Transcript Materialy/06/Lecture13- ICM Neuronal Nets 2

Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Intelligent Control
Methods
Lecture 14: Neuronal Nets (Part 2)
Decision (solution, active dynamics)
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The input xk=(x1k, x2k, ... xnk) is applied to input
layer.
Neurons in all layers work out output signals
according to their input(s), threshold and transfer
function. The output signals are multiplicated by
weights.
The neuron output signals lead to next layer.
The output of the last layer is yk=(y1k, y2k, ... ymk).
The net gives (declares, indicates, defines) with
decision yk what is xk.
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Decision example: – characters classification
w11
x1=1
x2=1
y1 = 0.84  1
y2 = 0.04  0
x3=0
x4=1
y3 = 0.75  1
y4 = 0.92  1
y5 = 0.12  0
x15=1
y = (1,0,1,1,0) responds to „R“
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Learning (adaptive dynamics)
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Theoretically possible ways of learning:
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construction of new connections
removing of connections
changing of neurons threshold values
changing of transfer functions
changing of layers number or number of neurons in layers
changing of synaptic weights (practical used only)
Base idea: Synaptic weights are adapted to values,
which guarantee the proper output vector y for each
input vector x.
The way (not only) of weight adaptation: Learning with
training set.
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Learning with training set:
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Training set:
 M = {(x1,b1), ... , (xn,bn)}
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xk – input vector
bk – correct output vector (response)
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The real net response to input xk: yk
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Adapted (learnt) net: yk = bk
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Learning with training set:
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Vector xk from training set is connected to net input. The
signal spreads in net and the net produces an output yk.
yk compares with bk.
The needed changes of synaptic weights are computed
according to differences (net mistake) between yk and
bk..The biggest weight changes are in connections,
where are the greatest differences (delta rule). Because the
difference is measured in output, the first calculations are performed in
output layer. The calculation process moves along to net from right to left
(back propagation).
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The global net mistake is calculated after complete
training net using. If it is in allowed range, the adaptation
process ends. Otherwise the training set must be used
again (perhaps thousands iterations).
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Learning with training set:
Net global mistake (with weights w):
E ( w)   ( ( ykj  bkj ))
k
j
mistake of element j
mistake of pattern k
training net global mistake
The sum of quadrates errors is used for mistake of
pattern k estimation, therefore:
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E ( w)  
2 k
2
(
y
kj

b
kj
)

j
wopt = arg min E(w)
w
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Learning with training set:
Iterative process. The initial synaptic weights are set up.
Changes wij are computed so, that the mistake of
pattern k is minimal:
E (k , w)
wij(k )  
wij(k )
j
yij, wij
yi
i
 - learning rate - defines a speed of learning process.
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Learning with training set:
The result of derivation: delta rule
For output layer:
wij(k )   (bi (k )  yi (k )) yij(k )
needed
change
of wij
learning
rate
mistake
of
output i
contribution
of neuron j
to input i
For hidden layer:
wij(k )   [ wmi(bm(k )  ym(k ))] yij(k )
m
all outputs into next
layer
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Learning with training set:
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The weights are adapted mostly in places,
where are the greatest mistakes.
The calculations are performed in direction from
output to input layer (back propagation).
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Example of neuronal net learning:
p
XOR:
w31
1
3
w53
w41
y
5
w32
q
2
w54
4
w42
p
q
y
1
1
0
0
1
1
1
0
1
0
0
0
= 0.01 (for all neurons), unit jump transfer functions,  = 1.
initial weights for 1. pattern (p=1, q=1) are:
w31(1) = -4.9, w41(1) = 4.6, w32(1) = 5.0, w42(1) = -5.1, w53(1) = 2.2, w54(1) = 2.5
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Example of neuronal net learning:
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b(1) = 0
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For the 1st pattern: y53(1) =1, y54(1) = 0, y5(1) = 1.
w53(1) = 1 (b5(1) – y5(1)) y53(1) = 1 (0 – 1) 1 = -1
w54(1) = 1 (b5(1) – y5(1)) y54(1) = 1 (0 – 1) 0 = 0
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In hidden layer:
w31(1) = 1 [w53(1) (b5(1) – y5(1))] y31(1) = 1 [2.2 (0 – 1)] 1 = -2.2
w41(1) = 1 [w54(1) (b5(1) – y5(1))] y41(1) = 1 [2.5 (0 – 1)] 1 = -2.5
w32(1) = 1 [w53(1) (b5(1) – y5(1))] y32(1) = 1 [2.2 (0 – 1)] 1 = -2.2
w42(1) = 1 [w54(1) (b5(1) – y5(1))] y42(1) = 1 [2.5 (0 – 1)] 1 = -2.5
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Example of neuronal net learning:
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New weights for pattern 2: (p=0, q=1)
wij(2) = wij(1) + wij(1)
 w31(2) = w31(1) + w31(1) = -4.9 + (-2.2) = -7.1
 w41(2) = w41(1) + w41(1) = 4.6 -2.5 = 2.1
 w32(2) = w32(1) + w32(1) = 5.0 - 2.2 = 2.8
 w42(2) = w42(1) + w42(1) = -5.1 - 2.5 = -7.6
 w53(2) = w53(1) + w53(1) = 2.2 - 1 = 1.2
 w54(2) = w54(1) + w54(1) = 2.5 + 0 = 2.5
Pattern 2 is used, the net produces an adequate output.
The weights are right (expected and real output are
equal), therefore they are not changed. The same for
patterns 3 and 4. The net is learnt.
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Applications of NNs:
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Pattern recognition (classification)
 example
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with characters recognition in raster
Optimisation
 There
are various input combinations and their
optimal outputs in training set. The learnt net can find
the optimum for another inputs, too. Used in cases,
where the analytical formulation input-output misses.
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Applications of NNs:
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data evaluation, state monitoring (system, organism, TP,
...) – example: chemical column
 inputs (linguistic variables):
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volume quantity of input hydrocarbons mixture
volume quantity of reflux (= backward) flow
volume quantity of heating steam
volume quantity of product output
 outputs:
 temperature
 pressure
 distilled material concentration in product
 contamination concentration on the column bottom
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Applications of NNs:
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Processes control
 Example: Abrasive
cutting. (Material with abrasium
circulate along a closed loop.
 Inputs:
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flow speed
hardness of abrasium, hardness of material
size of abrasium, size of material
number of cycles
 outputs:
 material decrease
 surface roughness
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Regulation
 The
controllers constants are estimated according to
combination of input, state, output and desired
values.
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Neuronal nets – concluding remarks:
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It do not exist rules for number of layers and for
number of neurons in layers estimation. (Nets with 0
- 2 hidden layers are used. The number of input neurons depends
on input number, the number of input neurons depends on the
needed outputs number n (example: 2i > n). The number of hidden
layers is 1 or 2, the number of neurons in hidden layers is low.)
Net
global
mistake
net size
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Neuronal nets – concluding remarks:
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I have not found recommedations for the
choice of transfer function, threshold
value, initial synaptic weights setup (average
values from allowed scope, random values?)
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Learning rate is selected from 0 - 1. (little
value needs more iterations but is more precise, the
bigger one learns rapid, but it can oscillate around the
extreme).
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It is not defined, when to stop the learning
process. After some iterations the net global mistake
can start to grow. (net overlearning).
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