""" **Description**
A principal component model analyzes an incident signal to define
transformation matrices which consume an incident signal to produce a
reference signal, normalized and ordered to define orthogonal axes of
descending variance.
A principal component model is a supervised learning model which
analyzes an incident signal representing a training set to learn a mean
vector, standard deviation vector, and a collection of eigenvectors
associated with an incident signal.
.. math::
\\vec{\mu_{i}} = \matrix{\ \\frac{\sum_{n=0}^{N}\\vec{x_{i,n}}}{N}}
.. math::
\\vec{\sigma_{i}} = \matrix{\ \\frac{\sum_{n=0}^{N}(\ \\vec{x_{i,n}} - \\vec{\mu_{i}}\ )^{2}}{N}}^{0.5}
.. math::
\Lambda_{n} = eig\matrix{\ cov\matrix{\ \matrix{\\frac{\ X_{n}^{T} - \\vec{\mu}\ }{\\vec{\sigma}}\ }\ }^{T}\ }^{T}
An incident signal which is not a part of an inital training set is
transformed without modifying a principal component model, by
translation, normalization, and rotation to produce a reference signal
which is a candidate for dimension reduction, in which higher order
dimensions are discarded, reducing the order of the reference signal,
while preserving significant and often sufficient information.
.. math::
Y_{n} = \Lambda_{n} \ \matrix{\\frac{\ X_{n}^{T} - \\vec{\mu}\ }{\\vec{\sigma}}\ }^{T}
Principal component analysis and dimension reduction has application in
clustering, classification, pattern recognition, and visualization.
**Example**
::
from diamondback import PrincipalComponentModel
# Create an instance.
obj = PrincipalComponentModel( )
# Model an incident signal and extract eigenvalue, eigenvector, mean, and deviation arrays.
x = numpy.random.rand( 3, 32 )
y = obj.model( x )
eigenvalue, eigenvector, mean, deviation = obj.eigenvalue, obj.eigenvector, obj.mean, obj.deviation
**License**
`BSD-3C. <https://github.com/larryturner/diamondback/blob/master/license>`_
© 2018 - 2022 Larry Turner, Schneider Electric Industries SAS. All rights reserved.
**Author**
Larry Turner, Schneider Electric, Analytics & AI, 2019-01-25.
"""
from typing import List, Union
import numpy
[docs]class PrincipalComponentModel( object ) :
""" Principal component model.
"""
@property
def deviation( self ) :
""" deviation : numpy.ndarray.
"""
return self._deviation
@property
def eigenvalue( self ) :
""" eigenvalue : numpy.ndarray.
"""
return self._eigenvalue
@property
def eigenvector( self ) :
""" eigenvector : numpy.ndarray.
"""
return self._eigenvector
@property
def mean( self ) :
""" mean : numpy.ndarray.
"""
return self._mean
def __init__( self ) -> None :
""" Initialize.
"""
super( ).__init__( )
self._deviation, self._eigenvalue = numpy.array( [ ] ), numpy.array( [ ] )
self._eigenvector, self._mean = numpy.array( [ ] ), numpy.array( [ ] )
[docs] def clear( self ) -> None :
""" Clears an instance.
"""
self._deviation, self._eigenvalue = numpy.array( [ ] ), numpy.array( [ ] )
self._eigenvector, self._mean = numpy.array( [ ] ), numpy.array( [ ] )
[docs] def model( self, x : Union[ List, numpy.ndarray ] ) -> numpy.ndarray :
""" Models an incident signal and produces a reference signal.
Arguments :
x : Union[ List, numpy.ndarray ] - incident signal.
Returns :
y : numpy.ndarray - reference signal.
"""
if ( ( not numpy.isscalar( x ) ) and ( not isinstance( x, numpy.ndarray ) ) ) :
x = numpy.array( list( x ) )
if ( ( len( x.shape ) != 2 ) or ( not len( x ) ) ) :
raise ValueError( f'X = {x}' )
if ( not len( self._deviation ) ) :
self._mean = x.mean( 1 )
self._deviation = x.std( 1 )
z = ( ( x.T - self._mean ) / self._deviation ).T
self._eigenvalue, self._eigenvector = numpy.linalg.eig( numpy.matmul( z, z.T ) / z.shape[ 1 ] )
self._eigenvector = self._eigenvector.T
else :
z = ( ( x.T - self._mean ) / self._deviation ).T
rows, cols = x.shape
if ( ( rows != len( self._mean ) ) or ( ( rows, rows ) != self._eigenvector.shape ) or ( cols <= 0 ) ):
raise ValueError( f'Rows = {rows} Columns = {cols}' )
return numpy.matmul( self._eigenvector, z )