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//Caption: Factor Analysis
//Centroid Method
//Example11.2
//Page439
clear;
clc;
X1 = [6,4,4,1,4,4,3,7,5,5];// Fuel Efficiency
X2 = [8,4,1,2,3,4,3,7,3,4];//Life of the two-wheeler
X3 = [9,6,6,6,5,6,0,6,1,2];//Handling convenience
X4 = [9,8,5,3,5,8,9,9,8,3];//Quality of original spare
X5 = [1,2,1,2,2,2,1,9,1,1];//Breakdown rate
X6 = [2,1,2,3,3,3,3,2,2,0];//Price
n = 6;//number of variables
m = length(X1);//number of sets of observations
F = 3;//number of factors to be identified
f = 1;//current factor number
Xcorr1 = [1,0.742,0.168,0.496,0.484,-0.430;0.742,1,0.424,0.568,0.474,-0.204;0.168,0.424,1,0.050,0.238,0.092]
Xcorr2 = [0.496,0.568,0.050,1,0.29,0.196;0.484,0.474,0.238,0.290,1,0.037;-0.430,-0.204,0.092,0.196,0.037,1];
Xcorr = [Xcorr1;Xcorr2]
Xcorr_ref = abs(Xcorr)
disp(Xcorr_ref,'Correlation Coefficient Matrix=')
for j = 1:n
S(j)= sum(Xcorr_ref(:,j))
end
disp(S,'column totals S=')
T = sum(S);
disp(T,'Grand total of column totals T=')
Tsqrt = sqrt(T);
for j = 1:n
L1(j)= S(j)/Tsqrt;
if j ==n then
L1(j)=-L1(j);
end
end
disp(L1,'Loading values of factor1 L1=')
for i = 1:n
for j =1:n
P2(i,j)= L1(i)*L1(j);
R2(i,j) = Xcorr(i,j)-P2(i,j)
if (R2(i,j)<0) then
R2_ref(i,j)= -R2(i,j);
else
R2_ref(i,j)= R2(i,j);
end
end
end
disp(P2,'Cross-Product matrix P2=')
disp(R2,'Residual matrix R2=')
disp(R2_ref,'Reflected residual matrix R2=')
for j = 1:n
S1(j)= sum(R2_ref(:,j))
end
disp(S1,'column totals S1=')
T1 = sum(S1);
disp(T1,'Grand total of column totals T1=')
Tsqrt1 = sqrt(T1);
for j = 1:n
L2(j)= S1(j)/Tsqrt1;
if S1(j)>1 then
L2(j)=-L2(j);
end
end
disp(L2,'Loading values of factor2 L2=')
for i = 1:n
for j =1:n
P3(i,j)= L2(i)*L2(j);
R3(i,j) = R2(i,j)-P3(i,j)
if (R3(i,j)<0) then
R3_ref(i,j)= -R3(i,j);
else
R3_ref(i,j)= R3(i,j);
end
end
end
disp(P3,'Cross-Product matrix P3=')
disp(R3,'Residual matrix R3=')
disp(R3_ref,'Reflected residual matrix R3=')
for j = 1:n
S2(j)= sum(R3_ref(:,j))
end
disp(S2,'column totals S2=')
T2 = sum(S2);
disp(T2,'Grand total of column totals T2=')
Tsqrt2 = sqrt(T2);
for j = 1:n
L3(j)= S2(j)/Tsqrt2;
if S2(j)>0.8 then
L3(j)=-L3(j);
end
end
disp(L3,'Loading values of factor3 L3=')
for i = 1:n
for j = 1:F
h(i) = L1(i)^2+L2(i)^2+L3(i)^2;
end
end
disp(h,'Communality h^2 =')
EigenValueL1 = sum(L1.^2);
EigenValueL2 = sum(L2.^2);
EigenValueL3 = sum(L3.^2);
EigenValueh = sum(h);
disp(EigenValueL1,'Eigen Value of L1 =')
disp(EigenValueL2,'Eigen Value of L2 =')
disp(EigenValueL3,'Eigen Value of L3 =')
disp(EigenValueh,'Eigen Value of h =')
disp([EigenValueL1/n,EigenValueL2/n,EigenValueL3/n],'Proportion of total variance=')
disp([EigenValueL1/EigenValueh,EigenValueL2/EigenValueh,EigenValueL3/EigenValueh],'Proportion of common variance=')
disp(round(EigenValueL1*100/n),'The proportion of total variance of the factor-1=')
disp(round(EigenValueL2*100/n),'The proportion of total variance of the factor-1=')
disp(round(EigenValueL3*100/n),'The proportion of total variance of the factor-1=')
disp(round(EigenValueL1*100/EigenValueh),'The proportion of common variance=')
disp(round(EigenValueL2*100/EigenValueh),'The proportion of common variance=')
disp(round(EigenValueL3*100/EigenValueh),'The proportion of common variance=')
//Result
// Correlation Coefficient Matrix=
//
// 1. 0.742 0.168 0.496 0.484 0.43
// 0.742 1. 0.424 0.568 0.474 0.204
// 0.168 0.424 1. 0.05 0.238 0.092
// 0.496 0.568 0.05 1. 0.29 0.196
// 0.484 0.474 0.238 0.29 1. 0.037
// 0.43 0.204 0.092 0.196 0.037 1.
//
// column totals S=
//
// 3.32
// 3.412
// 1.972
// 2.6
// 2.523
// 1.959
//
// Grand total of column totals T=
//
// 15.786
//
// Loading values of factor1 L1=
//
// 0.8356069
// 0.8587623
// 0.4963304
// 0.6543910
// 0.6350109
// - 0.4930584
//
// Cross-Product matrix P2=
//
// 0.6982389 0.7175877 0.4147371 0.5468136 0.5306195 - 0.4120030
// 0.7175877 0.7374727 0.4262298 0.5619663 0.5453235 - 0.4234200
// 0.4147371 0.4262298 0.2463438 0.3247941 0.3151752 - 0.2447199
// 0.5468136 0.5619663 0.3247941 0.4282275 0.4155454 - 0.3226530
// 0.5306195 0.5453235 0.3151752 0.4155454 0.4032389 - 0.3130975
// - 0.4120030 - 0.4234200 - 0.2447199 - 0.3226530 - 0.3130975 0.2431066
//
// Residual matrix R2=
//
// 0.3017611 0.0244123 - 0.2467371 - 0.0508136 - 0.0466195 - 0.0179970
// 0.0244123 0.2625273 - 0.0022298 0.0060337 - 0.0713235 0.2194200
// - 0.2467371 - 0.0022298 0.7536562 - 0.2747941 - 0.0771752 0.3367199
// - 0.0508136 0.0060337 - 0.2747941 0.5717725 - 0.1255454 0.5186530
// - 0.0466195 - 0.0713235 - 0.0771752 - 0.1255454 0.5967611 0.3500975
// - 0.0179970 0.2194200 0.3367199 0.5186530 0.3500975 0.7568934
//
// Reflected residual matrix R2=
//
// 0.3017611 0.0244123 0.2467371 0.0508136 0.0466195 0.0179970
// 0.0244123 0.2625273 0.0022298 0.0060337 0.0713235 0.2194200
// 0.2467371 0.0022298 0.7536562 0.2747941 0.0771752 0.3367199
// 0.0508136 0.0060337 0.2747941 0.5717725 0.1255454 0.5186530
// 0.0466195 0.0713235 0.0771752 0.1255454 0.5967611 0.3500975
// 0.0179970 0.2194200 0.3367199 0.5186530 0.3500975 0.7568934
//
// column totals S1=
//
// 0.6883406
// 0.5859465
// 1.6913123
// 1.5476123
// 1.2675222
// 2.1997807
//
// Grand total of column totals T1=
//
// 7.9805146
//
// Loading values of factor2 L2=
//
// 0.2436621
// 0.2074161
// - 0.5986988
// - 0.5478312
// - 0.4486835
// - 0.7786888
//
// Cross-Product matrix P3=
//
// 0.0593712 0.0505394 - 0.1458802 - 0.1334857 - 0.1093272 - 0.1897369
// 0.0505394 0.0430215 - 0.1241798 - 0.1136290 - 0.0930642 - 0.1615126
// - 0.1458802 - 0.1241798 0.3584402 0.3279858 0.2686263 0.4662000
// - 0.1334857 - 0.1136290 0.3279858 0.3001190 0.2458028 0.4265900
// - 0.1093272 - 0.0930642 0.2686263 0.2458028 0.2013169 0.3493849
// - 0.1897369 - 0.1615126 0.4662000 0.4265900 0.3493849 0.6063563
//
// Residual matrix R3=
//
// 0.2423899 - 0.0261272 - 0.1008569 0.0826720 0.0627076 0.1717400
// - 0.0261272 0.2195058 0.1219500 0.1196627 0.0217408 0.3809326
// - 0.1008569 0.1219500 0.3952159 - 0.6027800 - 0.3458015 - 0.1294801
// 0.0826720 0.1196627 - 0.6027800 0.2716535 - 0.3713482 0.0920630
// 0.0627076 0.0217408 - 0.3458015 - 0.3713482 0.3954442 0.0007126
// 0.1717400 0.3809326 - 0.1294801 0.0920630 0.0007126 0.1505371
//
// Reflected residual matrix R3=
//
// 0.2423899 0.0261272 0.1008569 0.0826720 0.0627076 0.1717400
// 0.0261272 0.2195058 0.1219500 0.1196627 0.0217408 0.3809326
// 0.1008569 0.1219500 0.3952159 0.6027800 0.3458015 0.1294801
// 0.0826720 0.1196627 0.6027800 0.2716535 0.3713482 0.0920630
// 0.0627076 0.0217408 0.3458015 0.3713482 0.3954442 0.0007126
// 0.1717400 0.3809326 0.1294801 0.0920630 0.0007126 0.1505371
//
// column totals S2=
//
// 0.6864936
// 0.8899191
// 1.6960844
// 1.5401794
// 1.1977549
// 0.9254655
//
// Grand total of column totals T2=
//
// 6.935897
//
// Loading values of factor3 L3=
//
// 0.2606665
// - 0.3379086
// - 0.6440153
// - 0.5848170
// - 0.4547960
// - 0.3514058
//
// Communality h^2 =
//
// 0.8255572
// 0.8946764
// 1.0195397
// 1.0703575
// 0.8113952
// 0.9729489
//
// Eigen Value of L1 =
//
// 2.7566285
//
// Eigen Value of L2 =
//
// 1.568625
//
// Eigen Value of L3 =
//
// 1.2692212
//
// Eigen Value of h =
//
// 5.5944748
//
// Proportion of total variance=
//
// 0.4594381 0.2614375 0.2115369
//
// Proportion of common variance=
//
// 0.4927413 0.2803883 0.2268705
//
// The proportion of total variance of the factor-1=
//
// 46.
//
// The proportion of total variance of the factor-1=
//
// 26.
//
// The proportion of total variance of the factor-1=
//
// 21.
//
// The proportion of common variance=
//
// 49.
//
// The proportion of common variance=
//
// 28.
//
// The proportion of common variance=
//
// 23.
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