Description
Problem 1
Part a)
Consider an n-ary fuzzy relation zyxR ,…),,( , which has n independent variables (coordinate axes),
x, y,z,… . How many different projections are possible for this relation, in various subspaces? Explain
how you arrived at this number.
Consider a discrete ternary ( n 3) fuzzy relation (rule base) given by the following “pages” of
binary ),( ji
yx relation along the z axis:
6.06.01.0
5.08.03.0
2.04.05.0
),(
1
jiz
yxR ;
8.06.02.0
6.00.14.0
3.05.06.0
),(
2
jiz
yxR ;
5.05.00.0
3.07.02.0
1.03.04.0
),(
3
jiz
yxR
Determine all possible projections of this relation
Part b)
Show that max[0, x+ y – 1] is a t-norm. Also, determine the corresponding t-conorm (i.e., s-norm).
Problem 2
Part a)
Show that the optimal learning rate for minimizing the quadratic error function is:
ηopt=(w
TQ
2w/ wTQ
3w)|(k)
(Hint: Express the error function at step , and minimize this expression with respect to the learning rate .
Use the fact that for quadratic forms )
Part b)
Using results in problem 1, perform two iterations of the optimal gradient algorithm to find the minimum of
, where is the weight vector of dimension 2. The starting
point is . Find and show that the gradients and are
perpendicular.
Problem 3
We need to train a Radial Basis Function network for obtaining the output of the following twoto-one mapping function:
y(x , x ) sin( x ) cos( . x ) 1 2 2 1 0 5 2
a) Set up two sets of data, each of which consists of 100 input-output patterns, one for
network training and the other for testing (with which to compare performance with
the true function). The input-output data are obtained by randomly varying the input
variables (x , x )
1 2 within the interval [-1,1].
b) Use two type RBF networks: one with Gaussian MF’s and the other with exponential
MF’s (logistic). Analyse the results by changing the number of nodes from 10 to 100 (in
steps of 10). Compare the outcome of both networks (with different basis functions).
Compare the performance of the networks in terms of execution time and accuracy
(mean squared errors.). Discuss your results
Problem 4
The dynamics of a time series is represented by the following nonlinear difference equation:
2
2
( ) (0.3- 0.6exp(- ( -1))) ( -1) –
(0.8 0.9exp(- ( -1))) ( – 2) 0.3sin( ( -1))
y t y t y t
y t y t y t
a) Considering y(t) as the output of the time series and y t( -1) and y t( – 2) as its inputs with
2
( ( -1), ( – 2)) -1.8 1.8 y t y t display the surface output. Note the linear behaviour of y t( ) as
a function of y t( – 2) when y t( -1) is held at0 .
b) Design a feed forward neural network to estimate the output surface. You may wish to use 500
sample data for training and testing purposes (70% of which for training and 30% for testing).
Use several scenarios to find the best network (varying the number of hidden layers from 1 to 3
and varying the number of nodes in each layer let say from 2 to 58 in steps of 8). Display your
results in a table and discuss the results.
c) Compare the results obtained in (b) with those you obtain using a RBF network with 30, 60, 90
hidden nodes, respectively. Comparison is made in terms of Root Mean Square Error (RMSE).
Problem 5 (Linear Classifiers)
We need to develop a neurocomputing based classifier for three various but related products. The collected data
are the results of a chemical analysis of liquid products grown in the same region in Italy but derived from three
different cultivars. The analysis determined the quantities of 12 constituents found in each of the three types of
products. The data file provided is in text format and has thirteen dimensions, the first of which determines the
class of products (product ‘1’, product ‘2’, product ‘3’), which should serve as the output of the neurocomputing
classifier. The remaining ones determine the input of the classifier and has 12 constituents as:
1)Malic acid
2) Ash
3) Alcalinity of ash
4) Magnesium
5) Total phenols
6) Flavanoids
7) Nonflavanoid phenols
8) Proanthocyanins
9)Color intensity
10)Hue
11)OD280/OD315 of diluted liquid
12)Proline
5.1 Design a classifier of your choice (neural network based or SVM based), vary its parameters and try to find
the best possible classification performance (a table illustrating various results as parameters are varied would
be preferred).
5.2 Once this is done, classify (determine to which product they belong) the following entries each of which
has 12 attributes:
a) 13.72; 1.43; 2.5; 16.7; 108; 3.4; 3.67; 0.19; 2.04; 6.8; 0.89; 2.87; 1285
b) 12.04; 4.3; 2.38; 22; 80; 2.1; 1.75; 0.42; 1.35; 2.6; 0.79; 2.57; 580
c) 14.13; 4.1; 2.74; 24.5; 96; 2.05; 0.76; 0.56; 1.35; 9.2; 0.61; 1.6; 560
Hint for implementation: You may wish to calibrate all input data to be all between 0 and one. Also from the set of
data choose 75% of the data from product 1, product 2 and product 3 as training data and remaining 25%
remaining as testing. You can use existing Matlab libraries or other libraries to create the classifier, which code
needs to be appended to the solution. In case you make use of SOM (Self Organizing Map), please disregard the
classes and assume three clusters.
