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Statsample::Factor::PrincipalAxis

Principal Axis Analysis for a covariance or correlation matrix.

For PCA, use Statsample::Factor::PCA

Usage:

require 'statsample'
a=[2.5, 0.5, 2.2, 1.9, 3.1, 2.3, 2.0, 1.0, 1.5, 1.1].to_scale
b=[2.4,0.7,2.9,2.2,3.0,2.7,1.6,1.1,1.6,0.9].to_scale
ds={'a'=>a,'b'=>b}.to_dataset
cor_matrix=Statsample::Bivariate.correlation_matrix(ds)
pa=Statsample::Factor::PrincipalAxis.new(cor_matrix)
pa.iterate(1)
pa.m
=> 1
pca.component_matrix
=> GSL::Matrix
[  9.622e-01 
   9.622e-01 ]
pca.communalities
=> [0.962964636346122, 0.962964636346122]

References:

Constants

DELTA

Minimum difference between succesive iterations on sum of communalities

MAX_ITERATIONS

Maximum number of iterations

Attributes

name[RW]

Name of analysis

m[RW]

Number of factors. Set by default to the number of factors with eigenvalues > 1 (Kaiser criterion).

Warning: Kaiser criterion overfactors! Give yourself some time and use Horn's Parallel Analysis.

iterations[R]

Number of iterations required to converge

initial_eigenvalues[R]

Initial eigenvalues

epsilon[RW]

Tolerance for iterations

smc[RW]

Use SMC(squared multiple correlations) as diagonal. If false, use 1

max_iterations[RW]

Maximum number of iterations

eigenvalues[R]

Eigenvalues of factor analysis

Public Class Methods

new(matrix, opts=Hash.new) click to toggle source
# File lib/statsample/factor/principalaxis.rb, line 65
def initialize(matrix, opts=Hash.new)
  @matrix=matrix
  if @matrix.respond_to? :fields
    @fields=@matrix.fields
  else
    @fields=@matrix.row_size.times.map {|i| _("Variable %d") % (i+1)}
  end
  @n_variables=@matrix.row_size
  @name=""
  @m=nil
  @initial_eigenvalues=nil
  @initial_communalities=nil
  @component_matrix=nil
  @delta=DELTA
  @smc=true
  @max_iterations=MAX_ITERATIONS
  opts.each{|k,v|
    self.send("#{k}=",v) if self.respond_to? k
  }
  if @matrix.respond_to? :fields
    @variables_names=@matrix.fields
  else
    @variables_names=@n_variables.times.map {|i| "V#{i+1}"}
  end
  if @m.nil?
    pca=PCA.new(::Matrix.rows(@matrix.to_a))
    @m=pca.m
  end
  
  @clean=true
end
separate_matrices(matrix, y) click to toggle source

Returns two matrixes from a correlation matrix with regressors correlation matrix and criteria xy matrix.

# File lib/statsample/factor/principalaxis.rb, line 178
def self.separate_matrices(matrix, y)
  ac=[]
  matrix.column_size.times do |i|
    ac.push(matrix[y,i]) if i!=y
  end
  rxy=Matrix.columns([ac])
  rows=[]
  matrix.row_size.times do |i|
    if i!=y
      row=[]
      matrix.row_size.times do |j|
        row.push(matrix[i,j]) if j!=y
      end
      rows.push(row)
    end
  end
  rxx=Matrix.rows(rows)
  [rxx,rxy]
end

Public Instance Methods

communalities(m=nil) click to toggle source

Communality for all variables given m factors

# File lib/statsample/factor/principalaxis.rb, line 97
def communalities(m=nil)
  if m!=@m or @clean
    iterate(m)
    raise "Can't calculate comunality" if @communalities.nil?
  end
  @communalities
end
component_matrix(m=nil) click to toggle source

Component matrix for m factors

# File lib/statsample/factor/principalaxis.rb, line 105
def component_matrix(m=nil)
  if m!=@m  or @clean
    iterate(m)
  end
  @component_matrix
end
compute(m=nil) click to toggle source
Alias for: iterate
initial_communalities() click to toggle source
# File lib/statsample/factor/principalaxis.rb, line 154
def initial_communalities
  if @initial_communalities.nil?
    
    if @smc
      # Based on O'Connors(2000)
      @initial_communalities=@matrix.inverse.diagonal.map{|i| 1-(1.quo(i))}
    @initial_communalities=@matrix.column_size.times.collect {|i|          rxx , rxy = PrincipalAxis.separate_matrices(@matrix,i)          matrix=(rxy.t*rxx.inverse*rxy)          matrix[0,0]        }=end
    else
      @initial_communalities=[1.0]*@matrix.column_size
    end
  end      
  @initial_communalities
end
iterate(m=nil) click to toggle source

Iterate to find the factors

# File lib/statsample/factor/principalaxis.rb, line 112
def iterate(m=nil)
  @clean=false
  m||=@m
  @m=m
  t = @max_iterations
  work_matrix=@matrix.to_a
  
  prev_com=initial_communalities
  
  pca=PCA.new(::Matrix.rows(work_matrix))
  @initial_eigenvalues=pca.eigenvalues
  prev_sum=prev_com.inject(0) {|ac,v| ac+v}
  @iterations=0
  t.times do |i|
    "#{@name}: Iteration #{i}" if $DEBUG
    @iterations+=1
    prev_com.each_with_index{|v,it|
      work_matrix[it][it]=v
    }
    pca=PCA.new(::Matrix.rows(work_matrix))
    @communalities=pca.communalities(m)
    @eigenvalues=pca.eigenvalues
    com_sum = @communalities.inject(0) {|ac,v| ac+v}
    #jump=true
    
    break if (com_sum-prev_sum).abs < @delta
    @communalities.each_with_index do |v2,i2|
      raise "Variable #{i2} with communality > 1" if v2>1.0
    end
    prev_sum=com_sum
    prev_com=@communalities
    
  end
  @component_matrix=pca.component_matrix(m)
  @component_matrix.extend CovariateMatrix
  @component_matrix.name=_("Factor Matrix")
  @component_matrix.fields_x = @variables_names
  @component_matrix.fields_y = m.times.map {|i| "factor_#{i+1}"}
  
end
Also aliased as: compute
report_building(generator) click to toggle source
# File lib/statsample/factor/principalaxis.rb, line 197
def report_building(generator)
  iterate if @clean
  generator.section(:name=>@name) do |s|
    s.text _("Number of factors: %d") % m
    s.text _("Iterations: %d") % @iterations
    s.table(:name=>_("Communalities"), :header=>[_("Variable"),_("Initial"),_("Extraction")]) do |t|
      communalities(m).each_with_index {|com,i|
        t.row([@fields[i], sprintf("%0.4f", initial_communalities[i]), sprintf("%0.3f", com)])
      }
    end
    s.table(:name=>_("Total Variance"), :header=>[_("Factor"), _("I.E.Total"), _("I.E. %"), _("I.E.Cum. %"),
    _("S.L.Total"), _("S.L. %"), _("S.L.Cum. %")
      ]) do |t|
    ac_eigen,ac_i_eigen=0,0
      @initial_eigenvalues.each_with_index {|eigenvalue,i|
        ac_i_eigen+=eigenvalue
        ac_eigen+=@eigenvalues[i]
        new_row=[
        _("Factor %d") % (i+1), 
        sprintf("%0.3f",eigenvalue),
        sprintf("%0.3f%%", eigenvalue*100.quo(@n_variables)),
        sprintf("%0.3f",ac_i_eigen*100.quo(@n_variables))
        ]
        if i<@m
          new_row.concat [
            sprintf("%0.3f", @eigenvalues[i]),
            sprintf("%0.3f%%", @eigenvalues[i]*100.quo(@n_variables)),
            sprintf("%0.3f",ac_eigen*100.quo(@n_variables))              
          ]
        else
          new_row.concat ["","",""]
        end
        
        t.row new_row
      }
    end
    s.parse_element(component_matrix)
  end
end

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