module Statsample module Bivariate class Polychoric # Provides statistics for a given combination of rho, alpha and beta and contingence table. class Processor attr_reader :alpha, :beta, :rho, :matrix EPSILON=1e-10 def initialize(alpha,beta,rho,matrix=nil) @alpha=alpha @beta=beta @matrix=matrix @nr=@alpha.size+1 @nc=@beta.size+1 @rho=rho @pd=nil end def bipdf(i,j) Distribution::NormalBivariate.pdf(a(i), b(j), rho) end def loglike rho=@rho if rho.abs>0.9999 rho= (rho>0) ? 0.9999 : -0.9999 end loglike=0 @nr.times do |i| @nc.times do |j| res=pd[i][j]+EPSILON loglike+= @matrix[i,j] * Math::log( res ) end end -loglike end def a(i) raise "Index #{i} should be <= #{@nr-1}" if i>@nr-1 i < 0 ? -100 : (i==@nr-1 ? 100 : alpha[i]) end def b(j) raise "Index #{j} should be <= #{@nc-1}" if j>@nc-1 j < 0 ? -100 : (j==@nc-1 ? 100 : beta[j]) end def eq12(u,v) Distribution::Normal.pdf(u)*Distribution::Normal.cdf((v-rho*u).quo( Math::sqrt(1-rho**2))) end def eq12b(u,v) Distribution::Normal.pdf(v) * Distribution::Normal.cdf((u-rho*v).quo( Math::sqrt(1-rho**2))) end # Equation(8) from Olsson(1979) def fd_loglike_cell_rho(i, j) bipdf(i,j) - bipdf(i-1,j) - bipdf(i, j-1) + bipdf(i-1, j-1) end # Equation(10) from Olsson(1979) def fd_loglike_cell_a(i, j, k) =begin if k==i Distribution::NormalBivariate.pd_cdf_x(a(k),b(j), rho) - Distribution::NormalBivariate.pd_cdf_x(a(k),b(j-1),rho) elsif k==(i-1) -Distribution::NormalBivariate.pd_cdf_x(a(k),b(j),rho) + Distribution::NormalBivariate.pd_cdf_x(a(k),b(j-1),rho) else 0 end =end if k==i eq12(a(k),b(j))-eq12(a(k), b(j-1)) elsif k==(i-1) -eq12(a(k),b(j))+eq12(a(k), b(j-1)) else 0 end end def fd_loglike_cell_b(i, j, m) if m==j eq12b(a(i),b(m))-eq12b(a(i-1),b(m)) elsif m==(j-1) -eq12b(a(i),b(m))+eq12b(a(i-1),b(m)) else 0 end =begin if m==j Distribution::NormalBivariate.pd_cdf_x(a(i),b(m), rho) - Distribution::NormalBivariate.pd_cdf_x(a(i-1),b(m),rho) elsif m==(j-1) -Distribution::NormalBivariate.pd_cdf_x(a(i),b(m),rho) + Distribution::NormalBivariate.pd_cdf_x(a(i-1),b(m),rho) else 0 end =end end # phi_ij for each i and j # Uses equation(4) from Olsson(1979) def pd if @pd.nil? @pd=@nr.times.collect{ [0] * @nc} pc=@nr.times.collect{ [0] * @nc} @nr.times do |i| @nc.times do |j| if i==@nr-1 and j==@nc-1 @pd[i][j]=1.0 else a=(i==@nr-1) ? 100: alpha[i] b=(j==@nc-1) ? 100: beta[j] #puts "a:#{a} b:#{b}" @pd[i][j]=Distribution::NormalBivariate.cdf(a, b, rho) end pc[i][j] = @pd[i][j] @pd[i][j] = @pd[i][j] - pc[i-1][j] if i>0 @pd[i][j] = @pd[i][j] - pc[i][j-1] if j>0 @pd[i][j] = @pd[i][j] + pc[i-1][j-1] if (i>0 and j>0) end end end @pd end # First derivate for rho # Uses equation (9) from Olsson(1979) def fd_loglike_rho rho=@rho if rho.abs>0.9999 rho= (rho>0) ? 0.9999 : -0.9999 end total=0 @nr.times do |i| @nc.times do |j| pi=pd[i][j] + EPSILON total+= (@matrix[i,j].quo(pi)) * (bipdf(i,j)-bipdf(i-1,j)-bipdf(i,j-1)+bipdf(i-1,j-1)) end end total end # First derivative for alpha_k # Uses equation (6) def fd_loglike_a(k) fd_loglike_a_eq6(k) end # Uses equation (6) from Olsson(1979) def fd_loglike_a_eq6(k) rho=@rho if rho.abs>0.9999 rho= (rho>0) ? 0.9999 : -0.9999 end total=0 @nr.times do |i| @nc.times do |j| total+=@matrix[i,j].quo(pd[i][j]+EPSILON) * fd_loglike_cell_a(i,j,k) end end total end # Uses equation(13) from Olsson(1979) def fd_loglike_a_eq13(k) rho=@rho if rho.abs>0.9999 rho= (rho>0) ? 0.9999 : -0.9999 end total=0 a_k=a(k) @nc.times do |j| #puts "j: #{j}" #puts "b #{j} : #{b.call(j)}" #puts "b #{j-1} : #{b.call(j-1)}" e_1=@matrix[k,j].quo(pd[k][j]+EPSILON) - @matrix[k+1,j].quo(pd[k+1][j]+EPSILON) e_2=Distribution::Normal.pdf(a_k) e_3=Distribution::Normal.cdf((b(j)-rho*a_k).quo(Math::sqrt(1-rho**2))) - Distribution::Normal.cdf((b(j-1)-rho*a_k).quo(Math::sqrt(1-rho**2))) #puts "val #{j}: #{e_1} | #{e_2} | #{e_3}" total+= e_1*e_2*e_3 end total end # First derivative for b # Uses equation 6 (Olsson, 1979) def fd_loglike_b_eq6(m) rho=@rho if rho.abs>0.9999 rho= (rho>0) ? 0.9999 : -0.9999 end total=0 @nr.times do |i| @nc.times do |j| total+=@matrix[i,j].quo(pd[i][j]+EPSILON) * fd_loglike_cell_b(i,j,m) end end total end # First derivative for beta_m. # Uses equation 6 (Olsson,1979) def fd_loglike_b(m) fd_loglike_b_eq14(m) end # First derivative for beta_m # Uses equation(14) from Olsson(1979) def fd_loglike_b_eq14(m) rho=@rho if rho.abs>0.9999 rho= (rho>0) ? 0.9999 : -0.9999 end total=0 b_m=b(m) @nr.times do |i| e_1=@matrix[i,m].quo(pd[i][m]+EPSILON) - @matrix[i,m+1].quo(pd[i][m+1]+EPSILON) e_2=Distribution::Normal.pdf(b_m) e_3=Distribution::Normal.cdf((a(i)-rho*b_m).quo(Math::sqrt(1-rho**2))) - Distribution::Normal.cdf((a(i-1)-rho*b_m).quo(Math::sqrt(1-rho**2))) #puts "val #{j}: #{e_1} | #{e_2} | #{e_3}" total+= e_1*e_2*e_3 end total end # Returns the derivative correct according to order def im_function(t,i,j) if t==0 fd_loglike_cell_rho(i,j) elsif t>=1 and t<=@alpha.size fd_loglike_cell_a(i,j,t-1) elsif t>=@alpha.size+1 and t<=(@alpha.size+@beta.size) fd_loglike_cell_b(i,j,t-@alpha.size-1) else raise "incorrect #{t}" end end def information_matrix total_n=@matrix.total_sum vars=@alpha.size+@beta.size+1 matrix=vars.times.map { vars.times.map {0}} vars.times do |m| vars.times do |n| total=0 (@nr-1).times do |i| (@nc-1).times do |j| total+=(1.quo(pd[i][j]+EPSILON)) * im_function(m,i,j) * im_function(n,i,j) end end matrix[m][n]=total_n*total end end m=::Matrix.rows(matrix) end end # Processor end # Polychoric end # Bivariate end # Statsample