t_icpbf.m

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function [t_out, T_out, error1_out, error2_out, count_out]=t_icpbf(alpha, nu)
% % inverse cumulative probability function, t-distribution
% %
% % Syntax='';
% %
% % [t_out, T_out, error1_out, error2_out, count_out]=t_icpbf(alpha, nu);
% %
% % ***********************************************************
% %
% % Description
% %
% % This program calculates the t-statistic given a level of
% % significance alpha.  alpha can be a vector.  nu must be a constant.
% %
% % In more technical language, this program is the inverse cumulative
% % probability density function for the t-distribution.
% %
% % The accuracy is generally 6 significant digits or more.
% % tol=10^(-8) is hard programmed.
% %
% % This program is an inverse cumulative probability function
% % for the t-distribution for a level of significance alpha and
% % nu degrees of freedom.
% %
% % For alpha less than 0.5, the statistic is negative.
% % For alpha greater than 0.5, the statistic is positive.
% %
% % alpha=0.95 is the default level of significance or in other words
% % cumulative probability.
% %
% % The default value of nu is Inf (infinity) is is approximated by the
% % normal distribution).
% %
% %
% % ***********************************************************
% %
% % Input Variables
% %
% % alpha is the level of significance, cumulative probability
% %       alpha is a constant.
% %
% % nu is the number of degrees of freedom.
% %        nu is a constant.
% %
% % ***********************************************************
% %
% % Output Variables
% %
% % t_out       %  estimated statistic value of the student-t distribution
% %
% % T_out       % estimated cumulative probability function value
% %
% % error1_out  % error estimate of the significance level
% %
% % error2_out  % error estimate of the statistic value
% %
% % count_out   % number of iterations to reach tolerance or fail
% %
% % ***********************************************************
% %
%
% Example='';
% alpha=0.0005;         % level of significance
% nu=1;                 % number of degrees of freedom
%
% [t0005]=t_icpbf(alpha, nu);
%
% t0005exact=-636.6192487;
% Perror0005=100*(1 - t0005/t0005exact);
%
% alpha=0.95;           % level of significance
% nu=Inf;               % number of degrees of freedom (standard normal)
% [t95]=t_icpbf(alpha, nu);
%
% t95exact=1.64485362695;
% Perror95=100*(1 - t95/t95exact);
%
% %
% %
% % ***********************************************************
% %
% %
% % This program was written by Edward L. Zechmann
% %
% %     date  20 January 2008
% %
% %  modified 21 January 2008   updated comments
% %                             added atan and reciprocal
% %                             interpolation methods
% %                             added the offset to the interpolation
% %
% %  modified  9 March   2008   Modifed code to adjust the inerpolated
% %                             value.  Updated comments to reflect changes
% %                             in t_alpha.  Vectorized the program to
% %                             accept alpha as a vector.
% %
% % ***********************************************************
% %
% %
% % Feel free to modify this code.
% %


% The 95 percent confidence interval is the default
if nargin < 1 || isempty(alpha) || ~isnumeric(alpha)
    alpha=0.95;
end

% alpha can be a scalar, vector, or matrix
[m2, n2]=size(alpha);

% The default is the normal distibution
if nargin < 2 || isempty(nu) || ~isnumeric(nu)
    nu=inf;
end

% nu can be a scalar, vector, or matrix
[m1, n1]=size(nu);

if isequal(m1, m2) && isequal(n1, n2)

elseif isequal(length(alpha), length(nu))
    nu=nu';
elseif numel(alpha) > numel(nu)
    % nu must be a constant integer and be >= 1
    nu=round(nu);
    if nu(1,1) < 1
        nu=1;
    end

    nu1=round(nu(1,1));
    nu=nu1*ones(size(alpha));
else
    alpha1=alpha(1,1);
    alpha=alpha1*ones(size(nu));
end

% alpha can be a scalar, vector, or matrix
[m2, n2]=size(alpha);

t_out=zeros(size(alpha));
T_out=zeros(size(alpha));
error1_out=zeros(size(alpha));
error2_out=zeros(size(alpha));
count_out=zeros(size(alpha));

% check special cases of alpha

[set_ix, set_iy]=meshgrid(1:m2, 1:n2);
set_ix=set_ix';
set_iy=set_iy';

[ix, iy]=find( alpha <= 0 );
alpha(ix, iy)=0;
t_out(ix, iy)=-Inf;
T_out(ix, iy)=0;
set_ix(ix, iy)=0;
set_iy(ix, iy)=0;

[ix, iy]=find( alpha >= 1 );
alpha(ix, iy)=1;
t_out(ix, iy)=Inf;
T_out(ix, iy)=1;
set_ix(ix, iy)=0;
set_iy(ix, iy)=0;

[ix, iy]=find( alpha == 0.5 );
alpha(ix, iy)=0.5;
t_out(ix, iy)=0;
T_out(ix, iy)=0.5;
set_ix(ix, iy)=0;
set_iy(ix, iy)=0;


% set the required precision for convergence.
tol=10^(-8);

max_count=100;

for e1=1:m2;
    for e2=1:n2;
        
        ix=set_ix(e1, e2);
        iy=set_iy(e1, e2);

        if ix > 0 && logical(iy > 0)

            alpha2=alpha(ix, iy);
            nu2=nu(ix, iy);
            
            % make a small table so that the initial guess is generally within a factor of 10
            if nu2 < 2
                t=-fliplr(-1+logspace(0, 9, 10));
            elseif logical(nu2 >= 2) && logical(nu2 <= 5)
                t=-fliplr(-1+logspace(0, 3, 10));
            elseif logical(nu2 > 5) && logical(nu2 <= 20)
                t=-fliplr(-1+logspace(0, 2, 10));
            else
                t=-fliplr(-1+logspace(0, 1, 10));
            end

            n11=length(t);
            T=zeros(size(t));

            for e3=1:n11;
                T(e3)=t_alpha(t(e3), nu2);
            end
            
            t=[t -fliplr(t)];
            T=[T fliplr(1-T)];

            % find the two closest values
            [buf ixt]=unique(t);
            [buf ixT]=unique(T);

            ixtT=intersect(ixT, ixt);

            t=t(ixtT);
            T=T(ixtT);

            n11=length(t);

            % Rebuild the arrays for each value of alpha
            t2=1:n11;
            T2=t2;
            t2(1:n11)=t;
            T2(1:n11)=T;

            % set threshold values to switch from one type of interpolation to another
            % set an offset for the inverse tangent interpolation of alpha
            thresh1=0.3;
            thresh2=0.7;
            offset=sqrt(2)*10^(-15)-alpha2;

            if logical(alpha2 < thresh1)
                [X IX]=unique(1./(T2+sqrt(2)*10^(-15)));
                t3=interp1(X, t2(IX), 1./(alpha2+sqrt(2)*10^(-15)));
            elseif logical(alpha2 >= thresh1) && logical(alpha2 <= thresh2)
                [X IX]=unique(atan(T2+offset));
                t3=interp1(X, t2(IX), atan(alpha2+offset));
            else
                [X IX]=unique(-1./(T2-1-sqrt(2)*10^(-15)));
                t3=interp1(X, t2(IX), -1./(alpha2-1-sqrt(2)*10^(-15)));
            end


            t11=t2(find(t2 <= t3, 1, 'last' ));
            t22=t2(find(t2 >= t3, 1 ));

            T3=t_alpha(t3, nu2);
            error1=abs(T3-alpha2);
            error2=max(abs([t3-t11, t22-t3]));

            count=0;
            flag=0;

            if isequal(alpha2, 0.5)
                t3=0;
                T3=05;
                error1=0;
            else

                % % use a while loop to get within the tolerance
                while  (logical(error1 > tol) || logical(error2 > tol)) && (logical( count < max_count) &&  isequal(flag, 0 ))
                    count=count+1;

                    if count > 5
                        interp_type='cubic';
                    else
                        interp_type='linear';
                    end

                    if logical(alpha2 < thresh1)
                        [X IX]=unique(1./(T2+sqrt(2)*10^(-15)));
                        t3=interp1(X, t2(IX), 1./(alpha2+sqrt(2)*10^(-15)), interp_type);
                    elseif logical(alpha2 >= thresh1) && logical(alpha2 <= thresh2)
                        [X IX]=unique(atan(T2+offset));
                        t3=interp1(X, t2(IX), atan(alpha2+offset), interp_type);
                    else
                        [X IX]=unique(-1./(T2-1-sqrt(2)*10^(-15)));
                        t3=interp1(X, t2(IX), -1./(alpha2-1-sqrt(2)*10^(-15)), interp_type);
                    end


                    % find the nearest neighbors of the interpolated value
                    t11=t2(find(t2 <= t3, 1, 'last' ));
                    t22=t2(find(t2 >= t3, 1 ));

                    [error2 ix2]=max(abs([t3-t11, t22-t3]));

                    if ix2 == 1
                        pp=min(abs(abs(t3/t11)-1), 0.3);
                        t33=t3-pp*(t3-t11);
                    else
                        pp=min(abs(abs(t3/t22)-1), 0.3);
                        t33=t3+pp*(t22-t3);
                    end

                    q=t_alpha(t33, nu2);
                    t2(n11+count)=t33;
                    T2(n11+count)=q;

                    [t2 ixt2]=unique(t2);
                    T2=T2(ixt2);
                    [T2 ixt2]=unique(T2);
                    t2=t2(ixt2);
                    T3=q;

                    if logical(q <= 1) && logical(q >= -1)

                        t11=t2(find(t2 <= t3, 1, 'last' ));
                        t22=t2(find(t2 >= t3, 1 ));
                        error1=abs(T3-alpha2);
                        error2=max(abs([t3-t11, t22-t3]));

                    else
                        flag=1;
                    end


                end

                % Final Interpolation
                if isequal(flag, 0 )

                    if count > 5
                        interp_type='cubic';
                    else
                        interp_type='linear';
                    end

                    if logical(alpha2 < thresh1)
                        [X IX]=unique(1./(T2+sqrt(2)*10^(-15)));
                        t3=interp1(X, t2(IX), 1./(alpha2+sqrt(2)*10^(-15)), interp_type);
                    elseif logical(alpha2 >= thresh1) && logical(alpha2 <= thresh2)
                        [X IX]=unique(atan(T2+offset));
                        t3=interp1(X, t2(IX), atan(alpha2+offset), interp_type);
                    else
                        [X IX]=unique(-1./(T2-1-sqrt(2)*10^(-15)));
                        t3=interp1(X, t2(IX), -1./(alpha2-1-sqrt(2)*10^(-15)), interp_type);
                    end

                end
            end

            t_out(ix, iy)=t3;
            T_out(ix, iy)=T3;
            error1_out(ix, iy)=error1;
            error2_out(ix, iy)=error2;
            count_out(ix, iy)=count;
        end
    end
end