PCA.zip_pca_pca降维_主成分分析

function [A,B] = kmo(X) %KMO Kaiser-Meyer-Olkin Measure of Sampling Adequacy. % Factor analysis can be used as a guide to how coherently a set of variables % relate to a hypothesized underlying dimension that they are all being used % to measure. External validity analysis assesses whether the scale that has % been constructed performs as theoretically expected in correlation with % other variables to which it is expected to be related. % There are some assumptions about the characteristics of factors that are % extracted and defined that are unobserved common dimensions that may be % listed to account for the correlations among observed variables. Sampling % adequacy predicts if data are likely to factor well, based on correlation % and partial correlation. Is used to assess which variables to drop from the % model because they are too multicollinear. % It has been suggested that inv(R) should be a near-diagonal matrix in order % to successfully fit a factor analysis model. To assess how close inv(R) % is to a diagonal matrix, Kaiser (1970) proposed a measure of sampling % adequacy, now called KMO (Kaiser-Meyer-Olkin) index. The common part, called % the image of a variable, is defined as that part which is predictable by % regressing each variable on all other variables. % The anti-image is the specific part of the variable that cannot be predicted. % Examining the anti-image of the correlation matrix. That is the negative of the % partial correlations, partialling out all other variables. % There is a KMO statistic for each individual variable and their sum is % the overall statistic. If it is not > 0.6 drop the indicator variables with % the lowest individual statistic value until the overall one rises above 0.6: % factors which is meritorious. The diagonal elements on the Anti-image % correlation matrix are the KMO individual statistics for each variable. A KMO % index <= 0.5 indicates the correlation matrix is not suitable for factor % analysis. % % Syntax: function kmo(X) % % Input: % X - Input matrix can be a data matrix (size n-data x p-variables) % Output(s): % - Kaiser-Meyer-Olkin Index. % - Degree of Common Variance Report (shared by a set of variables % and thus assesses the degree to which they measure a common % underlying factor). % optional(s): % - Anti-image Covariance Matrix. % - Anti-image Correlation Matrix % % Example: From the example given on the web page % http://www.ncl.ac.uk/iss/statistics/docs/factoranalysis.html % We are interested to calculate the Kaiser-Meyer-Olkin measure of sampling % adequacy in order to see if proceeds a satisfactory factor analysis to % investigate the reasons why customers buy a product such as a particular % brand of soft drink (e.g. coca cola). Several variables were identified % which influence customer to buy coca cola. Some of the variables identified % as being influential include availability of product (X1), cost of product % (X2), experience with product (X3), popularity of product (X4), prestige % attached to product (X5), quality of product (X6), quantity of product (X7), % and respectability of product (X8). From this, you designed a questionnaire % to solicit customers' view on a seven point scale, where 1 = not important % and 7 = very important. The results from your questionnaire are show on the % table below. Only the first twelve respondents (cases) are used in this % example. % % Table 1: Customer survey % -------------------------------------------------- % X1 X2 X3 X4 X5 X6 X7 X8 % -------------------------------------------------- % 4 1 4 5 2 3 6 7 % 4 2 7 6 6 3 3 4 % 6 4 3 4 2 5 7 7 % 5 3 5 4 3 4 6 7 % 5 2 4 5 2 5 5 6 % 6 3 3 5 4 4 7 7 % 6 2 4 4 4 3 4 5 % 4 1 3 4 3 3 5 6 % 5 3 4 3 4 3 6 6 % 5 4 3 4 4 4 6 7 % 6 2 4 4 4 3 7 5 % 5 2 3 3 3 3 7 6 % -------------------------------------------------- % % Data matrix must be: % X=[4 1 4 5 2 3 6 7;4 2 7 6 6 3 3 4;6 4 3 4 2 5 7 7;5 3 5 4 3 4 6 7; % 5 2 4 5 2 5 5 6;6 3 3 5 4 4 7 7;6 2 4 4 4 3 4 5;4 1 3 4 3 3 5 6; % 5 3 4 3 4 3 6 6;5 4 3 4 4 4 6 7;6 2 4 4 4 3 7 5;5 2 3 3 3 3 7 6]; % % Calling on Matlab the function: % kmo(X) % % Answer is: % % Kaiser-Meyer-Olkin Measure of Sampling Adequacy: 0.4172 % The KMO test yields a degree of common variance unacceptable (Don't Factor). % % % Created by A. Trujillo-Ortiz, R. Hernandez-Walls, A. Castro-Perez, % K. Barba-Rojo and A. Otero-Limon % Facultad de Ciencias Marinas % Universidad Autonoma de Baja California % Apdo. Postal 453 % Ensenada, Baja California % Mexico. % [email protected] % % Copyright. October 10, 2006. % % To cite this file, this would be an appropriate format: % Trujillo-Ortiz, A., R. Hernandez-Walls, A. Castro-Perez, K. Barba-Rojo % and A. Otero-Limon (2006). kmo:Kaiser-Meyer-Olkin Measure of Sampling % Adequacy. A MATLAB file. [WWW document]. URL http://www.mathworks.com/ % matlabcentral/fileexchange/loadFile.do?objectId=12736 % % References: % Rencher, A. C. (2002), Methods of Multivariate Analysis. 2nd. ed. % New-Jersey:John Wiley & Sons. Chapter 13 (pp. 408-450). % error(nargchk(1,1,nargin)); msg = nargoutchk(1, 2, nargout); X = corrcoef(X); iX = inv(X); S2 = diag(diag((iX.^-1))); AIS = S2*iX*S2; %anti-image covariance matrix IS = X+AIS-2*S2; %image covariance matrix Dai = diag(diag(sqrt(AIS))); IR = inv(Dai)*IS*inv(Dai); %image correlation matrix AIR = inv(Dai)*AIS*inv(Dai); %anti-image correlation matrix a = sum((AIR - diag(diag(AIR))).^2); AA = sum(a); b = sum((X - eye(size(X))).^2); BB = sum(b); MSA = b./(b+a); %measures of sampling adequacy AIR = AIR-eye(size(AIR))+diag(MSA); %Examine the anti-image of the correlation matrix. That is the negative of the partial correlations, %partialling out all other variables. N = BB; D = AA+BB; kmo = N/D; disp(' ') fprintf('Kaiser-Meyer-Olkin Measure of Sampling Adequacy: %3.4f\n', kmo); if (kmo >= 0.00 && kmo < 0.50); disp('The KMO test yields a degree of common variance unacceptable (Don''t Factor).') elseif (kmo >= 0.50 && kmo < 0.60); disp('The KMO test yields a degree of common variance miserable.') elseif (kmo >= 0.60 && kmo < 0.70); disp('The KMO test yields a degree of common variance mediocre.') elseif (kmo >= 0.70 && kmo < 0.80); disp('The KMO test yields a degree of common variance middling.') elseif (kmo >= 0.80 && kmo < 0.90); disp('The KMO test yields a degree of common variance meritorious.') else (kmo >= 0.90 && kmo <= 1.00); disp('The KMO test yields a degree of common variance marvelous.') end if nargout == 1; disp(' ') disp('A = Anti-image covariance matrix.'); A = AIS; elseif nargout > 1; disp(' ') disp('A = Anti-image covariance matrix.'); A = AIS; disp('B = Anti-image correlation matrix.'); B = AIR; end return,