Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Transcript

DEALING WITH UNCERTAINTY (2) WEEK 6 CHAPTER 3 1 Bayesian Approaches Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. 2 Bayesian Approaches derive the probability of an event given another event Often useful for diagnosis: If X are (observed) effects and Y are (hidden) causes, We may have a model for how causes lead to effects (P(X | Y)) We may also have prior beliefs (based on experience) about the frequency of occurrence of effects (P(Y)) Which allows us to reason abductively from effects to causes (P(Y | X)). has gained importance recently due to advances in efficiency more computational power available better methods 3 Bayes’ Rule for Single Event single hypothesis H, single event E P(H|E) = (P(E|H) * P(H)) / P(E) or P(H|E) = (P(E|H) * P(H) / (P(E|H) * P(H) + P(E|H) * P(H) ) 4 Bayes theorem gives us a way of calculating P(E|F) from a knowledge of P(F|E). Example E = the event that an anabolic steroid detection test gives a positive result F = the event that the athlete uses steroids then P(E|F) = probability that the test is positive for an athlete who uses steroids. P(F|E) = probability that an athlete uses steroids given that the test is positive. 5 Example Two Boxes A & B. A has 3 red ball & 2 white ball, while B has 2 red ball & 1 white ball. If we move one ball from A to B then we chose a ball form A. 1.find the probability of chosen the red ball. 2.If the chosen ball was red, what is the propability for the moved ball to be red. 6 1. find the probability of chosen the red ball. E1 : the moved ball is white E2 : the moved ball is red H : the chosen ball is red P(H)= P(H|E1) + P(H)= (2/5 * 1/4) + P( H|E2) (3/5 * 2/4) P(H)= 2/20 + 6/20 P(H)= 0.40 7 1. If the chosen ball was red, what is the probability for the moved ball to be red. P(E2|H) = P ( E2 H) / P ( H) P(E2|H) = (P(E2) * P(H/E2)) / P(H) P(E2|H) = ( 3/5 * 2/4) / 0.65 P(E2|H) = 0.30 / 0.65 P(E2|H) = 5/13 Or 0.37 8 Advantages and Problems Of Bayesian Reasoning advantages sound theoretical foundation well-defined semantics for decision making problems requires large amounts of probability data sufficient sample sizes subjective evidence may not be reliable independence of evidences assumption often not valid relationship between hypothesis and evidence is reduced to a number explanations for the user difficult high computational overhead 9 Contents 10 Possibility theory: fuzzy sets and fuzzy logic Note that: Bayesian updating and certainty theory - from statistical variations or randomness. Possibility theory handles vagueness in the use of language. Also called fuzzy logic Developed by Lotfi Zadeh. Builds upon his theory of fuzzy sets. 11 Conventional Set Theory In Conventional set theory: The Set Temperature = {high, medium, low} Elements of the set is mutually exclusive. If a temperature value (say 300°C) is considered high, it cannot be medium or low. Values are crisp or non-fuzzy If the boundary between medium and high is 300°C, then 301°C is high 299°C is medium. This is a rather artificial distinction A small change of 2°C from 299°C to 301°C completely change the rule-firing A huge change of 699°C from 301°C to 1000°C has no effect at all. 12 Crisp Set for temperature 13 Fuzzy Set Fuzzy sets smooth the boundaries. Fuzzy set theory expresses imprecision quantitatively Use characteristic membership functions with degrees of membership from 0 (“not a member”) through to 1 (“a full member”). For a fuzzy set F, the membership function μF (x) measures the degree to which an absolute value x belongs to F (possibility that x is described by F) The process of Getting the membership function or deriving these possibility values for a given value of x is called fuzzification. 14 Membership Function If we are given an imprecise statement that the temperature is low. If LT is the fuzzy set of low temperatures, then we might define the membership function μLT such that: μLT (250°C) = 0.0 μLT (200°C) = 0.0 μLT (150°C) = 0.25 μLT (100°C) = 0.5 μLT (50°C) = 0.75 μLT (0°C) = 1.0 μLT (–50°C) = 1.0 15 Crisp vs Fuzzy Sets Fuzzy sets might be applied in handling uncertainties caused by the use of vague language. Examples of vague language phrases: water level is low. temperature is high. pressure is high. 16 Fuzzy Set Example membership tall short medium 1 0.5 0 0 50 100 150 200 height (cm) 250 17 Fuzzy vs. Crisp Set membership short 1 tall medium 0.5 0 0 50 100 150 200 height (cm) 250 18 Crisp Set vs Fuzzy Set The key characteristics of fuzzy sets (that makes it different from crisp sets) are that: an element has a degree of membership (0–1) of a fuzzy set; membership of one fuzzy set does not prevent membership of another set 19 Fuzzy Set Temperature 350°C may have some (non-zero) degree of membership to both fuzzy sets high and medium. This is represented by the overlap between the fuzzy sets. Sum of the membership functions for a given value can be arranged to equal 1. 350°C is 0.25 Medium and 0.75 High 20 Terminologies Terminologies of fuzzy sets: fuzzy set - low temperature fuzzy variable - temperature fuzzy statement - temperature is low 21 Crisp Rules vs Fuzzy Rules CRISP RULES FUZZY RULES Variable value change in steps as different rules fire. Input variable values alter, causing smooth changes in the outputs. To smooth the steps require many rules. Require not as many rules. (depends on the no. of variables, the no. of fuzzy sets, and the ways in which the variables are combined in the fuzzy rule conditions). Numerical information is explicit e.g Numerical information is implicit in the chosen shape of the fuzzy membership functions. 22 Example Assume a rule base that contains the following fuzzy rules: /* IF /* IF /* IF Rule 3.6f */ temperature is high THEN pressure is high Rule 3.7f */ temperature is medium THEN pressure is medium Rule 3.8f */ temperature is low THEN pressure is low Suppose temperature is 350°C. This is a member of both fuzzy sets high and medium Rules 3.6f and 3.7f will both fire. The pressure, will be somewhat high and somewhat medium. 23 Using the membership functions for temperature given; the possibility that the temperature is high, μHT, is 0.75 the possibility that the temperature is medium, μMT, is 0.25. As a result of firing the rules, the possibilities that the pressure is high and medium, μHP and μMP, are set as follows: μ = max[μHT, μHP] μ = max[μMT, μMP] HP MP 24 The initial possibility values for pressure are assumed to be zero if these are the first rules to fire, and thus µHP and µMP become 0.75 and 0.25, respectively. These values can be passed on to other rules that might have pressure is high or pressure is medium in their condition clauses. 25 Compound Conditions Rules 3.6f, 3.7f and 3.8f contain only simple conditions. Fuzzy logic allows for compound conditions similar to those in certainty theory discussed earlier. 26 Example: AND Conjunction Suppose water level has the fuzzy membership functions shown below Suppose also that Rule 3.6f is redefined as follows: /* Rule 3.9f */ IF temperature is high AND water level is NOT low THEN pressure is high For a water level of 1.2m, the possibility of the water level being low, µLW(1.2m), is 0.6. The possibility of the water level not being low is therefore 0.4. As this is less than 0.75, the combined possibility for the temperature being high and the water level not being low is 0.4. Thus the possibility that the pressure is high, µHP, becomes 0.4 if it has not already been set to a higher value. 27 Example: OR Disjunction If several rules affect the same fuzzy set of the same variable, they are equivalent to a single rule whose conditions are joined by the disjunction OR. For example, these two rules: /* IF /* IF Rule 3.6f */ temperature is high THEN pressure is high Rule 3.10f */ water level is high THEN pressure is high are equivalent to this single rule: /* Rule 3.11f */ IF temperature is high OR water level is high THEN pressure is high 28 Dependent OR We can treat OR differently when it involves two fuzzy sets of the same fuzzy variable, for example, high and medium temperature. In such cases, the memberships are clearly dependent on each other. Therefore, we can introduce a new operator DOR for dependent OR. For example, given the rule: /* Rule 3.12f */ IF temperature is low DOR temperature is medium THEN pressure is lowish the combined possibility for the condition becomes: 29 Example DOR vs OR Given the fuzzy sets for temperature as below left, the combined possibility would be the same for any temperature below 200°C, as shown below right. This is consistent with the intended meaning of fuzzy Rule 3.12f. If the OR operator had been used, the membership would dip between 0°C and 200°C, with a minimum at 100°C, as shown below. 30 The problem A temperature control system has four settings Cold, Cool, Warm, and Hot Humidity can be defined by: Change the speed of a heater fan, based off the room temperature and humidity. Low, Medium, and High Using this we can define the fuzzy set. Nonlinear and dynamic in nature Inputs for Intel Fuzzy ABS are derived from Brake 4 WD Feedback Wheel speed Ignition Outputs Pulsewidth Error lamp Fuzzy rule-based system A Rule Base (RB) of fuzzy rules A Data Base (DB) of linguistic terms and their membership functions Together the RB and DB are the knowledge base (KB) A fuzzy inference system which maps from fuzzy inputs to a fuzzy output Fuzzification and defuzzification processes 33 34 FUZZIFICATION Fuzzification is the process of making a crisp quantity fuzzy. In the real world, hardware such as a digital voltmeter generates crisp data, but these data are subject to experimental error. 35 FUZZIFICATION The fuzzification comprises the process of transforming crisp values into grades of membership for linguistic terms of fuzzy sets. The membership function is used to associate a grade to each linguistic term. 36 For the fuzzification of the car speed value X0=70 km/h the two membership functions A and B from the Figure below can be used, which characterize a low and a medium speed fuzzy set, respectively. The given speed value of X0=70 km/h belongs with a grade of A (X0)= 0.75 to the fuzzy set ``low'' and with a grade of B (X0)= 0.25 to the fuzzy set ``medium''. 37 Example Measurement devices in technical systems provide crisp measurements, like 110.5 Volt or 31,5 °C. At first, these crisp values must be transformed into linguistic terms (fuzzy sets) . This is called fuzzification. The membership functions of the linguistic variable speed will be determined below. 38 Defuzzification At 350°C ( µHP = 0.75, µMP = 0.25, µLP = 0) /* IF /* IF Rule 3.6f */ temperature is high THEN pressure is high Rule 3.7f */ temperature is medium THEN pressure is medium These values can be passed on to other rules that might have pressure is high or pressure is medium in their condition clauses without any further manipulation. However, to interpret the membership values in numerical value of pressure, they need to be defuzzified. Defuzzification is important especially if a control action must be performed like “set current,” where a specific value setting is required. 39 Defuzzification Defuzzification takes place in two stages, described below. Stage 1: scaling the membership functions adjust the fuzzy sets in accordance with the calculated possibilities Stage 2: finding the centroid 40 Defuzzification - Stage 1 Larsen’s product operation rule - the membership functions are multiplied by their respective possibility values. The effect is to compress the fuzzy sets so that the peaks equal the calculated possibility values Alternative approach - truncate the fuzzy sets 41 Defuzzification - Stage 1 For most shapes of fuzzy set, the difference between the two approaches is small. But Larsen’s product operation rule has the advantages of simplifying the calculations and allowing fuzzification followed by defuzzification to return the initial value. 42 Defuzzification – Stage 2 Centroid method The most commonly used method sometimes called the center of gravity, center of mass, or center of area method. Defuzzified value = the point along the fuzzy variable axis that is the centroid, or balance point, of all the scaled membership functions taken together for that variable Imagine the cut out from stiff card and pasted together with overlap. Defuzzified value = the balance point along the fuzzy variable axis of this composite shape. When two membership functions overlap, both overlapping regions contribute to the mass of the composite shape. 43 Fuzzy Reasoning In order to implement a fuzzy reasoning system you need For each variable, a defined set of values for membership Can be numeric (1 to 10) Can be linguistic really no, no, maybe, yes, really yes tiny, small, medium, large, gigantic good, okay, bad And you need a set of rules for combining them Good and bad = okay. 44 Advantages and Problems of Fuzzy Logic advantages general theory of uncertainty wide applicability, many practical applications natural use of vague and imprecise concepts helpful for commonsense reasoning, explanation problems membership functions can be difficult to find multiple ways for combining evidence problems with long inference chains 45 Uncertainty: Conclusions In AI we must often represent and reason about uncertain information This is no different from what people do all the time! There are multiple approaches to handling uncertainty. Probabilistic methods are most rigorous but often hard to apply; Bayesian reasoning Fuzzy logic provides an alternate approach which better supports ill-defined or non-numeric domains. Empirically, it is often the case that the main need is some way of expressing "maybe". Any system which provides for at least a three-valued logic tends to yield the same decisions. 46