mcorran2.m

这是一个matlab版的基础响应分析功能文件

function mcorran2(X)
%MCORRAN2 Multiple Correspondence Analysis Based on the Burt Matrix.
% Statistics fundamentals of the Correspondence Analysis (CA) is presented
% in the CORRAN and MCORRAN1 m-files you can find in this FEX author''s page. 
% CA can be extended to more than two categorical variables, called Multiple 
% Correspondence Analysis (MCA). CA and MCA are graphical techniques for
% representing the information in a two-way or higher-order multiway
% contingency table. They contain the counts (frequencies) of items for a
% cross-classifications of the categorical variables (Rencher, 2000).
%
% Karl Pearson (1913) developed the antecedent of CA used by Procter&Gamble
% (Horst 1935). R.A. Fisher (1940) named the approach 'reciprocal averaging'
% because is reciprocally averages row and column percents in table data
% until they are reconciled. Since reciprocal averaging was inefficient,
% Europeans such as Mosaier (1946) and Benzecri (1969) related table data
% with computer programs for principal component (factor) analysis. Burt 
% (1953) developed MCA (homogeneity analysis) of a binary indicator.
%
% Here, MCA is applied to the Burt matrix (B), the matrix of all two-way
% cross-tabulations of the categorical variables. The Burt matrix has a
% square block on the diagonal for each variable (the frequencies for the
% categories in the corresponding variable) and a rectangular block 
% off-diagonal for each pair of variables (a two-way contingency table for
% the corresponding pair of variables). In the dual eigenanalysis or
% Singular Value Decomposition (SVD) we get the squares of the singular
% values, or principal inertias. 
%
% The so-called 'percentage of inertia problem' can be improved by using 
% adjusted inertias procedure or eigenvalue correction. The adjusted inertias
% are calculated only for each singular value that satisfies the inequality
% >= 1/number of variables. They are expressed as a percentage of the average
% off-diagonal inertia, which can be calculated either by direct calculation
% on the off-diagonal tables in the Burt matrix. The adjusted solution not
% only does it considerably improve the measure of fit, but it also removes
% the inconsistency about the Burt matrix to analyse. This inconsistency is
% due to artificial dimensions added because one categorical variable is 
% coded with several columns. As a consequence, the inertia (i.e., variance)
% of the solution space is artificially inflated and therefore the percentage
% of inertia explained by the first dimension is severely underestimated.
%
% A complete statistics fundamentals explanation is found on Greenacre (2006).
%
% A MCA yields only rows or columns coordinates and each point represents a
% category (attribute) of one of the variables.
%
% Syntax: function mcorran2(X) 
%      
%     Input:
%          X - Data matrix=Burt matrix. Size: categorical variables x 
%              categorical variables (>2).
%     Outputs:
%          Complete Multiple Correspondence Analysis
%          By default the adjusted inertias table is given
%          Pair-wise Dimensions Plots. For the vertical and horizonal lines
%              we use the hline.m and vline.m files kindly published on FEX
%              by Brandon Kuczenski [http://www.mathworks.com/matlabcentral/
%              fileexchange/1039]. For connecting lines to the originwe use
%              the plot2org published on FEX by Jos
%              [http://www.mathworks.com/matlabcentral/fileexchange/11337]
%
%  Example: From the table 2.2 (Greenacre, 2006) we have the data on attitudes
%  to science and the environment from the International Social Survey Program
%  on environment (ISSP), showing the complete Burt matrix of all pairwise 
%  cross-tables of the four variables. Each variable with five categories. The
%  number of respondents were 871. 
% ----------------------------------------------------------------------------------    
%        Variable A          Variable B          Variable C          Variable D
% ----------------------------------------------------------------------------------    
%    A1  A2  A3  A4  A5  B1  B2  B3  B4  B5  C1  C2  C3  C4  C5  D1  D2  D3  D4  D5
% ----------------------------------------------------------------------------------    
% A1 119  0   0   0   0  27  28  30  22  12  49  40  18   7   5  15  25  17  34  28
% A2  0  322  0   0   0  38  74  84  96  30  67  142 60  41  12  22  102 76  68  54
% A3  0   0  204  0   0   3  48  63  73  17  18  75  70  34   7  10  44  68  58  24
% A4  0   0   0  178  0   3  21  23  79  52  16  50  40  56  16   9  52  28  54  35
% A5  0   0   0   0  48   0   3   5  11  29   2   9   9  16  12   4   9  13  12  10
% B1 27  38   3   3   0  71   0   0   0   0  43  19   4   3   2   9  17  10  10  25
% B2 28  74  48  21   3   0  174  0   0   0  36  88  34  15   1  16  51  42  45  20
% B3 30  84  63  23   5   0   0  205  0   0  37  90  57  19   2  10  53  63  51  28
% B4 22  96  73  79  11   0   0   0  281  0  27  88  75  74  17   6  66  70  92  47
% B5 12  30  17  52  29   0   0   0   0  140  9  31  27  43  30  19  45  17  28  31
% C1 49  67  18  16   2  43  36  37  27   9  152  0   0   0   0  25  24  15  38  50
% C2 40  142 75  50   9  19  88  90  88  31   0  316  0   0   0  15  97  67  89  48
% C3 18  60  70  40   9   4  34  57  75  27   0   0  197  0   0   5  51  83  41  17
% C4  7  41  34  56  16   3  15  19  74  43   0   0   0  154  0   6  44  30  51  23
% C5  5  12   7  16  12   2   1   2  17  30   0   0   0   0  52   9  16   7   7  13
% D1 15  22  10   9   4   9  16  10   6  19  25  15   5   6   9  60   0   0   0   0
% D2 25  102 44  52   9  17  51  53  66  45  24  97  51  44  16   0  232  0   0   0
% D3 17  76  68  28  13  10  42  63  70  17  15  67  83  30   7   0   0  202  0   0
% D4 34  68  58  54  12  10  45  51  92  28  38  89  41  51   7   0   0   0  226  0
% D5 28  54  24  35  10  25  20  28  47  31  50  48  17  23  13   0   0   0   0  151
% ----------------------------------------------------------------------------------    
%
%  Data matrix must be:
%     X=[119 0 0 0 0 27 28 30 22 12 49 40 18 7 5 15 25 17 34 28;
%     0 322 0 0 0 38 74 84 96 30 67 142 60 41 12 22 102 76 68 54;
%     0 0 204 0 0 3 48 63 73 17 18 75 70 34 7 10 44 68 58 24;
%     0 0 0 178 0 3 21 23 79 52 16 50 40 56 16 9 52 28 54 35;
%     0 0 0 0 48 0 3 5 11 29 2 9 9 16 12 4 9 13 12 10;
%     27 38 3 3 0 71 0 0 0 0 43 19 4 3 2 9 17 10 10 25;
%     28 74 48 21 3 0 174 0 0 0 36 88 34 15 1 16 51 42 45 20;
%     30 84 63 23 5 0 0 205 0 0 37 90 57 19 2 10 53 63 51 28;
%     22 96 73 79 11 0 0 0 281 0 27 88 75 74 17 6 66 70 92 47;
%     12 30 17 52 29 0 0 0 0 140 9 31 27 43 30 19 45 17 28 31;
%     49 67 18 16 2 43 36 37 27 9 152 0 0 0 0 25 24 15 38 50;
%     40 142 75 50 9 19 88 90 88 31 0 316 0 0 0 15 97 67 89 48;
%     18 60 70 40 9 4 34 57 75 27 0 0 197 0 0 5 51 83 41 17;
%     7 41 34 56 16 3 15 19 74 43 0 0 0 154 0 6 44 30 51 23;
%     5 12 7 16 12 2 1 2 17 30 0 0 0 0 52 9 16 7 7 13;
%     15 22 10 9 4 9 16 10 6 19 25 15 5 6 9 60 0 0 0 0;
%     25 102 44 52 9 17 51 53 66 45 24 97 51 44 16 0 232 0 0 0;
%     17 76 68 28 13 10 42 63 70 17 15 67 83 30 7 0 0 202 0 0;
%     34 68 58 54 12 10 45 51 92 28 38 89 41 51 7 0 0 0 226 0;
%     28 54 24 35 10 25 20 28 47 31 50 48 17 23 13 0 0 0 0 151];
%
%  Calling on Matlab the function: 
%             mcorran2(X)
%  Answer is:
%
%  Give me the vector of categories of each variable. Please, use [ ]: [5 5 5 5]
% 
%  Inertias of the Multiple Correspondence Analysis
%  for the Burt matrix given.
%  ----------------------------------------
%   Eigenvalue     Percent     Cummulative
%                 (Inertia)      Percent
%  ----------------------------------------
%     0.2092        18.55         18.55
%     0.1857        16.47         35.02
%     0.1036         9.19         44.21
%     0.0939         8.33         52.54
%     0.0760         6.74         59.28
%     0.0635         5.63         64.91
%     0.0588         5.22         70.12
%     0.0552         4.90         75.02
%     0.0508         4.51         79.53
%     0.0487         4.32         83.84
%     0.0440         3.90         87.75
%     0.0389         3.45         91.20
%     0.0316         2.81         94.00
%     0.0286         2.54         96.54
%     0.0234         2.07         98.61
%     0.0157         1.39        100.00
%  ----------------------------------------
%    Total         100.00
%  Variable categories = 5  5  5  5
%  ----------------------------------------
% 
%  Adjusted inertias of the Multiple Correspondence
%  Analysis for the Burt matrix given.
%  ----------------------------------------
%   Eigenvalue     Percent     Cummulative
%                 (Inertia)      Percent
%  ----------------------------------------
%    0.07646        44.91         44.91
%    0.05822        34.20         79.11
%    0.00920         5.40         84.51
%    0.00567         3.33         87.84
%    0.00117         0.69         88.53
%    0.00001         0.00         88.53
%  ----------------------------------------
%    Total          88.53
%  Variable categories = 5  5  5  5
%  ----------------------------------------
% 
%  Are you interested to get the dimensions plots? (y/n): y
%  --------------
%  The pair-wise plots you can get are: 120
%  --------------
%  Dm =
%      1     2
%      1     3
%      1     4
%      .     .
%      .     .
%      1    15
%      1    16
%      2     3
%      2     4
%      .     .
%      .     .
%     13    15
%     13    16
%     14    15
%     14    16
%     15    16
%  --------------
%  Give me the interested dimensions to plot. Please, use [a b]:[1 2]
%