# Class: Statsample::Factor::PCA

Inherits:
Object
• Object
show all
Includes:
Summarizable
Defined in:
lib/statsample/factor/pca.rb

## Overview

Principal Component Analysis (PCA) of a covariance or correlation matrix..

NOTE: Sign of second and later eigenvalues could be different using Ruby or GSL, so values for PCs and component matrix should differ, because extendmatrix and gsl's methods to calculate eigenvectors are different. Using R is worse, cause first eigenvector could have negative values! For Principal Axis Analysis, use Statsample::Factor::PrincipalAxis

## Usage:

``````require 'statsample'
a = Daru::Vector.new([2.5, 0.5, 2.2, 1.9, 3.1, 2.3, 2.0, 1.0, 1.5, 1.1])
b = Daru::Vector.new([2.4,0.7,2.9,2.2,3.0,2.7,1.6,1.1,1.6,0.9])
ds = Daru::DataFrame.new({:a => a,:b => b})
cor_matrix = Statsample::Bivariate.correlation_matrix(ds)
pca=  Statsample::Factor::PCA.new(cor_matrix)
pca.m
=> 1
pca.eigenvalues
=> [1.92592927269225, 0.0740707273077545]
pca.component_matrix
=> GSL::Matrix
[  9.813e-01
9.813e-01 ]
pca.communalities
=> [0.962964636346122, 0.962964636346122]
``````

## Instance Attribute Summary collapse

• Number of factors.

• Returns the value of attribute matrix_type.

• Name of analysis.

• Type of rotation.

• Add to the summary a parallel analysis report.

• Add to the summary a rotation report.

• Use GSL if available.

## Instance Method Summary collapse

• Matrix with correlations between components and variables.

• Matrix with correlations between components and variables.

• Array with eigenvalues.

• Feature matrix for `m` factors Returns `m` eigenvectors as columns.

• constructor

A new instance of PCA.

• Returns Principal Components for `input` matrix or dataset The number of PC to return is equal to parameter `m`.

• :nodoc:.

#summary

## Constructor Details

### #initialize(matrix, opts = Hash.new) ⇒ PCA

Returns a new instance of PCA.

 ``` 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87``` ```# File 'lib/statsample/factor/pca.rb', line 54 def initialize(matrix, opts=Hash.new) @use_gsl = opts[:use_gsl] opts.delete :use_gsl @name=_("Principal Component Analysis") @matrix=matrix @n_variables=@matrix.column_size @variables_names=(@matrix.respond_to? :fields) ? @matrix.fields : @n_variables.times.map {|i| "VAR_#{i+1}".to_sym } @matrix_type = @matrix.respond_to?(:_type) ? @matrix._type : :correlation @m=nil @rotation_type=Statsample::Factor::Varimax opts.each{|k,v| self.send("#{k}=",v) if self.respond_to? k } if @use_gsl.nil? @use_gsl=Statsample.has_gsl? end if @matrix.respond_to? :fields @variables_names=@matrix.fields else @variables_names=@n_variables.times.map {|i| "V#{i+1}".to_sym} end calculate_eigenpairs if @m.nil? # Set number of factors with eigenvalues > 1 @m=@eigenpairs.find_all {|ev,ec| ev>=1.0}.size end end```

## Instance Attribute Details

### #m ⇒ Object

Number of factors. Set by default to the number of factors with eigen values > 1

 ``` 44 45 46``` ```# File 'lib/statsample/factor/pca.rb', line 44 def m @m end```

### #matrix_type ⇒ Object

Returns the value of attribute matrix_type.

 ``` 53 54 55``` ```# File 'lib/statsample/factor/pca.rb', line 53 def matrix_type @matrix_type end```

### #name ⇒ Object

Name of analysis

 ``` 40 41 42``` ```# File 'lib/statsample/factor/pca.rb', line 40 def name @name end```

### #rotation_type ⇒ Object

Type of rotation. By default, Statsample::Factor::Rotation::Varimax

 ``` 52 53 54``` ```# File 'lib/statsample/factor/pca.rb', line 52 def rotation_type @rotation_type end```

### #summary_parallel_analysis ⇒ Object

Add to the summary a parallel analysis report

 ``` 50 51 52``` ```# File 'lib/statsample/factor/pca.rb', line 50 def summary_parallel_analysis @summary_parallel_analysis end```

### #summary_rotation ⇒ Object

Add to the summary a rotation report

 ``` 48 49 50``` ```# File 'lib/statsample/factor/pca.rb', line 48 def summary_rotation @summary_rotation end```

### #use_gsl ⇒ Object

Use GSL if available

 ``` 46 47 48``` ```# File 'lib/statsample/factor/pca.rb', line 46 def use_gsl @use_gsl end```

## Instance Method Details

### #communalities(m = nil) ⇒ Object

 ``` 188 189 190 191 192 193 194 195 196 197 198 199``` ```# File 'lib/statsample/factor/pca.rb', line 188 def communalities(m=nil) m||=@m h=[] @n_variables.times do |i| sum=0 m.times do |j| sum += (@eigenpairs[j][0].abs*@eigenpairs[j][1][i]**2) end h.push(sum) end h end```

### #component_matrix(m = nil) ⇒ Object

 ``` 145 146 147 148``` ```# File 'lib/statsample/factor/pca.rb', line 145 def component_matrix(m=nil) var="component_matrix_#{matrix_type}" send(var,m) end```

### #component_matrix_correlation(m = nil) ⇒ Object

Matrix with correlations between components and variables

 ``` 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187``` ```# File 'lib/statsample/factor/pca.rb', line 170 def component_matrix_correlation(m=nil) m||=@m raise "m should be > 0" if m<1 omega_m=::Matrix.build(@n_variables, m) {0} gammas=[] m.times {|i| omega_m.column=i, @eigenpairs[i][1] gammas.push(Math::sqrt(@eigenpairs[i][0])) } gamma_m=::Matrix.diagonal(*gammas) cm=(omega_m*(gamma_m)).to_matrix cm.extend CovariateMatrix cm.name=_("Component matrix") cm.fields_x = @variables_names cm.fields_y = m.times.map { |i| "PC_#{i+1}".to_sym } cm end```

### #component_matrix_covariance(m = nil) ⇒ Object

Matrix with correlations between components and variables. Based on Härdle & Simar (2003, p.243)

 ``` 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167``` ```# File 'lib/statsample/factor/pca.rb', line 151 def component_matrix_covariance(m=nil) m||=@m raise "m should be > 0" if m<1 ff=feature_matrix(m) cm=::Matrix.build(@n_variables, m) {0} @n_variables.times {|i| m.times {|j| cm[i,j]=ff[i,j] * Math.sqrt(eigenvalues[j] / @matrix[i,i]) } } cm.extend NamedMatrix cm.name=_("Component matrix (from covariance)") cm.fields_x = @variables_names cm.fields_y = m.times.map {|i| "PC_#{i+1}".to_sym } cm end```

### #eigenvalues ⇒ Object

Array with eigenvalues

 ``` 201 202 203``` ```# File 'lib/statsample/factor/pca.rb', line 201 def eigenvalues @eigenpairs.collect {|c| c[0] } end```

### #eigenvectors ⇒ Object

 ``` 204 205 206 207 208``` ```# File 'lib/statsample/factor/pca.rb', line 204 def eigenvectors @eigenpairs.collect {|c| @use_gsl ? c[1].to_gsl : Daru::Vector.new(c[1]) } end```

### #feature_matrix(m = nil) ⇒ Object

Feature matrix for `m` factors Returns `m` eigenvectors as columns. So, i=variable, j=component

 ``` 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122``` ```# File 'lib/statsample/factor/pca.rb', line 106 def feature_matrix(m=nil) m||=@m if @use_gsl omega_m=GSL::Matrix.zeros(@n_variables,m) ev=eigenvectors m.times do |i| omega_m.set_column(i,ev[i]) end omega_m else omega_m=::Matrix.build(@n_variables, m) {0} m.times do |i| omega_m.column= i, @eigenpairs[i][1] end omega_m end end```

### #principal_components(input, m = nil) ⇒ Object

Returns Principal Components for `input` matrix or dataset The number of PC to return is equal to parameter `m`. If `m` isn't set, m set to number of PCs selected at object creation. Use covariance matrix

 ``` 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144``` ```# File 'lib/statsample/factor/pca.rb', line 128 def principal_components(input, m=nil) if @use_gsl data_matrix=input.to_gsl else data_matrix=input.to_matrix end m||=@m raise "data matrix variables<>pca variables" if data_matrix.column_size!=@n_variables fv=feature_matrix(m) pcs=(fv.transpose*data_matrix.transpose).transpose pcs.extend Statsample::NamedMatrix pcs.fields_y = m.times.map { |i| "PC_#{i+1}".to_sym } pcs.to_dataframe end```

### #report_building(builder) ⇒ Object

:nodoc:

 ``` 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238``` ```# File 'lib/statsample/factor/pca.rb', line 214 def report_building(builder) # :nodoc: builder.section(:name=>@name) do |generator| generator.text _("Number of factors: %d") % m generator.table(:name=>_("Communalities"), :header=>[_("Variable"),_("Initial"),_("Extraction"), _("%")]) do |t| communalities(m).each_with_index {|com, i| perc=com*100.quo(@matrix[i,i]) t.row([@variables_names[i], "%0.3f" % @matrix[i,i] , "%0.3f" % com, "%0.3f" % perc]) } end te=total_eigenvalues generator.table(:name=>_("Total Variance Explained"), :header=>[_("Component"), _("E.Total"), _("%"), _("Cum. %")]) do |t| ac_eigen=0 eigenvalues.each_with_index {|eigenvalue,i| ac_eigen+=eigenvalue t.row([_("Component %d") % (i+1), sprintf("%0.3f",eigenvalue), sprintf("%0.3f%%", eigenvalue*100.quo(te)), sprintf("%0.3f",ac_eigen*100.quo(te))]) } end generator.parse_element(component_matrix(m)) if (summary_rotation) generator.parse_element(rotation) end end end```

### #rotation ⇒ Object

 ``` 88 89 90``` ```# File 'lib/statsample/factor/pca.rb', line 88 def rotation @rotation_type.new(component_matrix) end```

### #total_eigenvalues ⇒ Object

 ``` 91 92 93``` ```# File 'lib/statsample/factor/pca.rb', line 91 def total_eigenvalues eigenvalues.inject(0) {|ac,v| ac+v} end```