Transcript Lecture 6 , Feb - 09

Machine Learning
Mehdi Ghayoumi
MSB rm 132
[email protected]
Ofc hr: Thur, 11-12 a
Machine Learning
THE NAÏVE BAYES CLASSIFIER
In the naïve Bayes classification scheme, the required
estimate of the pdf at a point x=[x(1),...,x(l)]T∈Rl is given as
That is, the components of the feature vector x are assumed
to be statistically independent.
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Example .
Generate a set X1 that consists of N1 = 50 5-dimensional data vectors that
stem from two classes, ω1 and ω2. The classes are modeled by Gaussian
distributions with means
m1 = [0,0,0,0,0]T and m2 = [1,1,1,1,1]T and respective covariance matrices
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Step 1. Classify the points of the test set X2 using the naive
Bayes classifier, where for a given x, p(x|ωi ) is estimated as
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Step 2. Compute the ML estimates of m1, m2, S1, and S2
using X1. Employ the ML estimates in the Bayesian classifier
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Step 3. Compare the results obtained in steps 1 and 2.
The
resulting
classification
errors—naive_error
and
Bayes_ML_error—are 0.1320 and 0.1426, respectively.
In other words, the naive classification scheme outperforms
the standard ML-based scheme.
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The techniques that are built around the optimal Bayesian
classifier rely on the estimation of the pdf functions
describing the data distribution in each class.
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The focus is on the direct design of a discriminant
function/decision surface that separates the classes in some
optimal sense according to an adopted criterion.
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Machine learning involves adaptive mechanisms that enable computers to
learn from experience, learn by example and learn by analogy. Learning
capabilities can improve the performance of an intelligent system over
time. The most popular approaches to machine learning are artificial
neural networks and genetic algorithms. This lecture is dedicated to
neural networks.
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• Cell structures
– Cell body
– Dendrites
– Axon
– Synaptic terminals
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• Networks of processing units (neurons) with connections
(synapses) between them
• Large number of neurons: 1010
• Large connectitivity: 105
• Parallel processing
• Distributed computation/memory
• Robust to noise, failures
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Understanding the Brain
•
Levels of analysis (Marr, 1982)
1. Computational theory
2. Representation and algorithm
3. Hardware implementation
•
Reverse engineering: From hardware to theory
•
Parallel processing: SIMD vs MIMD
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Real Neural Learning
• Synapses change size and strength with experience.
• Hebbian learning: When two connected neurons are
firing at the same time, the strength of the synapse
between them increases.
• “Neurons that fire together, wire together.”
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Synapse
Axon
Soma
Synapse
Dendrites
Axon
Soma
Dendrites
Synapse
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Biological Neural Network
Soma
Dendrite
Axon
Synapse
Artificial Neural Network
Neuron
Input
Output
Weight
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Neural Network Learning
• Learning approach based on modeling adaptation in
biological neural systems.
• Perceptron: Initial algorithm for learning simple neural
networks (single layer) developed in the 1950’s.
• Backpropagation: More complex algorithm for learning
multi-layer neural networks developed in the 1980’s.
Machine Learning
Perceptron Learning Algorithm
• First neural network learning model in the 1960’s
• Simple and limited (single layer models)
• Basic concepts are similar for multi-layer models so this
is a good learning tool
• Still used in many current applications
Machine Learning
Input Signals
W eights
Output Signals
x1
Y
w1
x2
w2
Neuron
wn
xn
Y
Y
Y
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Step function
Sign function
Sigmoid function Linear function
Y
Y
Y
Y
+1
+1
1
1
0
X
0
X
-1
-1
1, if X  0
step
Y


0, if X  0
0
-1
1, if X  0 sigmoid
sign
Y

Y

1, if X  0
X
0
-1
1
1  e X
Y linear X
X
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Inputs
x1
w1
Linear
Combiner
Hard
Limiter
Output
Y
w2
x2

Threshold
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Perceptron Node – Threshold Logic Unit
x1
w
1
x2
w
q
z
2
xn
n
w
n
1
if
åx w ³q
i =1
z=
i
i
n
0
if
åx w <q
i =1
i
i
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x1
w
1
x2
q
w
z
2
xn
w
n
• Learn
weights
such
that
an
n
objective
function is maximized.
• What objective function should we use?
• What learning algorithm should we use?
1
if
åx w ³q
i =1
z=
i
i
n
0
if
åx w <q
i =1
i
i
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Perceptron Learning Algorithm
x1
.4
z
.1
x2
-.2
n
x1 x2 t
.8 .3 1
.4 .1 0
1
if
åx w ³q
i =1
z=
i
i
n
0
if
åx w <q
i =1
i
i
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First Training Instance
.8
.4
z =1
.1
.3
-.2
net = .8*.4 + .3*-.2 = .26
n
x1 x2 t
.8 .3 1
.4 .1 0
1
if
åx w ³q
i =1
z=
i
i
n
0
if
åx w <q
i =1
i
i
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Second Training Instance
.4
.4
.1
.1
-.2
net = .4*.4 + .1*-.2 = .14
n
x1 x2 t
.8 .3 1
.4 .1 0
1
if
åx w ³q
i =1
z=
i
i
n
0
if
z =1
åx w <q
i =1
i
i
Thank you!