Module: Statsample::Bivariate

Defined in:
lib/statsample/bivariate.rb,
lib/statsample/bivariate/pearson.rb

Overview

Diverse methods and classes to calculate bivariate relations Specific classes:

  • Statsample::Bivariate::Pearson : Pearson correlation coefficient ®

  • Statsample::Bivariate::Tetrachoric : Tetrachoric correlation

  • Statsample::Bivariate::Polychoric : Polychoric correlation (using joint, two-step and polychoric series)

Defined Under Namespace

Classes: Pearson

Class Method Summary collapse

Class Method Details

.correlation_matrix(ds) ⇒ Object

Correlation matrix. Order of rows and columns depends on Dataset#fields order


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# File 'lib/statsample/bivariate.rb', line 199

def correlation_matrix(ds)
  vars, cases = ds.ncols, ds.nrows
  if !ds.has_missing_data? and Statsample.has_gsl? and prediction_optimized(vars,cases) < prediction_pairwise(vars,cases)
    cm=correlation_matrix_optimized(ds)
  else
    cm=correlation_matrix_pairwise(ds)
  end
  cm.extend(Statsample::CovariateMatrix)
  cm.fields = ds.vectors.to_a
  cm
end

.correlation_matrix_optimized(ds) ⇒ Object


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# File 'lib/statsample/bivariate.rb', line 211

def correlation_matrix_optimized(ds)
  s=covariance_matrix_optimized(ds)
  sds=GSL::Matrix.diagonal(s.diagonal.sqrt.pow(-1))
  cm=sds*s*sds
  # Fix diagonal
  s.row_size.times {|i|
    cm[i,i]=1.0
  }
  cm
end

.correlation_matrix_pairwise(ds) ⇒ Object


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# File 'lib/statsample/bivariate.rb', line 221

def correlation_matrix_pairwise(ds)
  cache={}
  vectors = ds.vectors.to_a
  cm = vectors.collect do |row|
    vectors.collect do |col|
      if row==col
        1.0
      elsif (ds[row].type!=:numeric or ds[col].type!=:numeric)
        nil
      else
        if cache[[col,row]].nil?
          r=pearson(ds[row],ds[col])
          cache[[row,col]]=r
          r
        else
          cache[[col,row]]
        end 
      end
    end
  end

  Matrix.rows cm
end

.correlation_probability_matrix(ds, tails = :both) ⇒ Object

Matrix of correlation probabilities. Order of rows and columns depends on Dataset#fields order


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# File 'lib/statsample/bivariate.rb', line 265

def correlation_probability_matrix(ds, tails=:both)
  rows=ds.fields.collect do |row|
    ds.fields.collect do |col|
      v1a,v2a=Statsample.only_valid_clone(ds[row],ds[col])
      (row==col or ds[row].type!=:numeric or ds[col].type!=:numeric) ? nil : prop_pearson(t_pearson(ds[row],ds[col]), v1a.size, tails)
    end
  end
  Matrix.rows(rows)
end

.covariance(v1, v2) ⇒ Object

Covariance between two vectors


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# File 'lib/statsample/bivariate.rb', line 13

def covariance(v1,v2)
  v1a,v2a=Statsample.only_valid_clone(v1,v2)

  return nil if v1a.size==0
  if Statsample.has_gsl?
    GSL::Stats::covariance(v1a.to_gsl, v2a.to_gsl)
  else
    covariance_slow(v1a,v2a)
  end
end

.covariance_matrix(ds) ⇒ Object

Covariance matrix. Order of rows and columns depends on Dataset#fields order


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# File 'lib/statsample/bivariate.rb', line 160

def covariance_matrix(ds)
  vars,cases = ds.ncols, ds.nrows
  if !ds.has_missing_data? and Statsample.has_gsl? and prediction_optimized(vars,cases) < prediction_pairwise(vars,cases)
    cm=covariance_matrix_optimized(ds)
  else
    cm=covariance_matrix_pairwise(ds)
  end
  cm.extend(Statsample::CovariateMatrix)
  cm.fields = ds.vectors.to_a
  cm
end

.covariance_matrix_optimized(ds) ⇒ Object


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# File 'lib/statsample/bivariate.rb', line 146

def covariance_matrix_optimized(ds)
  x=ds.to_gsl
  n=x.row_size
  m=x.column_size
  means=((1/n.to_f)*GSL::Matrix.ones(1,n)*x).row(0)
  centered=x-(GSL::Matrix.ones(n,m)*GSL::Matrix.diag(means))
  ss=centered.transpose*centered
  s=((1/(n-1).to_f))*ss
  s
end

.covariance_matrix_pairwise(ds) ⇒ Object


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# File 'lib/statsample/bivariate.rb', line 173

def covariance_matrix_pairwise(ds)
  cache={}
  vectors = ds.vectors.to_a
  mat_rows = vectors.collect do |row|
    vectors.collect do |col|
      if (ds[row].type!=:numeric or ds[col].type!=:numeric)
        nil
      elsif row==col
        ds[row].variance
      else
        if cache[[col,row]].nil?
          cov=covariance(ds[row],ds[col])
          cache[[row,col]]=cov
          cov
        else
          cache[[col,row]]
        end
      end
    end
  end
  
  Matrix.rows mat_rows
end

.covariance_slow(v1, v2) ⇒ Object

:nodoc:


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# File 'lib/statsample/bivariate.rb', line 33

def covariance_slow(v1,v2) # :nodoc:
  v1a,v2a=Statsample.only_valid(v1,v2)
  sum_of_squares(v1a,v2a) / (v1a.size-1)
end

.gamma(matrix) ⇒ Object

Calculates Goodman and Kruskal's gamma.

Gamma is the surplus of concordant pairs over discordant pairs, as a percentage of all pairs ignoring ties.

Source: faculty.chass.ncsu.edu/garson/PA765/assocordinal.htm


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# File 'lib/statsample/bivariate.rb', line 323

def gamma(matrix)
  v=pairs(matrix)
  (v['P']-v['Q']).to_f / (v['P']+v['Q']).to_f
end

.maximum_likehood_dichotomic(pred, real) ⇒ Object

Estimate the ML between two dichotomic vectors


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# File 'lib/statsample/bivariate.rb', line 24

def maximum_likehood_dichotomic(pred,real)
  preda,reala=Statsample.only_valid_clone(pred,real)                
  sum=0
  preda.each_index{|i|
     sum+=(reala[i]*Math::log(preda[i])) + ((1-reala[i])*Math::log(1-preda[i]))
  }
  sum
end

.min_n_valid(ds) ⇒ Object

Report the minimum number of cases valid of a covariate matrix based on a dataset


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# File 'lib/statsample/bivariate.rb', line 392

def min_n_valid(ds)
  min = ds.nrows
  m   = n_valid_matrix(ds)
  for x in 0...m.row_size
    for y in 0...m.column_size
      min=m[x,y] if m[x,y] < min
    end
  end
  min
end

.n_valid_matrix(ds) ⇒ Object

Retrieves the n valid pairwise.


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# File 'lib/statsample/bivariate.rb', line 246

def n_valid_matrix(ds)
  vectors = ds.vectors.to_a
  m = vectors.collect do |row|
    vectors.collect do |col|
      if row==col
        ds[row].only_valid.size
      else
        rowa,rowb = Statsample.only_valid_clone(ds[row],ds[col])
        rowa.size
      end
    end
  end

  Matrix.rows m
end

.ordered_pairs(vector) ⇒ Object


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# File 'lib/statsample/bivariate.rb', line 370

def ordered_pairs(vector)
  d = vector.to_a
  a = []
  (0...(d.size-1)).each do |i|
    ((i+1)...(d.size)).each do |j|
      a.push([d[i],d[j]])
    end
  end
  a
end

.pairs(matrix) ⇒ Object

Calculate indexes for a matrix the rows and cols has to be ordered


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# File 'lib/statsample/bivariate.rb', line 328

def pairs(matrix)
  # calculate concordant #p matrix
  rs=matrix.row_size
  cs=matrix.column_size
  conc=disc=ties_x=ties_y=0
  (0...(rs-1)).each do |x|
    (0...(cs-1)).each do |y|
      ((x+1)...rs).each do |x2|
        ((y+1)...cs).each do |y2|
          # #p sprintf("%d:%d,%d:%d",x,y,x2,y2)
          conc+=matrix[x,y]*matrix[x2,y2]
        end
      end
    end
  end
  (0...(rs-1)).each {|x|
    (1...(cs)).each{|y|
      ((x+1)...rs).each{|x2|
        (0...y).each{|y2|
          # #p sprintf("%d:%d,%d:%d",x,y,x2,y2)
          disc+=matrix[x,y]*matrix[x2,y2]
        }
      }
    }
  }
  (0...(rs-1)).each {|x|
    (0...(cs)).each{|y|
      ((x+1)...(rs)).each{|x2|
        ties_x+=matrix[x,y]*matrix[x2,y]
      }
    }
  }
  (0...rs).each {|x|
    (0...(cs-1)).each{|y|
      ((y+1)...(cs)).each{|y2|
        ties_y+=matrix[x,y]*matrix[x,y2]
      }
    }
  }
  {'P'=>conc,'Q'=>disc,'Y'=>ties_y,'X'=>ties_x}
end

.partial_correlation(v1, v2, control) ⇒ Object

Correlation between v1 and v2, controling the effect of control on both.


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# File 'lib/statsample/bivariate.rb', line 138

def partial_correlation(v1,v2,control)
  v1a,v2a,cona=Statsample.only_valid_clone(v1,v2,control)
  rv1v2=pearson(v1a,v2a)
  rv1con=pearson(v1a,cona)
  rv2con=pearson(v2a,cona)        
  (rv1v2-(rv1con*rv2con)).quo(Math::sqrt(1-rv1con**2) * Math::sqrt(1-rv2con**2))
end

.pearson(v1, v2) ⇒ Object Also known as: correlation

Calculate Pearson correlation coefficient ® between 2 vectors


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# File 'lib/statsample/bivariate.rb', line 46

def pearson(v1,v2)
  v1a,v2a=Statsample.only_valid_clone(v1,v2)
  return nil if v1a.size ==0
  if Statsample.has_gsl?
    GSL::Stats::correlation(v1a.to_gsl, v2a.to_gsl)
  else
    pearson_slow(v1a,v2a)
  end
end

.pearson_slow(v1, v2) ⇒ Object

:nodoc:


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# File 'lib/statsample/bivariate.rb', line 55

def pearson_slow(v1,v2) # :nodoc:
  v1a,v2a=Statsample.only_valid_clone(v1,v2)

  # Calculate sum of squares
  ss=sum_of_squares(v1a,v2a)
  ss.quo(Math::sqrt(v1a.sum_of_squares) * Math::sqrt(v2a.sum_of_squares))
end

.point_biserial(dichotomous, continous) ⇒ Object

Calculate Point biserial correlation. Equal to Pearson correlation, with one dichotomous value replaced by “0” and the other by “1”

Raises:

  • (TypeError)

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# File 'lib/statsample/bivariate.rb', line 283

def point_biserial(dichotomous,continous)
  ds = Daru::DataFrame.new({:d=>dichotomous,:c=>continous}).dup_only_valid
  raise(TypeError, "First vector should be dichotomous") if ds[:d].factors.size != 2
  raise(TypeError, "Second vector should be continous") if ds[:c].type != :numeric
  f0=ds[:d].factors.sort.to_a[0]
  m0=ds.filter_vector(:c) {|c| c[:d] == f0}
  m1=ds.filter_vector(:c) {|c| c[:d] != f0}
  ((m1.mean-m0.mean).to_f / ds[:c].sdp) * Math::sqrt(m0.size*m1.size.to_f / ds.nrows**2)
end

.prediction_optimized(vars, cases) ⇒ Object

Predicted time for optimized correlation matrix, in miliseconds See benchmarks/correlation_matrix.rb to see mode of calculation


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# File 'lib/statsample/bivariate.rb', line 115

def prediction_optimized(vars,cases)
  ((4+0.018128*cases+0.246871*vars+0.001169*vars*cases)**2) / 100
end

.prediction_pairwise(vars, cases) ⇒ Object

Predicted time for pairwise correlation matrix, in miliseconds See benchmarks/correlation_matrix.rb to see mode of calculation


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# File 'lib/statsample/bivariate.rb', line 109

def prediction_pairwise(vars,cases)
  ((-0.518111-0.000746*cases+1.235608*vars+0.000740*cases*vars)**2) / 100
end

.prop_pearson(t, size, tails = :both) ⇒ Object

Retrieves the probability value (a la SPSS) for a given t, size and number of tails. Uses a second parameter

  • :both or 2 : for r!=0 (default)

  • :right, :positive or 1 : for r > 0

  • :left, :negative : for r < 0


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# File 'lib/statsample/bivariate.rb', line 87

def prop_pearson(t, size, tails=:both)
  tails=:both if tails==2
  tails=:right if tails==1 or tails==:positive
  tails=:left if tails==:negative
  
  n_tails=case tails
    when :both then 2
    else 1
  end
  t=-t if t>0 and (tails==:both)
  cdf=Distribution::T.cdf(t, size-2)
  if(tails==:right)
    1.0-(cdf*n_tails)
  else
    cdf*n_tails
  end
end

.residuals(from, del) ⇒ Object

Returns residual score after delete variance from another variable


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# File 'lib/statsample/bivariate.rb', line 121

def residuals(from,del)
  r=Statsample::Bivariate.pearson(from,del)
  froms, dels = from.vector_standarized, del.vector_standarized
  nv=[]
  froms.reset_index!
  dels.reset_index!
  froms.each_index do |i|
    if froms[i].nil? or dels[i].nil?
      nv.push(nil)
    else
      nv.push(froms[i]-r*dels[i])
    end
  end
  Daru::Vector.new(nv)
end

.spearman(v1, v2) ⇒ Object

Spearman ranked correlation coefficient (rho) between 2 vectors


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# File 'lib/statsample/bivariate.rb', line 276

def spearman(v1,v2)
  v1a,v2a = Statsample.only_valid_clone(v1,v2)
  v1r,v2r = v1a.ranked, v2a.ranked
  pearson(v1r,v2r)
end

.sum_of_squares(v1, v2) ⇒ Object


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# File 'lib/statsample/bivariate.rb', line 37

def sum_of_squares(v1,v2)
  v1a,v2a=Statsample.only_valid_clone(v1,v2)
  v1a.reset_index!
  v2a.reset_index!        
  m1=v1a.mean
  m2=v2a.mean
  (v1a.size).times.inject(0) {|ac,i| ac+(v1a[i]-m1)*(v2a[i]-m2)}
end

.t_pearson(v1, v2) ⇒ Object

Retrieves the value for t test for a pearson correlation between two vectors to test the null hipothesis of r=0


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# File 'lib/statsample/bivariate.rb', line 65

def t_pearson(v1,v2)
  v1a,v2a=Statsample.only_valid_clone(v1,v2)
  r=pearson(v1a,v2a)
  if(r==1.0) 
    0
  else
    t_r(r,v1a.size)
  end
end

.t_r(r, size) ⇒ Object

Retrieves the value for t test for a pearson correlation giving r and vector size Source : faculty.chass.ncsu.edu/garson/PA765/correl.htm


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# File 'lib/statsample/bivariate.rb', line 77

def t_r(r,size)
  r * Math::sqrt(((size)-2).to_f / (1 - r**2))
end

.tau_a(v1, v2) ⇒ Object

Kendall Rank Correlation Coefficient (Tau a) Based on Hervé Adbi article


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# File 'lib/statsample/bivariate.rb', line 294

def tau_a(v1,v2)
  v1a,v2a=Statsample.only_valid_clone(v1,v2)
  n=v1.size
  v1r,v2r=v1a.ranked,v2a.ranked
  o1=ordered_pairs(v1r)
  o2=ordered_pairs(v2r)
  delta= o1.size*2-(o2  & o1).size*2
  1-(delta * 2 / (n*(n-1)).to_f)
end

.tau_b(matrix) ⇒ Object

Calculates Goodman and Kruskal’s Tau b correlation. Tb is an asymmetric P-R-E measure of association for nominal scales (Mielke, X)

Tau-b defines perfect association as strict monotonicity. Although it requires strict monotonicity to reach 1.0, it does not penalize ties as much as some other measures.

Reference

Mielke, P. GOODMAN–KRUSKAL TAU AND GAMMA. Source: faculty.chass.ncsu.edu/garson/PA765/assocordinal.htm


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# File 'lib/statsample/bivariate.rb', line 313

def tau_b(matrix)
  v=pairs(matrix)
  ((v['P']-v['Q']).to_f / Math::sqrt((v['P']+v['Q']+v['Y'])*(v['P']+v['Q']+v['X'])).to_f)
end