Statistical
learning methods
Definition:
The problem is to
learn a function mapping examples into two classes: positive and
negative. We are
given a database of examples already classified as positive or negative.
Concept learning: the
process of inducing a function mapping input examples into a
Boolean output.
Examples:
Classifying objects in astronomical images as
stars or galaxies
Classifying animals as vertebrates or
invertebrates
Class of Tasks: Predicting poisonous mushrooms
Performance: Accuracy of classification
Experience: Database describing mushrooms with
their class
Knowledge to learn: Function mapping mushrooms
to {0,1} where 0:not-poisonous and
1:poisonous
Representation of
target knowledge: conjunction of attribute values.
Learning mechanism: candidate-elimination
Representation of instances:
Features:
• color {red, brown,
gray}
• size {small, large}
• shape
{round,elongated}
• land {humid,dry}
• air humidity
{low,high}
• texture {smooth,
rough}
Input and Output Spaces:
X : The space of all
possible examples (input space).
Y: The space of
classes (output space).
An example in X is a
feature vector X.
For instance: X =
(red,small,elongated,humid,low,rough)
X is the cross
product of all feature values.
Only a small subset
of instances is available in the database of examples.
Learning with complete data and hidden variables
we must state some formal
definitions:
Definition 1: Let X be some
set of objects, with elements noted as x. Thus,X = {x}.
Definition 2: A fuzzy set A
in X is characterized by a membership function mA(x) which maps each point in X
onto the real interval [0.0, 1.0]. As mA(x) approaches 1.0, the "grade of
membership" of x in A increases.
Definition 3: A is EMPTY iff
for all x, mA(x) = 0.0.
Definition 4: A = B iff for
all x: mA(x) = mB(x) [or, mA = mB].
Definition 5: mA' = 1 - mA.
Definition 6: A is CONTAINED
in B iff mA <= mB.
Definition 7: C = A UNION B,
where: mC(x) = MAX(mA(x), mB(x)).
Definition 8: C = A
INTERSECTION B where: mC(x) = MIN(mA(x),mB(x)).
