Course Introduction Jan Jantzen Technical University of Denmark

Summary Fuzzy sets, fuzzy logic Fuzzy clustering Neural nets Neuro-fuzzy modelling

Course Objectives To teach the fundamental concepts To show some applications

General Approach 1.Get plenty of good data 2.Design a linear model 3.Replace it with a nonlinear model 4.Did the results improve? Else repeat step 3.

True Love Wife: Do you love me? Husband (Boolean logician): Yes. Wife: How much?

Seasons Time of the year Membership SpringSummerAutumnWinter

Tall Persons Height [cm] Membership fuzzy crisp

Zadeh’s Challenge Clearly, the “class of all real numbers which are much greater than 1,” or “the class of beautiful women,” or “the class of tall men,” do not constitute classes or sets in the usual mathematical sense of these terms (Zadeh, 1965).

Fuzzy ( Function: adjective Inflected Form(s): fuzz·i·er; -est Etymology: perhaps from Low German fussig loose, spongy Date: : marked by or giving a suggestion of fuzz 2 : lacking in clarity or definition - fuzz·i·ly / 'f&-z&-lE / adverb - fuzz·i·ness / 'f&-zE-n&s / noun

Fuzzy ( Function: adjective Synonyms: faint, bleary, dim, ill-defined, indistinct, obscure, shadowy, unclear, undefined, vague

Age Age [years] Membership Young Very young Not very young Age [years] Membership Old More or less old

Logic Wife: Do you like my girlfriend? Husband: Very much. Wife: Then you don’t love me.

A Warm Room Temperature [deg C] Truth Fuzzy Crisp

Fuzzy Logic Control Fuzzy logic control (FLC) may be viewed as a branch of intelligent control which serves as an emulator of human decision-making behaviour that is approximate rather than exact (C.C.Lee in Singh: Systems and Control Encyclopedia, 1992).

Rule Format R i : if x is A i and y is B i then z is C i

Implication IF room is warm THEN set cooling at 500 watts

Inference If room is warm then set cooling at 500 watts Temperature is 21 deg C Set cooling at 250 watts

Sets {Live dinosaurs in British Museum} = 

Fuzzy Sets {nice days} {adults}

Set Operations a) AB b) AB c) AB d) AB e) AB f) AB

Q: Why Logic? Example: If either the Pirates or the Cubs loose and the Giants win, then the Dodgers will be out of first place, and I will loose a bet. ((p  c)  g)  (d  b) A: Math proofs, computers

Boolean OR

Fuzzy OR

Tautologies 1.[p  (p  q)]  q 2.[(p  q)  (q  r)]  (p  r) 3.[p  (p  q)]  p  q

Q: Why fuzzy logic? A: Tolerant of imprecision

Papers with fuzzy in title (INSPEC+Math Reviews)

Example: Stopping a car 0y F m

PD Control

Rule base If distance is long and approach is fast, then brake zero If distance is long and approach is slow, then brake zero If distance is short and approach is fast, then brake hard If distance is short and approach is slow, then brake zero

Response Position [m] Time [s] x 10 4 Control [N] Time [s] PID fuzzy

Fuzzy Clustering Find clusters in data Extract rules from data E.g., bank customer segmentation, diagnosing cancer cells

Cluster analysis ( ) A statistical classification technique for discovering whether the individuals of a population fall into different groups by making quantitative comparisons of multiple characteristics.

Vehicle Example

Vehicle Clusters Top speed [km/h] Weight [kg] Sports cars Medium market cars Lorries

Example: Diagnose Cancer Cells Normal smearSeverely dysplastic smear Using a small brush, cotton stick, or wooden stick, a specimen is taken from the uterin cervix and smeared onto a thin, rectangular glass plate, a slide. The purpose of the smear screening is to diagnose pre-malignant cell changes before they progress to cancer. The smear is stained using the Papanicolau method, hence the name Pap smear. Different characteristics have different colours, easy to distinguish in a microscope. A cyto-technician performs the screening in a microscope. It is time consuming and prone to error, as each slide may contain up to cells. Dysplastic cells have undergone precancerous changes. They generally have longer and darker nuclei, and they have a tendency to cling together in large clusters. Mildly dysplastic cels have enlarged and bright nuclei. Moderately dysplastic cells have larger and darker nuclei. Severely dysplastic cells have large, dark, and often oddly shaped nuclei. The cytoplasm is dark, and it is relatively small.

The Perceptron Classification Learning

How Use A Neural Network? Classification or approximation ? Training data Examples and epoch Pattern or batch mode ? Test data Neural Network +- Compare d u y M Modifier e

Perceptron Hard limiter x f(x) (a)(b) f(x) w0w0 w1w1 w2w2 + 1

Single Layer Perceptron y 1 = sgn(w 1 T u), y 2 = sgn(w 2 T u), y 3 = sgn(w 3 T u) 1 2 u1u1 u2u2 y1y1 y2y2 y3y

Multilayer Perceptron Input layerHidden layerOutput layer

Case: Sunspot Cycles

Fuzzy Rules As A NN Pos Zero Neg 100 W1 0 W W / u e Input layerHidden layerOutput layer

MFs Before And After Learning

ANFIS net Layer u1u1 u2u2 A1A1 A2A2 B2B2 B1B1 AND N N u 1,u 2 + y

Summary Fuzzy sets, fuzzy logic Fuzzy clustering Neural nets Neuro-fuzzy modelling

Problems Attacked Nonlinear Multivariable Operator’s rules Learning