following observations can be made from Table-3 and are consistent with the Engles Principle in economics.

1) The Limit Spending Percentage (Ai) is largest for housing, increasing from 22% in 1985 to 35% in 2000.
2) The Limit Spending Percentage (Ai) is smallest for food, decreasing from 48.4% in 1985 to 40% in 2000.
3) The percentage on clothing will slightly decrease from 1985 to 2000
4) The percentage on supplies and entertainment will experience slow increase.

Table-3 Forecast on Limit Spending Percentage (Ai parameter in %)

 Category 1985 1990 1995 2000 Food 48.4 44.7 42.1 40.5 Clothing 9.2 7.2 5.8 4.9 Energy 6.2 4.8 3.8 3.2 Housing 21.9 28.2 32.4 35.3 Supplies 12.5 13.4 14 14.5 Entertainment 1.64 1.67 1.69 17
• 4.2 Forecast on Consumer's Satisfaction Level
Utility functions discussed above present the consumer's satisfaction level on individual and combined consumption. They can be used to forecast the customer satisfaction levels on various consumption categories.  Table-4 gives the results computed by using DataX.. The table shows that the satisfaction level for housing is lowest (U4 = 0.0788), therefore the demand on housing (living) will be high - more than double every 5 years from 1985 to 2000.

Table-4 Forecast on Consumption Utility (Consumer Satisfaction Level Ui)

 Utility Function 1985 1990 1995 2000 Total  Utotal 0.1477 0.2074 0.2877 0.3911 Food U1 0.184 0.2412 0.318 0.417 Clothing U2 0.3924 0.435 0.4922 0.5659 Energy U3 0.4177 0.4585 0.5133 0.584 Housing U4 0.0788 0.144 0.2301 0.3419 Supplies U5 0.1183 0.18 0.2631 0.3701 Entertainment U6 0.1358 0.1963 0.2777 0.3825

V.  Conclusions

This artical introduces a new data mining method for purchase preference analysis and trend forecast. A fuzzy set theoretical definition of consumption utility function is introduced, computation of model pareamaters is discussed. Examples are given to show the efficacy of this innovative approach to data mining in business and finance.