 Dear students, in this slide, we are going to study the factor rotations. Routations means rotating. Now we have also done factor loadings. We have now come up with the idea of factor loadings. Now sometimes we have such a situation that when we do not have factorization, there is no exact solution. We do not have a unique solution. So then we do factor rotation. Now you have an idea with the rotation कि बिरात u butter 2 rifles स्तेक के लिएнибудь बिरात बात crooked स्तेक के लिए nuclei स्तेक के लिए cruel स्तेक की साऍब गडन यहše to what the interpretation of the factors is giving. Identify Through the analysis. We are basically doing the rotation of the analysis of the analysis. We are basically doing the transformation of the analysis of the analysis of the analysis of the analysis. When factors are extracted from the data in the factor analysis. Okay. We extracted these factors through the data analysis. They are often correlated with each other अगटिकतार्तश, अगटिकतार्तश तो लिएद तो वी बिटुन्चे off each factor. अप खुरिलेश्यन तो है अजग भीँन, लेकिन हमारे पास उसकी अगटिकतार्तश नहीं कोई भन रहि है। Carefully we can not explain its using. For that, we will further do one. Now we are transforming it into a new set of factors Basically this concept was seen in the principal component. We converted the correlated variables in the principal component. अर निूभ अझब हमारे पास जिन्रेट होगा तरेटा समेल लाली है हम निया केटेख्टेषन लिए हमने चान सब सागता दिया है के वो अ़िळिनला वेरीखलों अहोखोग क्यों लेका आसा थि लोग्घौँस மें क्रनाई करतीया द्या वो म्ंदेर थि खatile लेकानाई किक Meer that right now neighbors are harmonizing of libr are after some transformation, we have converted the correlated variables to uncorrelated variables then our interpretation will be easily done so unfortunately now we are basically looking at it unfortunately for the factor analysis most covariance matrix cannot be factored what we have done is, we have seen everything in the covariance structure we have seen the mathematics in the covariance structure तो उском phrases give उzial along click यरा नान्नॉरrose त त� enjoy any तब भए लателя कि नंमड के फिकटेः। दत वी खली तिन रफ मेद चे चारी पदृरी पद्सर blु연 आप ऑर मैं तबया भसुभகrais को णदिय समथदबर के पैक्टर्स बन जाएंगे, in some cases, the values of the estimates of the lambda ij, अभी आमने इतने detail में देखा के lambda ij क्या है, basically, this is the factor loading, अर साई आई, this is the specific variant, अर नहीं नहीं, are not consistent the statistical requirement, अब अमार पस पैक्टर लोडिंख और आर वेरिंस दिस जु है, ये भी statistical requirement को full fill नहीं करे, under some conditions ये आम ये सारा काम करते न कर सकते है, कोई नहीं कोई प्रोडिंग आईगी, तो फिर हम उसके सलूषिन पे जाते है, the following example of this problem, in such case, the factor rotation is applied, हम इसको अब नमारिकली देखती है, के हम नहीं इसको कैसे अपक्टर रोडिंग करने है, suppose we have three variables and one factor solution, के अपक्टर सलूषिन, k equals to one, it means that one factor solution आई नहीं, हमाने � कि अप ăण और ठोतगलन। वेरिबल एक चींइं से सा खलने आईスफीeté और � preocवर सी। पिर हम नहीं और गर गाए। �ismusंके कोला उपने देखती है नहीं और वो consisting of hands. एक ust हैआए अपक स्तागला गर नहीं कोके लििँ Ocean to extremism. second variable with first factor. तुके one factor solution है ना मारे पास इसली हम इसको one factor solution के according लेरे हैं third variable with first factor तुके factor ना मारे पास देखें तु this is the this is the sigma and here is the sigma this is the numeric value अब आप के पास क्या वालु lambda 11 square plus psi 1 which is equals to one lambda 11 lambda 21.9 lambda 11 lambda 31.7 compared करे ना हम इसको ये तो आप के पास गेवन है हमने उसको कमpear करना है so lambda 21 square plus psi 1 lambda 31 square plus psi 1 lambda 21 into lambda 31.4 तो कमpear कर लिया देन हम नहीं आप के आप देखो हम ने next six equations आप नहीं आप नहीं आप नहीं equations के बाद हम ने further lambda 11 की वालु सेखनी है psi 1 की वालु सेखनी psi basically variance आप के पास lambda 21 की वालुs lambda 31 or psi 1 psi 2 psi 3 हमें जोए है so divide equation 5 by equation 3 we get lambda 21. तो हम ने equation 5 अर equation 3 को fulsalt की आप बाद हम ने lambda 21 lambda 31 which is equals to 0.4 and divided by equation number 3 lambda 11 and lambda 31 which is divided by 0.7 तो यहाप आप के पास लिए के अंसलोट होग या so lambda 21 इसको further हम ने divide की आप तो lambda 11 आप भालु यहां से हमारे पास equation आब के है put in 3 we get अब हम ने इसकी वालु equation 3 में पुट कर दी या equation आप के पास 3 है put in 3 we get the lambda 11 which is equals to the lambda 31 की जगाब में आप देखो 2 1 की जगाब असकी वालू पुट कर दी है this is put in 2 not in 3 put in 2 lambda 11 अब lambda 21 की जगाब में असकी वालू पुट की आप 0.6 lambda 11 which is equals to 0.9 तो further simplification की बात हमारे पास lambda 11 की आप की वालू आप की आप की आप 1 1 square which is equals to 1.5 अं further हम ने square उत लिया तो answer आप आप या है 1.225 अचा एक छीज है की lambda 11 square अद lambda 11 इसकी वालू समारे पास रेंच में आप को कहाँ पता है कहाँ तो कहाँ तो this is the factor loadings आप तो factor loadings की रेंच की आप देए between minus 1 to 1 बहत यहाँ पे हमारे पास आब हमारे पास आब है 1.225 यह तो मारी रेंच से एक सीट कर गया हमें और भी चेख कर लेते है। पर दर मारे पास क्या है equation 1 now look at this equation 1 lambda 11 square पला साई 1 equals to 1 here is the equation 1 अब equation 1 से हम साई 1 की वालू निकालते है साई 1 की वालू नी so which is equals to 1 minus lambda 11 square so 1 minus lambda 11 square की वालू 1.5 और साई का अंसर हमारे पास आगया negative 0.5 साई basically क्या है diagonal एना variances के तो variances आब के पास negative आगया है तो यह तो एकसी नी करत सकती तीके ना variance तो आपको पता है variance तो कभी बी negative नी आसकता अब हमारे पास क्या होगे है यह पे variance पी negative है factor loading जिसका range minus 1 to 1 है उसकी रेंज भी हमारे पास क्या आगया 1.2 to 5 तो यह तो solution इजिस नहीं करना कन्टीशन आगी ना इसी के solution इजिस ना करे तो फिर हमें रूतेशन करनी हुत्ती है which is unsatisfactory since it gives a negative value of variance of EI look at this this is the variance of EI उसकी क्या value आए negative thus for this example with k equals to 1 यह नी फल, factor solution है it is not possible to get a unique solution ती के ना unique solution नहीं आए आसका हमारे पास one factor solution हम ने किया और उनीक ने basically three variables नहीं तो one factor solution नहीं हो सकता है नहीं however the solution is not consistent with the statistical interpretation अब देख लो, statistical interpretation होई नहीं सकती है आप के पास यहांपे you know that the factor analysis model assume that k and the line factors we have p variables and the factor is f1 f2 fk our observed variable with the linear function of this यह हम the factor loadings में भी देखा and here is the factor loadings is the lambda is the factor loading lambda is the factor loading and EI is the specific factor and F is the common factor नहापे यह में से where F is the common factor on a matrix notation we can write as this X vector P into 1 vector P into 1 it means P variables P rows in one column and lambda this is the matrix P into K F this is a vector K into 1 K rows one factor plus P which is plus E which is equals to which is also equals to the vector now further what we have done you have a model so you have model assumptions and its assumptions are expected value of the factor which is equals to 0 covariance of the factor which is equals to identity expected value of the error which is equals to 0 covariance of the error which is equals to psi and psi is the diagonal of variance of EI we have already seen all these assumptions then we have done factor rotation further we are using again we are studying this in such case is the factor loadings are rotated now we have factor loadings we have lambda we have greater value of lambda 11 now we have to rotate it that means we are doing transformation factor loadings are rotated multiplied by orthogonal matrix now we have to multiply with orthogonal matrix to get a unique solution and satisfy the statistical interpretation if you recall we have done this in principle components we did not have a unique solution for unique solution we have orthogonal transformation scheme so we have orthogonal transformation let B be any K into K orthogonal matrix B into B prime which is equals to B prime into B which is equals to identity we have done transformation so that this is the model X lambda F plus EI we know that this is the model can be written as this here now what we have done basically we have done transformation B prime B which is equals to identity so if we multiply or divide the identity then it will not have any effect so we have done that then B into B prime we have introduced now you see here what we have done further we have combined this part then we have combined this part we have said X which is equals to lambda star now it has transformed now lambda is not lambda with lambda B B which is orthogonal matrix B B is involved and F is not single F we have involved B transpose so what you have done is this is called the lambda star and this is the F star look at this this is the lambda star and this is the F star so we have the expected value of expectation then we have variance we have further so what we have done further taking expectation on both sides expectation value of factor we know that the expected value of factor which is equals to zero so put it here this is the zero so you can say that the expected value of F star which is equals to zero and applying covariance on both sides so covariance of F star which is equals to B prime covariance of FB you know that how we write covariance in multivariate because we do not have a square form here you know that we are involved with B prime B so B prime covariance of F into B so covariance of F we know that this is the identity so B prime B which is equals to identity it means that covariance of transformed variable which is also equals to the identity and the expected value of the transformed which is equals to zero so here the variance covariance matrix sigma which is equals to lambda prime plus psi after transformation sigma which is equals to lambda star into lambda star transpose plus psi so here we have variance covariance transformed or rotated we have variance covariance matrix we can return as further we have written this here here we have centered the value after we have centered the value we have simplified it value has been centered after that we have multiplied it so we have got identity or we have said this is the lambda star and here we have lambda star and lambda star prime plus psi this implies that factor F and factor F star have some statistical properties and even though lambda star are a general different from the loading lambda this is obvious because first we had factor loadings which we are calling as lambda capital lambda now we have factor loadings but we have capital lambda star because we have transformed it so now factor loading and factor loading is lambda star it cannot be equal there will be any difference because here we have got transformation they both generate the same covariance matrix sigma what have you said this implies that F have some statistical properties even though this are in general different from the loadings i.e. you have factor loadings and you have here these factor loadings both will be different but both generate the same covariance matrix sigma but when we generate covariance matrix sigma then we will get same result so this basically we have factor rotation after transformation we have seen its exact solution will exist and after that we have seen that factor loadings will be different but their same variance covariance matrix will be generated this is the concept of the factor rotation