 शी kunfriends 對 is the covariance structure of the orthogonal factor modulus to please the orthogonal factor model with K common factor here it is the model X which is equal to the lambda F plus E अगर प्रीविय स्यादो दिस अस कोल थे एक्वेष्यनामभा वान कादा दास से थी मडल वेर एइ एस दी आयत श्पस्टपेक पक्तर, आप य य ते कोमन पक्तर लेम्टा एस दी लोडिंच लोडिंच of आयत वेरिबल on the कायत प्कटर, लम्दास is the loading of the i-th variable on the k-th factor and f, these are the factors, उसकी त्रान्सपोस लिये f1, f2 up to so on fk and here is the error e1, e2 up to so on ep. So, this is the structure of the lambda, this is the capital lambda अद या आपके पास, mattresses आगे लम्दास लिये. From the assumption of factor analysis model, we can see that, previous अभी हमने देखाता, assumptions अभाई रेपास लिया है, expected value of f equals to 0, यान इपक्तर की इपक्तेशन ले, which is equals to 0, factor की covariance which is equals to identity, error की अपक्तेशन इपक्तेशन इपक्तेशन equals to 0, error की covariance which is equals to psi and the psi which is equals to diagonal of ei. These are the assumptions of the factor analysis model, and the covariance of ef prime, which is equals to covariance of fe prime, both are the equal. This is the factor analysis model. अप्कतेशन लिया हमने किया की आपक्तेशन ले लिया थराउस लिया, तो तराउस भो लिया मने अपक्तेश लिया है, then x into x transpose, or tognell आपके पास तो उरतानट लिया एपक्तेशन लिया this is the taking expectation on both side and applying the expectation inside the bracket apply कर लेम्दा आपके पास कोंस्तें lambda expectation से हमने आपके पास कोंस्तें then expected value apply होगी f into f prime factors कर लेम्दा आपके पास कोंस्तें lambda prime as it is then आमने expectation apply की FE पे then expectation apply होगी EF पे and after that the expectation apply होगी E E prime कुके variable तो basically E आपके पास expected value X X prime and you know that F F prime which is equals to identity the expected value of F F prime which is equals to identity कुके covariance की से एक वल है covariance of F which is equals to identity यहां से आप उसकी expectation इगालो the covariance basically क्या होता आपके पास journal definition covariance आपके पास basically क्या होगे covariance of X अम लेए लेए तो expected value of X i के लिए for expected value of X j minus expected value of X i into expected value of X j this is the covariance value of i-th and the j-th unit so यहांपे इस केस में अगर में इसको covariance of F लेए लूँ so expected value of आपके पास F F prime आगर यहांपे minus expected value of F into expected value of F prime so expected value of F F prime की वालिए हमारे पास one covariance और यहांपे आपके पास basically क्या होगे है expected value of F zero expected value of F prime यहांपे पूट करो which is also equals to zero and the covariance of F F prime की वालिए होगे है की identity वो हम दे वालिए हम भे पूट कर लिए है expected value of F F prime identity so expected value of E which is equals to zero expected value again E prime equals to zero and the expected value of E E prime which is equals to variance and variance which is equals to psi now यहांपे पास यहांपे पास these two values eliminate होगी because zero multiplies any value which is equals to zero so expected value of X X prime हमाई देपास यह रगे है so expected value of X X prime which is equals to sigma कुके यह कोवेर्यंस की एक बलेए because the remaining values of covariance की minus expected value of X into expected value of X prime हो हमाई देपास पास फैखतर zero तो हमाई देपास खोवेर्यंस ताम क्या होगे expected value of X X prime which is equals to sigma कुके यामने लिक हाए, underscore value of x, zero as equals over L. आँ दे एकाशे नम्बो वान इसको या का? गूय इटियाक्तर एकाई खूव आप आप आप पच्षवरीन्त उसको इक आप इटिया किसे की है. आत्यान दे उ स्च्च्मा र आच्चिक कुरियटिग and which is equal to this मुन्ते इसकी जगा में। वोड़े पोड़े की ज़गा यहाप लिक ती। कोवेरिरिंच तो तो कोवेरिंच बच्या से एकपोल से ज़ग वर्ध़र खोवेरिल़श आंग और इंचटब अद्बच्या एक्ख आप अब हम नहीं किसकी लईव पाईडनी आप यहाप यहाप यहाप तो आमाडे पास इक्टेड वाल्यो ऐक्ट ऐक्ट प्राएंग, यह नी ये वाली वालिो किस की एक्ट बल ओगी, लेंदा अईटेंटेटिगी. तो विज्ट यह दीख्ट इक्टट बल तो को वेरियंस अप एक्ट एक्ट अप, हेर इस दीख्ट अब तो, यह वाली वाली प्राएंटा यह वाली वाली प्राएंटा यह और आमाडेग है, तू पी तुः, यह सब को में संवब करें तू सम में लिक है साम आम, हम ने आम एक नूतेषन लिए हैं कुए ब कोस अईच अल रहा हैं तू हम ने अदा नूतेषन चेआने कहाँने गे रहा है आम देरी स थू वान थृ भी लेम्दा आी आम् सक्यर उस Bloody... इसnięक म pronounced की अब चाहे. नहीं हुँओ vessels एक शाप्चडंताँन पर ड़ है मून उब फोर of these basic Montgomery hacerlo. पर आन्रई सिथ का तो एक आनक एक अवरुव हूँओ पर पी ड़ान. उन्उवरूँए की जोत् ہیں का बस्टी अदन्त् peloward other. this is the equation number 3 also covariance between xi and xj is given as this this is the general formula of the covariance as I said before the covariance which is equals to this and you know that the expected value of xi and the expected value of xj which is equals to 0 further expected value of xi xj which is equals to this expected value of xi xj which is equals to the covariance of the xi xj so we have taken the value of expected value of xi xj so xi in the model you know xi is equal to this is the xi and this is the xj for ith unit we have taken first then we have taken for the jth unit now its multiplication lambda i1 f1 multiplied by lambda j1 f1 so ff multiplied by f1 square so the lambda i1 lambda j1 further lambda i2 f2 lambda j2 f2 f2 f2 f2 square and we have further lambda i2 lambda j plus cross product terms will be in the cross product terms this multiplied by this value these are the cross product terms so cross product terms we have done plus as it is cross product of other terms now we have compiled it now we are simplifying it so the expected value will be applied on the factors expected value will be applied plus e c p t means cross product terms here we have the cross product terms now the expected value of this expected value of error do you know that it is equal to one expected value of error is one so the square of this is equal to one so what happened from here lambda i1 lambda j1 lambda i2 lambda j2 upto so on lambda i k lambda j k now we have sum up in this sum up m varies 1 to k again we have added m in the notation m varies 1 to k lambda i m lambda j m i m means for one lambda i1 lambda j1 so we have sum up this which is equal to lambda i m lambda j m and previous we have seen which is equal to the covariance now the covariance because we have covariance of x i x which is equal to expected value of i x i x j its value we have entered here is the equation number 4 this is the covariance it is seen that the variance of x i previous we have seen this partition into the portion of variance contributed by the k common factor this is the variance portion we have you have factor loading plus common factor specific factor which we are saying the portion of the variance contributed by the k common factor this is the portion called the i th communality and the portion of variance due to the specific variance this is called the psi i so psi i basically we have specific variance so variance of x i basically two parts combination the first part is the i th communality which we have all these and the second part is the specific variance specific variance in which we have psi i values these are the specific variance so variance is the combination of these two and here is the equation this is the covariance structure of the orthogonal factor model sigma which is equal to this so variance we have two combination here is the covariance matrix so this is basically we have seen covariance structure and this equation will be used in a lot of places its advantage and disadvantage we will check further and here is the importance of the factor analysis for two reasons now we have seen the correlation matrix sigma we determined the factor explain the of diagonal term of sigma namely covariance exactly size of this diagonal now understand it what is this basically you have this sigma what we have determined i have written further now in this sigma you have which factor these are the diagonal values because in this diagonal you have these variances those variances are added in this factor so the effect of off diagonal is coming what is it explaining the factor explain the of diagonal term of sigma namely covariance exactly size of this diagonal what is the size of this diagonal just diagonal values of diagonal values 0 it means the effect of diagonal is coming on this factor means the effect of diagonal is coming on this quantity of diagonal is not coming because of off diagonal values we have 0 it also establish that finding the factor loading what is the factor loading that is the lambda is essentially equal to the factorization the covariance of A is the particular case with the addition of the condition of the diagonal elements of size non-negative given a particular sigma because diagonal are coming they are coming with variances what do you have with variances non-negative after factorization if we do that suppose we have a variance negative it means that the solution will not exist because the effect of variance is coming which is a rare case but sometimes this thing exists to exist and if it does it is unique so we have to see when we have this effect of variances a negative is coming then how we will handle it further for that we will study it further then we go to the rotation site if there are any such values if we have such condition then we have to do rotation which means we will transform it the total parameter we have estimated the number of factor loading namely kp and here is the kp plus the number of communal variances p now there half p into p plus 1 separate variance and covariance in sigma so the equation corresponding the elements of the matrices on both side which is equals to this a further matter past care total parameter of the estimated number of factor that is kp and further we have half p into p plus 1 we have half p into p plus 1 equations will come to us how many equations total we have p into p plus 1 we generally require the number of parameters to be less than the number of equations that we have its basic condition that number of parameter will be less number of equations total number of equations which is equals to this total number of equations we have and condition which we generally check number of parameters we have kp plus something is less than the number of the equation so here is the plus p and further we can write it as p comma k plus 1 we can write it as p k plus p which is less than the number of equation further we have taken p comma after taking p comma it will cancel out so here it will be p into k plus 1 less than equals to p p plus 1 so p and p cancel out so k plus 1 half of p plus 1 further we have simplified it on the side of k equality here we have 1 by 2 p plus 1 minus 1 k less than 1 by 2 we are in total common so if p plus 1 minus 2 so what you have is p plus 1 minus 2 you have p minus 1 p minus 1 so this is the condition where k is less than half p minus 1 in other words k should be fairly small correspond with p but it does not guarantee that a solution will exist we have this condition but it is not guaranteed that solution will exist if it is not exist then further we will solve it thanks for watching