CorAna

CorAna performs correspondence analysis

Syntax

Description

example

out =CorAna(N) CorAna with all the default options.

example

out =CorAna(N, Name, Value) CorAna with name pairs.

Examples

expand all

  • CorAna with all the default options.
  •     load smoke
        [N,~,~,labels] =crosstab(smoke.data(:,1),smoke.data(:,2));
        [I,J]=size(N);
        if verLessThan('matlab','8.2.0')==0
        % Contingency table is supplied to CorAna in table format
            Ntable=array2table(N,'RowNames',labels(1:I,1),'VariableNames',labels(1:J,2))
            out=CorAna(Ntable);
        else
            out=CorAna(N);
        end
    

  • CorAna with name pairs.
  • Input is the contingency table, labels for rows and columns are supplied Data are read from the txt file

        load('smoke.txt')
        labels_rows= {'Senior-Managers' 'Junior-Managers' 'Senior-Employees' 'Junior-Employees' 'Secretaries'};
        labels_columns= {'None' 'Light' 'Medium' 'Heavy'};
        N=crosstab(smoke(:,1),smoke(:,2));
        out=CorAna(N,'Lr',labels_rows,'Lc',labels_columns);
    
    Summary
                 Singular_value     Inertia      Accounted_for    Cumulative
                 ______________    __________    _____________    __________
    
        dim_1        0.27342         0.074759        0.87756       0.87756  
        dim_2        0.10009         0.010017        0.11759       0.99515  
        dim_3       0.020337       0.00041357      0.0048547             1  
    
    ROW POINTS
    Results for dimension: 1
                             Scores      CntrbPnt2In    CntrbDim2In
                            _________    ___________    ___________
    
        Senior_Managers     -0.065768     0.0032977      0.092232  
        Junior_Managers       0.25896      0.083659        0.5264  
        Senior_Employees     -0.38059       0.51201       0.99903  
        Junior_Employees      0.23295       0.33097       0.94193  
        Secretaries          -0.20109      0.070064       0.86535  
    
    Results for dimension: 2
                             Scores     CntrbPnt2In    CntrbDim2In
                            ________    ___________    ___________
    
        Senior_Managers     -0.19374       0.21356        0.80034 
        Junior_Managers      -0.2433       0.55115        0.46468 
        Senior_Employees    -0.01066     0.0029976     0.00078372 
        Junior_Employees    0.057744       0.15177       0.057876 
        Secretaries         0.078911      0.080522        0.13326 
    
    COLUMN POINTS
    Results for dimension: 1
                   Scores     CntrbPnt2In    CntrbDim2In
                  ________    ___________    ___________
    
        None      -0.39331        0.654        0.99402  
        Light     0.099456      0.03085        0.32673  
        Medium     0.19632      0.16562        0.98185  
        Heavy      0.29378      0.14954         0.6844  
    
    Results for dimension: 2
                   Scores      CntrbPnt2In    CntrbDim2In
                  _________    ___________    ___________
    
        None      -0.030492      0.029336      0.0059745 
        Light       0.14106       0.46317        0.65729 
        Medium    0.0073591     0.0017368      0.0013796 
        Heavy      -0.19777       0.50575        0.31015 
    
    -----------------------------------------------------------
    Overview ROW POINTS
                              Mass       Score_1     Score_2      Inertia     CntrbPnt2In_1    CntrbPnt2In_2    CntrbDim2In_1    CntrbDim2In_2
                            ________    _________    ________    _________    _____________    _____________    _____________    _____________
    
        Senior_Managers     0.056995    -0.065768    -0.19374    0.0026729      0.0032977          0.21356        0.092232           0.80034  
        Junior_Managers     0.093264      0.25896     -0.2433     0.011881       0.083659          0.55115          0.5264           0.46468  
        Senior_Employees     0.26425     -0.38059    -0.01066     0.038314        0.51201        0.0029976         0.99903        0.00078372  
        Junior_Employees     0.45596      0.23295    0.057744     0.026269        0.33097          0.15177         0.94193          0.057876  
        Secretaries          0.12953     -0.20109    0.078911     0.006053       0.070064         0.080522         0.86535           0.13326  
    
    Overview COLUMN POINTS
                   Mass      Score_1      Score_2      Inertia     CntrbPnt2In_1    CntrbPnt2In_2    CntrbDim2In_1    CntrbDim2In_2
                  _______    ________    _________    _________    _____________    _____________    _____________    _____________
    
        None      0.31606    -0.39331    -0.030492     0.049186         0.654          0.029336         0.99402         0.0059745  
        Light     0.23316    0.099456      0.14106    0.0070588       0.03085           0.46317         0.32673           0.65729  
        Medium    0.32124     0.19632    0.0073591      0.01261       0.16562         0.0017368         0.98185         0.0013796  
        Heavy     0.12953     0.29378     -0.19777     0.016335       0.14954           0.50575          0.6844           0.31015  
    
    -----------------------------------------------------------
    Legend
    Row scores in principal coordinates
    Column scores in principal coordinates
    CntrbPnt2In = relative contribution of points to explain total Inertia of the latent dimension
                  The sum of the numbers in a column is equal to 1
    CntrbDim2In = relative contribution of latent dimension to explain total Inertia of a point
                  CntrbDim2In_1+CntrbDim2In_2+...+CntrbDim2In_K=1
    

    Related Examples

  • CorAna with original data matrix as input.
  •     load smoke
        out=CorAna(smoke.data,'datamatrix',true);
    

  • CorAna with supplementary rows and supplementary columns.
  • Children data Active rows = 1:15 Active columns = 1:5

        N=[51	64	32	29	17	59	66	70;
        53	90	78	75	22	115	117	86;
        71	111	50	40	11	79	88	177;
        1	7	5	5	4	9	8	5;
        7	11	4	3	2	2	17	18;
        7	13	12	11	11	18	19	17;
        21	37	14	26	9	14	34	61;
        12	35	19	6	7	21	30	28;
        10	7	7	3	1	8	12	8;
        4	7	7	6	2	7	6	13;
        8	22	7	10	5	10	27	17;
        25	45	38	38	13	48	59	52;
        18	27	20	19	9	13	29	53;
        35	61	29	14	12	30	63	58;
        2	4	3	1	4	nan  nan	nan	  ;
        2	8	2	5	2	nan  nan	nan;
        1	5	4	6	3	nan  nan	nan;
        3	3	1	3	4	nan  nan	nan];
        % rowslab = cell containing row labels
        rowslab={'money','future','unemployment','circumstances',...
        'hard','economic','egoism','employment','finances',...
        'war','housing','fear','health','work','comfort','disagreement',...
        'world','to_live'};
        % colslab = cell containing column labels
        colslab={'unqualified','cep','bepc','high_school_diploma','university',...
        'thirty','fifty','more_fifty'};
        tableN=array2table(N,'VariableNames',colslab,'RowNames',rowslab);
        % Extract just active rows and active columns
        Nactive=tableN(1:14,1:5);
        % Define tables containing supplementary rows and supplementary cols
        Nsupr=tableN(15:18,1:5);
        Nsupc=tableN(1:14,6:8);
        Sup=struct;
        Sup.r=Nsupr;
        Sup.c=Nsupc;
        out=CorAna(Nactive,'Sup',Sup);
    
    Summary
                 Singular_value     Inertia     Accounted_for    Cumulative
                 ______________    _________    _____________    __________
    
        dim_1        0.18815        0.035402       0.57043        0.57043  
        dim_2        0.11452        0.013115       0.21132        0.78175  
        dim_3       0.085447       0.0073011       0.11764        0.89939  
        dim_4       0.079018       0.0062439       0.10061              1  
    
    ROW POINTS
    Results for dimension: 1
                          Scores      CntrbPnt2In    CntrbDim2In
                         _________    ___________    ___________
    
        money             -0.11527      0.045499        0.42845 
        future             0.17645       0.17567        0.71562 
        unemployment      -0.21223       0.22616        0.87492 
        circumstances      0.40092      0.062745        0.58397 
        hard              -0.24998      0.029938        0.88369 
        economic           0.35396       0.12005        0.48362 
        egoism            0.059889     0.0068096       0.073339 
        employment        -0.13675      0.026215         0.1643 
        finances            -0.237      0.027904        0.27623 
        war                0.21682      0.021688        0.74907 
        housing          -0.006681    4.1183e-05     0.00072894 
        fear               0.20335       0.11666        0.90069 
        health             0.11165      0.020571        0.79911 
        work              -0.21168       0.12005        0.75402 
    
    Results for dimension: 2
                           Scores      CntrbPnt2In    CntrbDim2In
                         __________    ___________    ___________
    
        money             -0.020046     0.0037146       0.012958 
        future             0.097863       0.14587        0.22013 
        unemployment       0.070718      0.067786       0.097145 
        circumstances      -0.33099       0.11544          0.398 
        hard               -0.06765     0.0059184       0.064717 
        economic           -0.32072       0.26604        0.39705 
        egoism             0.025667     0.0033763       0.013471 
        employment         -0.21539       0.17555         0.4076 
        finances            0.20598      0.056902        0.20867 
        war                0.074663     0.0069419       0.088821 
        housing            -0.12824       0.04096        0.26858 
        fear               0.058068      0.025678       0.073446 
        health           -0.0042912    8.2025e-05      0.0011804 
        work               -0.10888      0.085745        0.19951 
    
    COLUMN POINTS
    Results for dimension: 1
                                Scores     CntrbPnt2In    CntrbDim2In
                               ________    ___________    ___________
    
        unqualified            -0.20932       0.2511        0.67619  
        cep                    -0.13858      0.18297        0.64492  
        bepc                    0.10876     0.067579         0.3119  
        high_school_diploma     0.27404      0.37976        0.75817  
        university              0.23123      0.11859        0.31171  
    
    Results for dimension: 2
                                Scores      CntrbPnt2In    CntrbDim2In
                               _________    ___________    ___________
    
        unqualified             0.080727      0.10082        0.10058  
        cep                    -0.056047     0.080794        0.10549  
        bepc                    0.028483     0.012512       0.021393  
        high_school_diploma      0.12134      0.20099        0.14865  
        university              -0.31786      0.60488          0.589  
    
    -----------------------------------------------------------
    Overview ROW POINTS
                           Mass       Score_1      Score_2       Inertia      CntrbPnt2In_1    CntrbPnt2In_2    CntrbDim2In_1    CntrbDim2In_2
                         ________    _________    __________    __________    _____________    _____________    _____________    _____________
    
        money             0.12123     -0.11527     -0.020046     0.0037595       0.045499        0.0037146          0.42845         0.012958  
        future            0.19975      0.17645      0.097863     0.0086904        0.17567          0.14587          0.71562          0.22013  
        unemployment      0.17776     -0.21223      0.070718     0.0091512        0.22616         0.067786          0.87492         0.097145  
        circumstances    0.013819      0.40092      -0.33099     0.0038038       0.062745          0.11544          0.58397            0.398  
        hard              0.01696     -0.24998      -0.06765     0.0011994       0.029938        0.0059184          0.88369         0.064717  
        economic          0.03392      0.35396      -0.32072     0.0087874        0.12005          0.26604          0.48362          0.39705  
        egoism           0.067211     0.059889      0.025667     0.0032871      0.0068096        0.0033763         0.073339         0.013471  
        employment       0.049623     -0.13675      -0.21539     0.0056484       0.026215          0.17555           0.1643           0.4076  
        finances         0.017588       -0.237       0.20598     0.0035763       0.027904         0.056902          0.27623          0.20867  
        war              0.016332      0.21682      0.074663      0.001025       0.021688        0.0069419          0.74907         0.088821  
        housing          0.032663    -0.006681      -0.12824     0.0020001     4.1183e-05          0.04096       0.00072894          0.26858  
        fear             0.099874      0.20335      0.058068     0.0045852        0.11666         0.025678          0.90069         0.073446  
        health           0.058417      0.11165    -0.0042912    0.00091131       0.020571       8.2025e-05          0.79911        0.0011804  
        work             0.094849     -0.21168      -0.10888     0.0056364        0.12005         0.085745          0.75402          0.19951  
    
    Overview COLUMN POINTS
                                 Mass      Score_1      Score_2      Inertia     CntrbPnt2In_1    CntrbPnt2In_2    CntrbDim2In_1    CntrbDim2In_2
                               ________    ________    _________    _________    _____________    _____________    _____________    _____________
    
        unqualified             0.20289    -0.20932     0.080727     0.013146        0.2511          0.10082          0.67619          0.10058   
        cep                     0.33731    -0.13858    -0.056047     0.010044       0.18297         0.080794          0.64492          0.10549   
        bepc                    0.20226     0.10876     0.028483    0.0076704      0.067579         0.012512           0.3119         0.021393   
        high_school_diploma     0.17902     0.27404      0.12134     0.017732       0.37976          0.20099          0.75817          0.14865   
        university             0.078518     0.23123     -0.31786     0.013468       0.11859          0.60488          0.31171            0.589   
    
    -----------------------------------------------------------
    Legend
    Row scores in principal coordinates
    Column scores in principal coordinates
    CntrbPnt2In = relative contribution of points to explain total Inertia of the latent dimension
                  The sum of the numbers in a column is equal to 1
    CntrbDim2In = relative contribution of latent dimension to explain total Inertia of a point
                  CntrbDim2In_1+CntrbDim2In_2+...+CntrbDim2In_K=1
    

    Input Arguments

    expand all

    N — Contingency table (default) or n-by-2 input dataset. Matrix or Table.

    Matrix or table which contains the input contingency table (say of size I-by-J) or the original data matrix.

    In this last case N=crosstab(N(:,1),N(:,2)). As default procedure assumes that the input is a contingency table.

    Data Types: single| double

    Name-Value Pair Arguments

    Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

    Example: 'k',3 , 'Lr',{'a' 'b' 'c'} , 'Lc',{'c1' c2' 'c3' 'c4'} , 'Sup',Sup=struct; Sup.c={'c2' 'c4'} , 'datamatrix',true , 'plots',1 , 'dispresults',false , 'd1',2 , 'd2',3

    k —Number of dimensions to retain.scalar.

    Scalar which contains the number of dimensions to retain.

    The default value of k is 2.

    Example: 'k',3

    Data Types: double

    Lr —Vector of row labels.cell.

    Cell containing the labels of the rows of the input contingency matrix N. This option is unnecessary if N is a table, because in this case Lr=N.Properties.RowNames;

    Example: 'Lr',{'a' 'b' 'c'}

    Data Types: cell array of strings

    Lc —Vector of column labels.cell.

    Cell containing the labels of the columns of the input contingency matrix N. This option is unnecessary if N is a table, because in this case Lc=N.Properties.VariableNames;

    Example: 'Lc',{'c1' c2' 'c3' 'c4'}

    Data Types: cell array of strings

    Sup —Structure containing indexes or names of supplementary rows or columns.structure.

    Structure with the following fields.

    Value Description
    r

    vector containing row indexes or vector of cell array of strings or table or 2D numeric array, containing supplementary rows.

    If indexes or cell array of strings are supplied in a vector, we assume that supplementary rows belong to contingency table N. For example:

    - if Sup.r=[2 5] (that is Sup.r is a numeric vector which contains row indexes) we use rows 2 and 5 of the input contingency table as supplementary rows.

    - if Sup.r={'Junior-Managers' 'Senior-Employees'} (that is Sup.r is a cell array of strings) we use rows named 'Junior-Managers' and 'Senior-Employees' of the input contingency table as supplementary rows. Of course the length of Sup.r must be smaller than the number of rows of the contigencey matrix divided by 2.

    - if Sup.r is a table, or a 2D array supplementary rows do not belong to N. Note that if Sup.r is a table, the labels of the rows are taken directly from the table. If on the other hand Sup.r is a matrix the names of the rows of the supplementary units can be given using Sup.Lr as a cell array of strings.

    Lr

    cell array of strings containing the labels of the supplementary units if Sup.r is a 2D numeric array.

    c

    vector containing column indexes or vector of cell array of strings or table or 2D numeric array use as supplementary columns, or table or 2D numeric array containing supplementary rows.

    If indexes or cell array of strings are supplied in a vector, we assume that supplementary columns belong to contingency table N. For example:

    - if Sup.c=[2 3] (that is Sup.c is a numeric vector which contains column indexes) we use columns 2 and 3 of the input contingency table as supplementary columns.

    - if Sup.c={'Smokers' 'NonSmokers'} (that is Sup.c is a cell array of strings) we use columns of the contingency table labelled 'Smokers' and 'NonSmokers' of the input contingency table N as supplementary columns.

    Of course the length of Sup.c must be smaller than the number of columns of the contigencey matrix divided by 2.

    - If Sup.c is a table, or a 2D array supplementary columns do not belong to N. Note that if Sup.c is a table, the labels of the columns are taken directly from the table. If on the other hand Sup.c is a matrix the names of the columns of the supplementary units can be given using Sup.Lc as a cell array of strings.

    Lc

    cell array of strings containing the labels of the supplementary units if Sup.r is a 2D numeric array.

    REMARK: The default value of Sup is a missing value that is we assume that there are no supplementary rows or columns.

    Example: 'Sup',Sup=struct; Sup.c={'c2' 'c4'}

    Data Types: struct

    datamatrix —Data matrix or contingency table.boolean.

    If datamatrix is true the first input argument N is forced to be interpreted as a data matrix, else if the input argument is false N is treated as a contingency table. The default value of datamatrix is false, that is the procedure automatically considers N as a contingency table

    Example: 'datamatrix',true

    Data Types: logical

    plots —Plot on the screen.scalar | structure.

    If plots = 1, a plot which shows the Principal coordinates of rows and columns is shown on the screen. If plots is a structure it may contain the following fields:

    Value Description
    alpha

    type of plot, scalar in the interval [0 1] or a string identifying the type of coordinates to use in the plot.

    If $plots.alpha='rowprincipal'$ the row points are in principal coordinates and the column coordinates are standard coordinates. Distances between row points are (approximated) chi-squared distances (row-metric-preserving). The position of the row points are at the weighted average of the column points.

    Note that 'rowprincipal' can also be specified setting plots.alpha=1.

    If $plots.alpha='colprincipal'$, the column coordinates are referred to as principal coordinates and the row coordinates as standard coordinates.

    Distances between column points are (approximated) chi-squared distances (column-metric-preserving). The position of the column points are at the weighted average of the row points.

    Note that 'colwprincipal' can also be specified setting plots.alpha=0.

    If $plots.alpha='symbiplot'$, the row and column coordinates are scaled similarly. The sum of weighted squared coordinates for each dimension is equal to the corresponding singular values. These coordinates are often called symmetrical coordinates. This representation is particularly useful if one is primarily interested in the relationships between categories of row and column variables rather than in the distances among rows or among columns. 'symbiplot' can also be specified setting plots.alpha=0.5;

    If $plots.alpha='bothprincipal'$, both the rows and columns are depicted in principal coordinates. Such a plot is often referred to as a symmetrical plot or French symemtrical model. Note that such a symmetrical plot does not provide a feasible solution in the sense that it does not approximate matrix $D_r^{-0.5}(P-rc')D_c^{-0.5}$.

    FontSize

    scalar which specifies the font size of row (column) labels. The default value is 10.

    MarkerSize

    scalar which specifies the marker size of symbols associated with rows or columns. The default value is 10.

    Example: 'plots',1

    Data Types: scalar double | struct

    dispresults —Display results on the screen.boolean.

    If dispresults is true (default) it is possible to see on the screen all the summary results of the analysis.

    Example: 'dispresults',false

    Data Types: Boolean

    d1 —Dimension to show on the horizontal axis.positive integer.

    Positive integer in the range 1, 2, .., K which indicates the dimension to show on the x axis. The default value of d1 is 1.

    Example: 'd1',2

    Data Types: single | double

    d2 —Dimension to show on the vertical axis.positive integer.

    Positive integer in the range 1, 2, .., K which indicates the dimension to show on the y axis. The default value of d2 is 2.

    Example: 'd2',3

    Data Types: single | double

    Output Arguments

    expand all

    out — description Structure

    A structure containing the following fields

    Value Description
    Lr

    cell of length $I$ containing the labels of active rows (i.e. the rows which participated to the fit).

    Lc

    cell of length $J$ containing the labels of active columns (i.e. the columns which participated to the fit).

    N

    $I$-by-$J$-array containing contingency table referred to active rows and active columns (i.e.

    referred to the rows/columns which participated to the fit). The $(i,j)$-th element is equal to $n_{ij}$, $i=1, 2, \ldots, I$ and $j=1, 2, \ldots, J$. The sum of the elements of out.P is $n$ (the grand total).

    Ntable

    Same as out.N but in table format (with row and column names).

    This output is present just if your MATLAB version is not<2013b.

    I

    Number of active rows of contingency table.

    J

    Number of active columns of contingency table.

    n

    Grand total. out.n is equal to sum(sum(out.N)).

    This is the number of observations.

    Nhat

    $I$-by-$J$-array containing contingency table referred to active rows (i.e. referred to the rows which participated to the fit) under the independence hypothesis.

    The $(i,j)$-th element is equal to $n_{i.}n_{.j}/n$, $i=1, 2, \ldots, I$ and $j=1, 2, \ldots, J$. The sum of the elements of out.Nhat is $n$ (the grand total).

    Nhattable

    Same as out.Nhat but in table format (with row and column names).

    P

    $I$-by-$J$-array containing correspondence matrix (proportions). The $(i,j)$-th element is equal to $n_{ij}/n$, $i=1, 2, \ldots, I$ and $j=1, 2, \ldots, J$. The sum of the elements of out.P is 1.

    Ptable

    Same as out.P but in table format (with row and column names).

    This output is present just if your MATLAB version is not<2013b.

    r

    Vector of length $I$ containing row masses.

    \[ r=(f_{1.}, f_{2.}, \ldots, f_{I.})' \] $r$ is also the centroid of column profiles.

    Dr

    Square matrix of size $I$ containing on the diagonal the row masses. This is matrix $D_r$.

    \[ D_r=diag(r) \]

    c

    Vector of length $J$ containing column masses.

    \[ c=(f_{.1}, f_{.2}, \ldots, f_{.J})' \] $c$ is also the centroid of row profiles.

    Dc

    Square matrix of size $J$ containing on the diagonal the column masses. This is matrix $D_c$.

    \[ D_c=diag(c) \]

    ProfilesRows

    $I$-by-$J$-matrix containing row profiles.

    The $i,j$-th element of this matrix is given by $f_{ij}/f_{i.}=n_{ij}/n_{i.}$.

    Written in matrix form:

    \[ ProfilesRows = D_r^{-1} \times P \]

    ProfilesCols

    $I$-by-$J$-matrix containing column profiles.

    The $i,j$-th element of this matrix is given by $f_{ij}/f_{.j}=n_{ij}/n_{.j}$.

    Written in matrix form:

    \[ ProfilesCols = P \times D_c^{-1} \]

    K

    Scalar integer containing the maximum number of dimensions. $K = \min(I-1,J-1)$.

    k

    Scalar integer containing the number of retained dimensions.

    Residuals

    $I$-by-$J$-matrix containing standardized residuals.

    \[ Residuals = D_r^{1/2} (D_r^{-1} P - r c') D_c^{-1/2} = D_r^{-1/2} (P - r c') D_c^{-1/2} \]

    With the singular value decomposition (SVD) we obtain that:

    \[ Residuals = U \Gamma V' \]

    TotalInertia

    Scalar containing total inertia. Total inertia is equal (for example) to the sum of the squares of the elements of matrix out.Residuals.

    Chi2stat

    Scalar containing Chi-square statistic for the contingency table. $Chi2stat= TotalInertia \times n$.

    CramerV

    Scalar containing Cramer's $V$ index.

    \[ V=\sqrt{Chi2stat/(n (\min(I,J)-1))} \] Cramer's index goes between 0 and 1.

    InertiaExplained

    matrix with 4 columnn.

    - First column contains the singular values (the sum of the squared singular values is the total inertia).

    - Second column contains the eigenvalues (the sum of the eigenvalues is the total inertia).

    - Third column contains the variance explained by each latent dimension.

    - Fourth column contains the cumulative variance explained by each dimension.

    RowsPri

    $I$-by-$K$ matrix containing principal coordinates of rows.

    \[ RowsPri = D_r^{-1/2} \times U \times \Gamma; \]

    ColsPri

    $J$-by-$K$ matrix containing Principal coordinates of columns.

    \[ ColsPri = D_c^{-1/2} \times V \times \Gamma; \]

    RowsSta

    $I$-by-$K$ matrix containing standard coordinates of rows.

    \[ RowsSta = RowsPri \times \Gamma^{-1} = D_r^{-1/2} U \Gamma \Gamma^{-1}= D_r^{-1/2} U \]

    ColsSta

    $J$-by-$K$ matrix containing standard coordinates of columns.

    \[ ColsSta = ColsPri \times \Gamma^{-1} = D_c^{-1/2} V \Gamma \Gamma^{-1}= D_c^{-1/2} V \]

    RowsSym

    $I$-by-$K$ matrix containing symmetrical coordinates of rows.

    \[ RowsSym = D_r^{-1/2} \times U \times \Gamma^{1/2} \]

    ColsSym

    $J$-by-$K$ matrix containing symmetrical coordinates of columns.

    \[ ColsSym = D_c^{-1/2} \times V \times \Gamma^{1/2} \]

    Symmetric plot represents the row and column profiles simultaneously in a common space (Bendixen, 2003). In this case, only the distance between row points or the distance between column points can be really interpreted.

    The distance between any row and column items is not meaningful! You can only make a general statements about the observed pattern. In order to interpret the distance between column and row points, the column profiles must be presented in row space or vice-versa. This type of map is called asymmetric biplot.

    InertiaRows

    $I$-by-$2$ matrix containing absolute and relative contribution of each row to total inertia.

    The inertia of a point is the squared distance of point $d_i^2$ to the centroid. The absolute contribution of a point to total inertia is the inertia of the point multiplied by the point mass.

    1st column = absolute contribution of each row to TotalInertia. The sum of values of the first column is equal to TotalInertia;

    2nd column = relative contribution of each row to TotalInertia. The sum of the values of the second column is equal to 1.

    InertiaCols

    $J$-by-$2$ matrix containing absolute and relative contribution of each column to total inertia.

    The inertia of a point is the squared distance of point $d_i^2$ to the centroid. The absolute contribution of a point to total inertia is the inertia of the point multiplied by the point mass.

    1st column = absolute contribution of each column to TotalInertia. The sum of values of the first column is equal to TotalInertia;

    2nd column = relative contribution of each column to TotalInertia. The sum of values of the second column is equal to 1.

    Point2InertiaRows

    $I$-by-$K$ matrix containing relative contributions of rows to inertia of the dimension. The inertia of first latent dimension is given by $\lambda_1=\gamma_{11}^2$. The inertia of second latent dimension is given by $\lambda_2=\gamma_{22}^2$ .... The sum of each column of matrix Point2InertiaRows is equal to 1.

    Remark: the points with the larger value of Point2Inertia are those which contribute the most to the definition of the dimension. If the row contributions were uniform, the expected value would be 1/size(contingeny_table,1) For a given dimension, any row with a contribution larger than this threshold could be considered as important in contributing to that dimension.

    Point2InertiaCols

    $J$-by-$K$ matrix containing relative contributions of columns to inertia of the dimension. The sum of each column of matrix Point2InertiaCols is equal to 1.

    Dim2InertiaRows

    $I$-by-$K$ matrix containing relative contributions of latent dimensions to inertia of the row points. These numbers can be interpreted as squared correlations and measures the degree of association between row points and a particular axis. The sum of each row of matrix Dim2InertiaRows is equal to 1.

    Dim2InertiaCols

    $J$-by-$K$ matrix containing relative contributions of latent dimensions to inertia of the column points. These numbers can be interpreted as squared correlations and measure the degree of association between columns points and a particular axis. The sum of each row of matrix Dim2InertiaCols is equal to 1.

    cumsumDim2InertiaRows

    $I$-by-$K$ matrix containing cumulative sum of the contributions of latent dimensions to inertia of the row points. These cumulative sums are equivalent to the communalities in PCA.

    The last column of matrix cumsumDim2InertiaRows is equal to 1.

    cumsumDim2InertiaCols

    $J$-by-$K$ matrix containing cumulative sum of the contributions of latent dimensions to inertia of the column points. These cumulative sums are equivalent to the communalities in PCA.

    The last column of matrix cumsumDim2InertiaCols is equal to 1.

    sqrtDim2InertiaRows

    $I$-by-$K$ matrix containing correlation of rows points with latent dimension axes. Similar to component loadings in PCA

    sqrtDim2InertiaCols

    $I$-by-$K$ matrix containing correlation of column points with latent dimension axes. Similar to component loadings in PCA.

    Summary

    $K$-times-4 table containing summary results for correpondence analysis.

    First column contains the singular values (the sum of the squared singular values is the total inertia).

    Second column contains the eigenvalues (the sum of the eigenvalues is the total inertia).

    Third column contains the variance explained by each latent dimension. Fourth column contains the cumulative variance explained by each dimension.

    This output is present just if your MATLAB version is not<2013b.

    OverviewRows

    $I$-times-(k*3+2) table containing an overview of row points. More precisely, if we suppose that $k=2$, First column contains the row masses (vector $r$).

    Second column contains the scores of first dimension.

    Third column contains the scores of second dimension.

    Fourth column contains the inertia of each point, where inertia of point is the squared distance of point $d_i^2$ to the centroid.

    Fifth column contains the relative contribution of each point to the explanation of the inertia of the first dimension. The sum of the elements of this column is equal to 1.

    Sixth column contains the relative contribution of each point to the explanation of the inertia of the second dimension. The sum of the elements of this column is equal to 1.

    Seventh column contains the relative contribution of the first dimension to the explanation of the inertia of the point.

    Eight column contains the relative contribution of the second dimension to the explanation of the inertia of the point.

    OverviewCols

    $J$-times-(k*3+2) table containing an overview of row points. More precisely if we suppose that $k=2$ First column contains the column masses (vector $c$).

    Second column contains the scores of first dimension.

    Third column contains the scores of second dimension.

    Fourth column contains the inertia of each point, where inertia of point is the squared distance of point $d_i^2$ to the centroid.

    Fifth column contains the relative contribution of each point to the explanation of the inertia of the first dimension. The sum of the elements of this column is equal to 1.

    Sixth column contains the relative contribution of each point to the explanation of the inertia of the second dimension. The sum of the elements of this column is equal to 1.

    Seventh column contains the relative contribution of the first dimension to the explanation of the inertia of the point.

    Eight column contains the relative contribution of the second dimension to the explanation of the inertia of the point.

    This output is present just if your MATLAB version is not<2013b.

    LrSup

    cell containing the labels of the supplementary rows (i.e. the rows whicg did not participate to the fit).

    LcSup

    cell containing the labels of supplementary columns (i.e. the columns which did not participate to the fit).

    SupRowsN

    matrix of size length(LrSup)-by-c referred to supplementary rows. If there are no supplementary rows this field is not present.

    SupRowsNtable

    Same as out.SupRowsN but in table format (with row and column names). This is the contingency table referred to supplementary rows. If there are no supplementary rows this field is not present.

    This output is present just if your MATLAB version is not<2013b.

    SupColsN

    matlab of size r-by-length(LcSup) referred to supplementary columns.

    If there are no supplementary columns this field is not present.

    SupColsNtable

    Same as out.SupColsN but in table format (with row and column names). This is the contingency table referred to supplementary columns.

    If there are no supplementary columns this field is not present.

    This output is present just if your MATLAB version is not<2013b.

    RowsPriSup

    Principal coordinates of supplementary rows.

    If there are no supplementary rows this field is not present.

    RowsStaSup

    Standard coordinates of supplementary rows.

    If there are no supplementary rows this field is not present.

    RowsSymSup

    Symmetrical coordinates of supplementary rows.

    If there are no supplementary rows this field is not present.

    ColsPriSup

    Principal coordinates of supplementary columns.

    If there are no supplementary columns this field is not present.

    ColsStaSup

    Standard coordinates of of supplementary columns.

    If there are no supplementary columns this field is not present.

    ColsSymSup

    Symmetrical coordinates of supplementary columns.

    If there are no supplementary columns this field is not present.

    References

    Benzecri, J.-P. (1992), Correspondence Analysis Handbook, New-York, Dekker.

    Benzecri, J.-P. (1980), L'analyse des donnees tome 2: l'analyse des correspondances, Paris, Bordas.

    Greenacre, M.J. (1993), Correspondence Analysis in Practice, London, Academic Press.

    Gabriel, K.R. and Odoroff, C. (1990), Biplots in biomedical research, Statistics in Medicine, 9, pp. 469-485.

    Greenacre, M.J. (1993), Biplots in correspondence Analysis, Journal of Applied Statistics, 20, pp. 251 - 269.

    Urbano L.-S., van de Velden M., Kiers H.A.L. (2009), CAR: A MATLAB Package to Compute Correspondence Analysis with Rotations, Journal of Statistical Software, Volume 31, Issue 8.

    Acknowledgements

    This function has been inspired by the code developed by:

    Urbano Lorenzo-Seva (Rovira i Virgili University, Tarragona, Spain), Michel van de Velden (Erasmus University, Rotterdam, The Netherlands), and Henk A.L. Kiers (University of Groningen, Groningen, The Netherlands) (See References).

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