Web Housing Expert Systems Using New Interval Type-2

Web Housing Expert Systems Using New Interval Type-2 Fuzzy Reasoning Ling Gu Thesis adviser: Dr. Yanqing Zhang

Outline § Introduction Ø Background, Motivation, Contributions § Fuzzy logic theory Ø Fuzzy sets, Type-1, Type-2, Interval type-2, Upper-lower method § System optimizations Ø Structure and parameter optimization, Least square method § Web Housing expert Ø Consideration, Implementation, Examples

Background (1) --- Decision support systems v A decision support system is a program that aids in making decisions and involves queries and constraints that need to be satisfied. v Fuzzy decision support systems, unlike classical logical systems (e. g. expert systems), aim at modeling the imprecise modes of reasoning to make rational decisions in an environment of uncertainty and imprecision [1].

Background (2) --- Fuzzy logic systems § Fuzzy logic system (FLS) provides a very useful solution for handling vague, ambiguous, and uncertain characteristics of information. § FLS is based on fuzzy set theory which is primarily formulated and developed by Zadeh [4]. § Two kinds of fuzzy sets: Type-1 and Type-2 defined by the membership functions are used in FLS.

Background (3) --- Related works § Wang et al. used a table-lookup scheme to generate fuzzy rules directly from numerical examples and proved that this fuzzy inference system is a universal approximation [11]. § Nozaki et al. presented a heuristic method for generating Takagi. Sugeno fuzzy rules from numerical data [10]. § Grauel et al. investigated the connection between the shape of transfer functions and the shape of membership functions [12]. § Klawonn applied to construct a fuzzy controller from the training data [13]. § Yager proposed the first decision support system using type-2 fuzzy sets [18]. Decision support systems have been applied in various fields including medical diagnosis, business, military, industry, traffic control, and science research.

Motivations Example: For a dream house, he/she might have various criteria or constraints such that the house is not too far from a highway exit, within 15 -minute driving distance from his/her work place, which costs around $300, 000, the house is in a good location, the house has a big backyard, and so on. Furthermore, good location has different meaning for different people. Some people would like downtown areas, some people would like a good school district, and some people would like quiet places To develop a fuzzy decision support system which has the capability of handling multi-criteria and constraints and dealing with vague and imprecise information.

Contributions § Applied the interval type-2 fuzzy set into the decision support system to represent the high level uncertainty. § Developed a upper-lower method to handle the complex calculations in the interval type-2 FLS. § Introduced the least square method to optimize the parameters of membership functions to obtain a better performance.

Fuzzy Sets A fuzzy set A of a set X is defined as a set of ordered pairs, each with the first element from X and the second element from an interval of [0, 1]: A: X [0, 1]. This defines a mapping, A, between elements of the set X and values in the interval [0, 1]. The value zero is used to represent complete non-membership, the value one is used to represent complete membership, and values in between are used to represent intermediate degrees of membership. The mapping is described by the membership function.

Type-1 fuzzy sets Satisfaction is a linguistic variable which has three linguistic values (bad, fair, and good). The shape of a membership function can be triangle, Gaussian, or other shapes.

Type-2 fuzzy sets Primary membership function Secondary membership function Type-2 fuzzy sets are described by membership functions that are themselves fuzzy.

Interval type-2 fuzzy sets Primary membership function Secondary membership function For an interval type-2 FLS, has the same probability (equals to 1) to be any values in its interval range

Upper-lower limit method avg = ( H + L ) / 2 , = H - avg = avg - L

Integrated Fuzzy output R = Ravg R

Rule base v Mamdani-Assilian rule base [23]: Ri: If x 1 is A 1, x 2 is A 2, …, xn is An then y = Bi. Used by most fuzzy systems. It is also used in this thesis. v Takagi and Sugeno rule base [24]: Ri: If x 1 is A 1, x 2 is A 2, …, xn is An then y = p 0 + p 1 A 1 + p 2 A 2 + … + pn. An. Takagi-Sugeno FLS usually needs a smaller number of rules, because their output is already a linear function of the inputs rather than a constant fuzzy set.

Type-1 FLS Reference 7

Type-2 FLS Reference 19

System optimizations § Structure optimization determines the fuzzy partition of input-output variables, and the set of rules to be used to generate mapping between input and output. ØBy considering the input variables separately § Parameter optimization, which are tuned on the experimental data through optimization procedures, are associated with the membership functions of input-output variables. ØThe least square method is used to optimize the parameters of the membership functions

Least square method The least square method is the most common technique in data analysis, which assumes that the best-fit curve of a given set of data is the curve that has the minimal sum of the deviations squared [31]. where xi is the independent variable and Yi is the corresponding dependent variable. f(x) is the fitting curve.

General Tuning Algorithm For a FLS with defined membership functions and rules, the general optimization algorithm using the least square will be: Begin 1. Read training data 2. Type-reduce for type-2 membership functions 3. Select tuning parameters, P 1, P 2, …, ( 1, 2, …)* 4. Initialize tuning parameters and 2 5. Determine appropriate rules to fire 6. Calculate out using prod and max operators defined in type-1 7. Calculate output using the method of the controid of area 8. Compute squared deviation 9. Calculate 2(P 1, P 2, …, 1, 2, …) 10. Adjust the tuning parameters to find minimum 2 (P 1, P 2, …, 1, 2, …) 11. Output tuning results 12. End 13. * For type-2 only; for type-1 = 0.

Structure of Web Housing Expert Training Process User Interface Fuzzy Logic System Database System FLS: interval type-2 fuzzy set Database: MS access Training process: least square in C++ User interface: Java, Servlet

Database No Address Price Location Year Distance S(Exp. ) 1 101 1 st Ave. 230 7 5 15 7 2 123 Lakeside Rd. 450 9 2 30 6 3 204 7 th St. 310 8 1 15 8 4 226 Wood St. 400 8 3 20 7 5 228 Wood St. 350 8 6 10 8 6 402 North Rd. 150 5 6 25 6 7 533 3 rd Ave. 180 6 8 15 6 8 666 Main St. 290 7 1 10 7 9 705 Main St. 260 7 7 10 8 10 717 South Rd. 200 5 5 5 7 UNITS Price: $1000, Year: Year, and Distance: mile

Linguistic variables § INPUT VARIABLES Price (0 – 500 k$): Low, Fair, High Location (0 – 10): Bad, Fair, Good Year (0 – 10 year): New, Old Distance (0 – 50 mile): Close, Far § OUTPUT VARIABLES Satisfaction (0 -10): Bad, Fair, Good

Membership Functions (1) The membership function of price is an interval type-2. It is reduced to type-1 using upperlow limit method. P and are the adjustable parameters.

Membership Functions (2)

Reduce Number of Rules The total number of possible rules is 32*22 = 36 Note that If P is {fair, high} and L is {fair, good} and Y is {new, old} and D is {close, far} is equivalent to P is {fair, high} and L is {fair, good} Because Y and D take every one of the elements from their domains. The inputs can be divided into two groups (price-location and year-distance) and considered separately, then combine these two groups. The total number of rules becomes 32+22+32 = 22

Fuzzy Rules First Group ( 1) L Second Group ( 2) P Low Fair High Good Fair Good Fair Poor Bad Fair Poor D Y Poor New Old Close Good Fair Poor Combination ( C) 1 st 2 nd Good Fair Poor Good Fair Good Fair Poor

Parameter Optimization (1) Read training data Initialize 2 For = 0 to 0. 2 For p = 100 to 400 For l = 3 to 7 2( , p, l) = 0 For each individual training data Calculate S using prod and max operators Update 2( , p, l) done If 2( , p, l) smaller than 2 Then 2 2( , p, l) done (p) done ( ) Output , p, l The product and maximum methods are used in the calculations of S: e. g. 1 = p* l, 2 = y* d, C = 1* 2, and s = max ( C 1, C 2, C 3)

Parameter Optimization (2) P changes from 250 to 180 p varies from 0. 05 to 0. 08 L shifts from 5 to 5. 1

Parameter Optimization (3) House # S (Exp) Sbefore (calc) Safter (calc) S_Range Sbefore Safter 1 7 5. 8 7. 3 0. 547 1. 2 0. 3 2 6 4. 6 5. 4 0. 482 1. 4 0. 6 3 8 4. 4 7. 7 0. 368 3. 6 0. 3 4 7 5. 5 6. 9 0. 258 1. 5 0. 1 5 8 7. 3 8. 3 0. 3 47 0. 3 6 6 7. 3 5. 8 0. 295 1. 3 0. 2 7 6 7. 2 6. 3 0. 496 1. 2 0. 3 8 7 5. 9 7. 0 0. 331 1. 1 0. 0 9 8 6. 6 8. 1 0. 415 1. 4 0. 1 10 7 7. 6 6. 8 0. 346 0. 2 Sbefore = |S(exp) – Sbefore(calc)| Safter = |S(exp) – Safter(calc)|

User Interface (1)

User Interface (2)

User Interface (3)

User Interface (4)

User Interface (5)

Procedure 0. Training system. 1. Obtain user’s queries through the user interface. 2. Search the database to find matched data. 3. Determine the fuzzy membership values activated by the inputs through the fuzzy inference system. 4. Decide which rules are fired in the rule set. 5. Combine the membership values for each activated rule using the AND operator. 6. Trace rule activation membership values back through the appropriate output fuzzy membership functions. 7. Utilize defuzzification to calculate the value for the output variable. 8. Rank decisions according to the output values using database system. 9. Display the output values on the user interface.

Type-2 vs. Type-1 Type-2 FLS: precise and reasonable

Interval Output Type-2 offers an interval satisfaction instead of the crisp output from type-1. The interval also gives more freedom to describe search results for a same query from different people because people will have different answer to a same question. Ø Some people might think the price of house 1 is too high and some people might think the price of house 1 is reasonable based on their own view about the house. Ø Dell Inspiron 9300 is a very good laptop to programmers but it may be too heavy (8. 2 pounds) for a traveler or too expansive ($2, 858) for a student.

A Real Case (1) No. Company Model Price ($) CPU Memory Display (GHz) (MB) (inch) 1 ABS Mayhem G 4 2199 2. 1 1024 15. 4 6. 2 2 Acer Aspire As 2003 Lmi 1599 1. 6 512 15. 4 5. 5 3 Acer Aspire 5000 1099 1. 6 512 15. 4 6. 8 4 Averatec 4200 1199 1. 6 512 15. 4 5. 6 5 Dell Inspiron 9300 2842 2 1024 17 7. 8 6 Gateway M 460 S 1359 1. 86 512 15 6. 4 * Computer Shopper (issues July – October, 2005) Rank*

A Real Case (2)

A Real Case (3)

Conclusions § An interval type-2 FLS based decision support system, the Web Housing Expert for online users is proposed. § A fuzzy output of the interval type-2 FLS based decision support system can be obtained using the upper-lower limit technique. § The total number of rules is reduced from 36 to 22 by separately considering the input variables. The membership functions of price and location are optimized using the least square technique to achieve the best performance. § It shows that the interval type-2 FLS not only provides a more reasonable result but also a more accurate result than type-1 FLS does

Future Work § Introduce more input variables to describe houses more precisely § Use more type-2 variables to handle the uncertainties of words effectively § Weight different input variables to present the importance of different variables § Apply interval inputs to solve the problem of the precise search criteria

ACKNOWLEDGEMENTS § I would like to send special thanks to my thesis advisor Dr. Zhang for his help, guidance and support throughout the duration of thesis. § I also would like to thank my thesis committee members, Dr. Sunderraman and Dr. Zhu.