5. PROPAGATION OF VARIANCES APPLIED TO LEAST SQUARES ADJUSTMENT OF INDIRECT OBSERVATIONS., the inverse of the normal equation coefficient matrix is Q

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1 RMI University 5. PROPAGAION OF VARIANCES APPLIED O LEAS SQUARES ADJUSMEN OF INDIREC OBSERVAIONS A most important outcome of a least squares adjustment is that estimates of the precisions of the quantities sought, the elements of, the unknowns or the parameters, are easily obtained from the matri equations of the solution. Application of the Law of Propagation of Variances demonstrates that equal to the cofactor matri N, the inverse of the normal equation coefficient matri is Q that contains estimates of the variances and covariances of the elements of. In addition, estimates of the precisions of the residuals and adjusted observations may be obtained. his most useful outcome enables a statistical analysis of the results of a least squares adjustment and provides the practitioner with a degree of confidence in the results Cofactor matrices for adjustment of indirect observations he observation equations for adjustment of indirect observations is given by v + B f (5.1) f is an (n,1) vector of numeric terms derived from the (n,1) vector of observations l and the (n,1) vector of constants d as f d l (5.) Associated with the vector of observations l is a variance-covariance matri as we as a cofactor matri Q and a weight matri W Q. Remember that in most practical applications of least squares, the matri is unknown, but estimated a priori by Q that contains estimates of the variances and covariances and σ Q where σ is the reference variance or variance factor. Note: In the derivations that foow, the subscript "" is dropped from Q and W If equation (5.) is written as f ( I) l + d (5.3) 5, R.E. Deakin Notes on Least Squares (5) 5 1

2 RMI University then (5.3) is in a form suitable for employing the Law of Propagation of Variances developed in Chapter 3; i.e., if y A + b and y and are random variables linearly related and b is a vector of constants then Q yy AQ A. Hence, the cofactor matri of the numeric terms f is Q ff I Q I Q hus the cofactor matri of f is also the a priori cofactor matri of the observations l. he solution "steps" in the least squares adjustment of indirect observations are set out Chapter and restated as N t B WB B Wf N t v f B ˆl l + v o apply the Law of Propagation of Variances, these equations may be re-arranged in the form y A + b where the terms in parenthesis constitute the A matri. t B W f (5.4) N t (5.5) v f B f f ( BN t BN B Wf ) I BN B W f (5.6) ˆ l l + v l+ f B d B B + d (5.7) Applying the Law of Propagation of Variances to equations (5.4) to (5.7) gives the foowing cofactor matrices 5, R.E. Deakin Notes on Least Squares (5) 5

3 RMI University tt ff Q B W Q B W N (5.8) tt Q N Q N N vv (5.9) ff Q I BN B W Q I BN B W ˆˆ Q BN B (5.1) Q B Q B BN B vv Q Q (5.11) Variance-covariance matrices for t,, v and by the variance factor σ. ˆl are obtained by multiplying the cofactor matri 5.. Calculation of the quadratic form vwv he a priori estimate of the variance factor may be computed from ˆ σ v Wv (5.1) r where v Wv r n u is the quadratic form, and is the degrees of freedom where n is the number of observations and u is the number of unknown parameters. r is also known as the number of redundancies. A derivation of equation (5.1) is given below. he quadratic form in the foowing manner. v Wv may be computed Remembering, for the method of indirect observations, the foowing matri equations N t B WB B Wf N t v f B then 5, R.E. Deakin Notes on Least Squares (5) 5 3

4 RMI University v Wv ( f B) W( f B) ( f B ) W( f B) ( f W B W)( f B) + fwf fwb BWf BWB f Wf f WB+ B WB f Wf t + N f Wf t+ t and vwv fwf t (5.13) 5.3. Calculation of the Estimate of the Variance Factor ˆ σ he variance-covariance matrices of residuals, adjusted observations and computed vv ˆˆ parameters are calculated from the general relationship σ Q (5.14) Cofactor matrices Q, Q and Q are computed from equations (5.9) to (5.11) and so it vv ˆˆ remains to determine an estimate of the variance factor ˆ σ. he development of a matri epression for computing ˆ σ is set out below and foows Mikhail (1976, pp.85-88). Some preliminary relationships wi be useful. 1. If A is an (n,n) square matri, the sum of its diagonal elements is a scalar quantity caed the trace of A and denoted by tr ( A ) he foowing relationships are useful ( + ) + tr A B tr A tr B for A and B of same order (5.15) tr ( A ) tr ( A) (5.16) and for the quadratic form A where A is symmetric A tr ( A) (5.17) 5, R.E. Deakin Notes on Least Squares (5) 5 4

5 RMI University. he variance-covariance matri given by equation (3.1) can be epressed in the foowing manner, remembering that is a vector of random variables and vector of means. {( m )( m ) } {( m)( m) } { m m mm} { } { } { } { { } {} { } E E E + Now from equation (3.18) m E{ } E E m E m + E m m } E E m m E + m m hence m is a { } { } E mm mm + mm E m m (5.18) or { } E + 3. he epected value of the residuals is zero, i.e., mm (5.19) E { v} mv (5.) 4. By definition (see Chapter ) the weight matri W, the cofactor matri Q and the variance-covariance matri are related by W Q σ (5.1) Now, for the least squares adjustment of indirect observations the foowing relationships are recaed v + B f, N B WB, t B Wf Q Q, Q N, Q N ff tt Bearing in mind equation (5.1), the foowing relationships may be introduced and from these foow 1 W, M B B σ 5, R.E. Deakin Notes on Least Squares (5) 5 5

6 RMI University, σ M, M 4 ff tt In addition, the epectation of the vector f is the mean Now since E { } m and { } m f and so we may write { } { } { } { } mf E f E v+ B E v + BE E v m Bm (5.) f Now the quadratic form and from equation (5.13) σ 1 vwv v v (5.3) Using the relationships above vwv fwf t f Wf N v v f f M Now the epected value of this quadratic form is E { v v} E{ f f M} E{ } E{ f f M} Recognising that the terms on the right-hand-side are both quadratic forms, equation (5.17) can be used to give Now using equation (5.19) { v v} { ( ff )} { ( M) } tr ( E{ ff }) tr ( E{ M} ) tr ( E{ ff } ) tr ( E{ } M) E E tr E tr { v v} ( ff + mf mf ) ( + mm M) E tr tr ( Inn mf mf ) tr ( Iuu mmm) ( Inn Iuu ) tr ( mf mf mmm) tr + + tr + m m + m Mm f f 5, R.E. Deakin Notes on Least Squares (5) 5 6

7 RMI University From equation (5.) mf Bm hence using the rule for matri transpose mf Bm mb, then E{ v v} ( n u) mb Bm + mmm mmm + mmm hus according to equation (5.3) and the epression above from which foows E { vwv} σ E{ v v} σ σ E { vwv} n u Consequently, an unbiased estimate of the variance factor ˆ σ can be computed from vwv vwv ˆ σ n u r (5.4) r n u is the number of redundancies in the adjustment and is known as the degrees of freedom Using equation (5.13) an unbiased estimate of the variance factor ˆ σ can be computed from ˆ σ f Wf t (5.5) r 5, R.E. Deakin Notes on Least Squares (5) 5 7