THE KALMAN FILTER: A LOOK BEHIND THE SCENE

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1 HE KALMA FILER: A LOOK BEHID HE SCEE R.E. Deain School of Matheatical and Geospatial Sciences, RMI University eail: rod.deain@rit.edu.au Presented at the Victorian Regional Survey Conference, Mildura, -5 June, 6 ABSRAC he Kalan filter is a set of equations, applied recursively, that can be used in surveying applications to obtain position, velocity and acceleration of a oving object fro traditional surveying easureents. he ai of this paper is to introduce the surveyor to the Kalan filter by eaination of two siple applications, (i) Electroagnetic Distance Measureent (EDM) and (ii) the position and velocity of a ship in a navigation channel. IRODUCIO A Kalan filter is a set of equations that are applied recursively to estiate the state of a syste fro a sequence of noisy easureents at ties t, t, t, etc. he "state" of the syste is its value or values at ties t, t, t, etc and a "syste" ay have a single value or ultiple values. Say, for instance, the syste is a ship steaing on a particular heading in a shipping channel and the state of the syste (the ship) is its east and north coordinates ( E, ) and its E,. We say that this syste (the velocity ( ) ship) has a state vector = E,, E, containing four eleents and the subscript indicates a value at tie. t On the other hand, a syste ay be a process such as EDM by phase coparison of eitted and reflected light beas. he state of this syste is a single value, the distance ( D ), deterined at ties t, t, t, etc, and this syste (the EDM) has a state vector = containing a single [ D ] eleent and the subscript indicates a value at tie. t "oisy" easureents are easureents that contain sall rando errors assued to be norally distributed, i.e., the aggregation of errors in size groupings would for the failiar syetric bell-shaped histogra with positive and negative errors equally liely and sall errors ore frequent than large errors. Surveyors usually tal of residuals (or corrections) rather than errors, where a residual is the sae agnitude as an error but of opposite sign. A Kalan filter gives the best estiates (in a least squares sense) of the state of a dynaic syste at a particular instant of tie. And a dynaic syste can be one whose values are changing with tie, due to the otion of the syste and easureent errors, or one whose values are easured at various instants of tie and appear to change due to easureent errors. Dynaic systes do not have a single state (consisting of one or any values) that can be deterined fro a finite set of easureents but instead have a continuously changing state that has values sapled at different instants of tie. In this paper, vectors are taen to ean colunvectors. A row-vector containing one row and n coluns, and shown as [ a a an ] is the transpose of the colun vector containing n rows and one colun. o save space, colun-vectors are often shown as [ a a a ] where the n notation [ ] indicates the transpose. A vector can also contain a single eleent. his paper ais to provide soe insight into the Kalan filter and its ipleentation by studying two eaples (i) the deterination of a theoretical distance by an EDM; a dynaic syste with a state vector containing a single value, and (ii) the deterination of the position and velocity of a ship in a navigation channel; a dynaic syste having a state vector containing four paraeters.

2 HE KALMA FILER he Kalan filter equations were published in 96 by Dr. R.E. Kalan in his faous paper describing a new approach to the solution of linear filtering and prediction (Kalan 96). Since that tie, papers on the application of the technique have been filling nuerous scientific journals and it is regarded as one of the ost iportant algorithic techniques ever devised. It has been used in applications ranging fro navigating the Ranger and Apollo spacecraft in their lunar issions to predicting short-ter fluctuations in the stoc aret. Sorenson (97) shows Kalan's technique to be an etension of C.F. Gauss' original ethod of least squares developed in 795 and provides an historical coentary on its practical solution of linear filtering probles studied by th century atheaticians. he derivation of the Kalan filter equations can be found in any tetboos related to signal processing that is the usual doain of Electrical Engineers, e.g., Brown and Hwang (99). hese derivations often use terinology that is unfailiar to surveyors, but two authors, Kraiwsy (975) and Cross (99) both with geodesy and surveying bacgrounds, have derivations, eplanations, terinology and eaples that would be failiar to any surveyor. his paper uses terinology siilar to Cross and Kraiwsy. he Kalan filter equations and the associated easureent and dynaic odels are given below with a brief eplanation of the ters. It is hoped that the study of the two eaples will help to ae the Kalan filter a relatively easily understood process. he priary odels (or easureent odels) at ties and t, and the secondary odel (or t dynaic odel) lining the states at are given by the atri equations v + B = f v + B = f = + v priary t priary t dynaic t and t () where is the state vector v is the vector of residuals associated with the easureents B is a coefficient atri f is a vector of nueric ters derived fro the easureents is the transition atri v is a vector of residuals associated with the dynaic odel. Enforcing the least squares condition that the su of the squares of the residuals, (ultiplied by coefficients reflecting their precisions) be a iniu, gives rise to the set of recursive equations (the Kalan filter) that are applied as follows: With initial estiates of the state vector and the state cofactor atri a Kalan filter has the following five general steps () Copute the predicted state vector at t = ˆ () () Copute the predicted state cofactor atri at t = + () () Copute the Kalan Gain atri K = B + B B (4) (4) Copute the filtered state vector by updating the predicted state with the easureents at t K f B (5) ˆ = + (5) Copute the filtered state cofactor atri I K B (6) = Go to step () and repeat the process for the net easureent epoch t +. ^ above the vector he "hat" sybol indicates that it is an estiate of the true (but unnown) state of the syste derived fro the Kalan filter. his is also nown as the filtered state. he "prie" sybol () ' indicates a predicted quantity. he superscript (), e.g., B denotes the atri transpose where the rows of B becoe the coluns of B. If B contains a single eleent, say = B = B =. he superscript B [ ] then [ ], e.g., A denotes the inverse of a atri and atri inversion is defined by AA = I, where I is the diagonal Identity atri (ones on the leading-diagonal). his relationship gives rise to atri inversion routines that soe coputer languages offer as standard atri functions.

3 It would be assued that any coputer ipleentation of a Kalan filter would use a language (C++, Visual Basic, etc.) that had soe standard atri routines attached that offered atri transposition, ultiplication and inversion. A = If a atri has only a single eleent, say [ ] then A = [ ]. Cofactor atrices, designated (often called covariance atrices) contain estiates of variances, denoted sa, sb, sc, etc and covariances, denoted sab, sac, sbc, etc associated with rando quantities A, B, C, etc. he cofactor atri of the easureents is denoted by, the cofactor atri of the state vector is denoted by and the cofactor atri of the dynaic odel residuals is denoted by. It should be noted that the cofactor atri following anner. he dynaic odel in equation () = + v is derived in the (7) is an estiation of the true (but unnown) changes in the eleents of the state vector fro tie t to tie t and as such we assue that there are corrections to these estiations that are contained in the vector, the dynaic odel residuals; v v and the eleents of are assued to be sall, rando and norally distributed with a ean of zero. Also, we assue that the vector v is the product of two atrices, a coefficient atri H and a vector w nown as the syste driving noise v = Hw (8) he syste driving noise w is a vector of rando variables having variances and covariances contained in the cofactor atri w and applying the general law of propagation of variances to equation (8) gives = H H w Deterining, H, w, w and will be discussed in the eaples below. (9) he Kalan filter equations are relatively easy to ipleent on odern coputers (a reason for its popularity) and the eaples studied below are suppleented by MALAB coputer code available fro the author. MALAB, a registered tradear of he MathWors, Inc., is a high-perforance language for DEERMIAIO OF A DISACE BY A EDM he EDM coponent of a otal Station easures distances by phase coparison of an eitted and reflected odulated light bea. he easureent is an electro/optical process and the distance we see displayed after pressing the easure button on the otal Station is the "filtered" value of any hundreds of individual easureents, since a easureent taes only a nuber of illiseconds. his value could be the result of a Kalan filter process. Consider the following sequence of easureents at ties t, t, t, etc, 55.46, 55.4, 55.4, 55.4, 55.49, he variation in the easureents is assued to be due to norally distributed rando errors arising fro the internal easureent process; often called the process noise. [he easureent sequence above, was generated by adding norally distributed rando errors with ean zero and standard deviation. to a constant value of 55.4.] How will a Kalan filter produce the "filtered" value fro this sequence? First, let us assue that the easureent odel is l ˆ + v = l () l is the ( n ) vector of easureents, v is the ( n ) vector of residuals (sall unnown corrections to the easureents) and ˆ l are estiates of the true (but unnown) value of the easureents. n is the nuber of easureents at each epoch, that in this case is one. he priary easureent odel can be epressed in ters of the filtered state vector at tie t as v + B ˆ = f () ˆ In this case contains the eleents of l, both vectors containing single quantities and f = l also both containing single quantities (the easured distance at ). he atri B will contain a single quantity, B =. t ˆ [ ] technical coputing. It integrates coputation, visualization, and prograing in an easy-to-use environent where probles and solutions are epressed in failiar atheatical notation. ˆ

4 Secondly, the dynaic odel lining the eleents of the state vector at ties t and t is = + v () he state vector contains a single eleent that should reain unchanged between t and t (any change is siply due to easureent errors) then the transition atri will contain a single eleent = and there are no assued [] () Copute the Kalan Gain atri noting that = (.) ( ) K = B + B B = (.) [ ] ( (.) + [ ] (.) [ ] ) (.) (.) = =.5 corrections to this odel; hence v =, H =, w = and fro equation (9) =. (4) Copute the filtered state vector by Lastly, an estiate of the cofactor atri of the easureents and the eleents of the state vector ust be ade. Let us assue (guess) that the easureents have a standard deviation of (. ) and hence their estiated variance is (.) and = (.). Since our priary easureent odel has a state vector containing a single value (the easureent), then will only contain a single value, and we have as a starting estiate (.) as. =, the sae ow we can now start the Kalan filter at epoch t using the values at t as filtered estiates. () Copute the predicted state vector at epoch t using the easureent at t as the filtered estiate ˆ ˆ updating the predicted state with the easureents at t ( ) [ 55.46] + {[.5] ([ 55.4] [ ][ 55.46] )} ˆ = + K f B = = 55.4 (5) Copute the filtered state cofactor atri at t I K B = ([] [.5][ ] ) (.) = =.5 Go to step () and repeat the process for the net easureent epoch. t = ˆ he values fro the Kalan filter for epochs = [][ 55.46] = () Copute the predicted state cofactor atri at t using = (.) as the filtered estiate = + []( ) [] =. + = (.) t, t and t 4 5 are epoch t = 55.4 =.5 K epoch t = =. ˆ = =. epoch t 5 = =.5 K =. ˆ = =. 4 =. K =.5 ˆ = =.5 4

5 So, the sequence of easureents is 55.46, 55.4, 55.4, 55.4, 55.48, and the Kalan filter estiates ˆ (the filtered values) are 55.46, 55.4, 55.49, 55.45, 55.46, Soething that should be noted is that the filtered state cofactor atri contains the estiate of the variance of the filtered value (variance is standard deviation squared). We started with an estiated value = =. =. and after five epochs the estiated value had reduced to =. equivalent to an 5 estiated standard deviation of.45. So the Kalan filter gives estiates with a better precision than the assued precision of the easureent sequence; as we should epect fro a least squares process. A Kalan filter progra ed., written in the MALAB language and available fro the author, processes 5 EDM easureents that are obtained by adding norally distributed rando errors, with ean zero and standard deviation., to a constant value Figure below shows two plots fro this progra (i) the filtered estiate of the distance (the filtered state) as a blac solid line and the 5 easureents as blac dots and (ii) a plot of the standard deviation of the filtered distance. After processing the 5 easureents the filtered distance was 55.4 with a standard deviation of.6. Figure. MALAB plots of filtered state (EDM distance) and standard deviation of filtered state. It is interesting to note that if all 5 easureents,,,, 5 (each with standard deviation s =. ) had been recorded and the ean = 5 coputed, then propagation of variances gives the standard deviation of the ean as s s = =.6 which is the sae as 5 the Kalan filter result. 5

6 DEERMIAIO OF POSIIO AD VELOCIY OF A SHIP I A AVIGAIO CHAEL Figure shows the path of a ship as it oves down a shipping channel at a constant heading and speed. avigation equipent on board autoatically easures distances to transponders at three navigation beacons A, B and C at 6- second intervals. he easured distances are nown to have a standard deviation of etre and the solid line in Figure represents solutions of the ship's position for each set of easureents at the 6-second tie intervals. he true path of the ship is shown as the dashed line. he transponder easureents, fro the ship to the navigation beacons A, B and C at 6-second tie intervals are shown in able below. he data were generated by assuing the starting coordinates of the ship were East and 69.9 orth and the ship was travelling at 5 nots on a heading of 64º ( not = nautical ile per hour and nautical ile = 85 etres). At 6-second intervals, the true ship position and distances to the beacons were coputed. hese distances were then "disturbed" by the addition of norally distributed rando errors (with zero ean and standard deviation etre) and then rounded to the nearest.. Epoch ransponder easureents to navigation beacons A B C able. ransponder easureents at 6-second tie intervals A true path B - C How will a Kalan filter produce an estiated position, speed and heading of the ship fro the transponder easureents? ote that in our Kalan filter, the state vector will be = E,, E, containing n = 4 E, are the eleents (or paraeters) where ship's position and ( E, ) the ship's velocity coponents. he speed of the ship at tie t is Figure. Path of a ship in a navigation channel. (A, B and C are nown navigation beacons) ( ) ( ) speed = E + () and the heading of the ship (bearing fro orth) at tie is t he coordinates of the three navigation beacons are: A :. E. B : 88. E 5. C : 555. E 76. tan ( heading) E = (4) 6

7 he easureent odel (priary odel) Let us assue that the priary or easureent odel is l ˆ + v = l (5) where l is the ( ) vector of easureents (the transponder distances), v is the ( ) vector of residuals (sall unnown corrections to the easureents) and ˆ l are estiates of the true (but unnown) value of the easureents. is the nuber of easureents, that in this case is three at each easureent epoch. he estiates ˆ l are non-linear functions of the coordinates E, of the beacons A, B and C and the filtered state coordinates Eˆ, ˆ of the ship at tie t (,,, ) lˆ = lˆ Eˆ ˆ E j j j for j = A, B, C ( Eˆ ) ( ˆ Ej j) = + (6) Epanding equation (6) into a series using aylor's theore gives ˆ ˆ ˆ l l = l + Eˆ E + ˆ Eˆ ˆ l ( ) ( ) + higher order ters where E, are approiate coordinates of the ship at t, l is an approiate distance coputed using E, and the coordinates of the beacon, and the partial derivatives are lˆ Eˆ lˆ ˆ E E j = = l j j = = l j d c j j for j = A, B, C (7) Re-arranging equation (5) for a single distance gives v lˆ = l and substituting the aylor series approiation for ˆl (ignoring higher-order ters) and rearranging gives the linearized for of the priary easureent odel as v d Eˆ c ˆ = l l + d E c (8) j j j j j j j for j = A, B, C his priary easureent odel can be epressed in ters of the filtered state vector at tie t in the atri for as Eˆ va da ca ˆ v B + db cb E vc dc cc E l A la da ca = l B l B + db cb E l C lc dc cc or v + B ˆ = l l + B = f (9) ow in step (4) of the Kalan filter algorith [see equation (5)], the filtered state vector ˆ is obtained fro K f B () ˆ = + and substituting for f fro equation (9) gives ( ) ˆ = + K l l + B B = + K l l ote: (i) the ter ( ) ˆ () f B in equation () is often called the predicted residuals v where, in our case v = f B = l l () (ii) he ter K( l l ) in equation () is often called the corrections to the predicted state where, in our case ( ) = K l l () he dynaic odel (secondary odel) A dynaic odel that is etreely siple and often used in navigation probles can be developed by considering a continuous function of y = y t. Following the developent tie, say by Cross (987), we can use aylor's theore to y t about the point t = t epand the function into the series ( t t ) ( t t ) y t = y t + t t y t + y t! + y t +! 7

8 ,,, etc are derivatives of y where y( t ) y( t ) y( t ) with respect to t evaluated at t = t. Letting t = t + and then t = t t we ay write ( + ) = + y( t) y( t ) t ( t ) y t t y t y t t + + +!! (4) We now have a power series epression for the y t at the point t = t + continuous function involving the function y and its derivatives y, y, etc, (all evaluated at t ) and the tie difference t = t t. In a siilar anner, if we assue y() t y( t) to be continuous functions of t, then,, etc ( ) y t y ( t + t) = y ( t) + y( t) t+ t +! y t + t = y t + y t t+ etc (5) t t ow consider two tie epochs and separated by a tie interval, we can cobine equations (4) and (5), with a change of subscripts for t, into the general atri fors: (i) involving ters up to y y y [ y] y = + y (ii) involving ters up to y ( ) 6 ( t) [ y] y t y y = t y + y y (6) (7) In any navigation probles the continuous y = y t is siply position, so function of tie y( t) { E, }( t) coordinates. Here y( t) { }( t) = where E, are east and north = eans y is a function of tie t where the function contains the variables within the braces {}. he derivatives are velocity: y t = E, t, { } acceleration: y ( t) = { E, }( t) jer: y( t) = { E, }( t) and which is the rate of change of acceleration. In equations (6) and (7), we can consider the vector on the left-hand-side of the equals sign to be the vector, the state vector, or the state of the syste at tie. he atri on the right-hand-side is the transition atri and the eleents of this atri contain the lins between the state vector at ties and t, i.e., =. he second ter in the equations above is the product of two atrices and the result will be the vector of odel residuals (containing the sae nuber of eleents as v the state vector). v is a reflection of the fact that the transition atri does not fully describe the eact physical lins between the states at ties and t and v = Hw where H is a coefficient atri and w is the syste driving noise. In equations (6) and (7), the syste driving noise is acceleration and jer respectively. We can now use these general fors to define a suitable dynaic odel. In our siple case (the ship in the channel) the state vector contains four eleents = E,, E, and the appropriate dynaic odel in the for of equation (6) is E E = E E ( ) ( ) E + t t t (8) or = + v (9) where is the ( n n) the ( n ) vector of odel residuals. v transition atri and is If we epand equation (8) we see that it is really just the atri for of the two equations of rectilinear otion; (i) v = u+ at and (ii) s = ut + at where s is distance, u is initial velocity, v is final velocity, a is acceleration and t is tie. 8

9 In our notation they are: (i) (ii) E = E + E = + and E = E + E t+ E = + t+ he dynaic odel residuals ( ) v = Hw are ve v ( t) E = v E () v where the coefficient atri H and the syste driving noise w are ( ) ( ) E H= and w = () In this siple navigation proble it is assued that the syste driving noise w contains sall rando accelerations caused by the sea and wind conditions, the steering of the ship, the engine speed variation, etc. ow we can start the Kalan filter, but first soe initial values ust be set. hese initialisation steps will be designated as (a), (b), (c) etc followed by the Kalan filter steps (), (), () etc. (a) (b) (c) Set the eleents of the transition atri = In this eercise t = 6sec Set the cofactor atri of the easureents s la = sl B s lc In this case s = s = s =. la lb lc Set the cofactor atri of the syste driving noise w s E = s In this case s = s =.7 s E 4 he cofactor atri of the dynaic odel given by = H H w w is () where, the cofactor atri of the syste driving noise, is w s E = s and s, E s are the estiates of the variances of the accelerations in the east and north directions and the covariance is assued to be zero. Using the coefficient atri H in equation () we have ( ) ( ) s E s = ( ) () ( ) 9 (d) (e) (f) Set the coefficient atri of the syste driving noise ( ) H = ( ) Copute the cofactor atri of the dynaic odel = H H w Set the starting estiates of the state vector. his will be the filtered state vector for epoch t E = = E 7 s s

10 (g) Set the starting estiates of the state cofactor atri. his will be the filtered state cofactor atri for epoch () Copute the Kalan Gain atri t K = B ( + B B ) s Using fro step (b) and B whose eleents E have been deterined using equations (7). s he for of B is given in equation (9). = s E s (4.) Copute the nueric ters "coputed observed" distances [see equation ()] In this case se = s = and s = s =.5 s (4.) Copute the filtered state vector ˆ by E updating the predicted state [see equation ()] ow start the Kalan filter at epoch t ˆ = + K( l l ) () Copute the predicted state vector at epoch t using the filtered estiate ˆ (5) Copute the filtered state cofactor atri at = ˆ t () Copute the predicted state cofactor atri at t using fro step (e) = + I K B = Go to step () and repeat the process for the net easureent epoch. t he following output fro a MALAB progra alship. (available fro the author) processes the data in able in a Kalan filter beginning at epoch. epoch = Filtered State Corrns Filtered State cofactor atri epoch = Filtered State Corrns Filtered State cofactor atri : : epoch = 9 Filtered State Corrns Filtered State cofactor atri epoch = Filtered State Corrns Filtered State cofactor atri

11 Filtered Values Epoch Distance Speed Heading (etres) (/s) (degrees) he speed and heading of the ship at each epoch have been coputed using equations () and (4) respectively. If the first two epochs are ignored, on the assuption that the starting values are not very close to the truth and the filter needs soe tie to "stabilise", then the average of the reaining values are 7.7 /s and 64.8 respectively with ranges.4 /s (epochs 8 and 9) and. (epochs and 4). Bearing in ind that the data were generated for a ship travelling at 5 nots (equivalent to 7.7 /s) on a heading of 64, these are very close estiates of the "true" values. Also, the "true" position of the ship at the th epoch (using the initial coordinates and 5 nots on a heading of 64 ) is E and he Kalan filter gives the ship's position as E and (see the filtered state for epoch in MALAB output above), which is within.48 and.465 respectively of its "true" location. he output fro the MALAB progra also gives the cofactor atri of the filtered state, i.e., the estiates of variances and covariances of the four eleents of the state vector. he initial estiates for epoch (see step (g) above) were s = s = E t and se = s = = s giving and for epoch t (after one pass through the filter) = and finally for epoch t = Here the variance estiates are s =.5989, s =.847 and E E s =.5855 s s =.6445 s. hese equate to standard deviations in position of.78 E and.9 and standard deviations in velocity of.6 /s E and.6 /s. Using equation () and the law of propagation of variances gives the estiated standard deviation of the ship's speed as.6 /s. We can see fro this very liited analysis that the Kalan filter gives quite reasonable estiates of the position and velocity of the ship fro a sequence of noisy distance easureents. his is a feature of a Kalan filter: you obtain inforation about the dynaics of your easureent platfor. COCLUSIO Kalan filtering is an etension of the least squares technique as forulated by C.F. Gauss. Indeed, Kraiwsy (975) shows that the Kalan filter equations can be derived fro the basic least squares principle (iniizing sus of weighted squares of easureent and odel residuals) used in deriving conventional ethods of least squares adjustent of survey data. In this paper we have stated the filter equations and the priary and dynaic odels upon which they are based and then used two eaples to deonstrate their use. he first eaple, the deterination of a distance by an EDM, is an attept to show how a Kalan filter could be used to obtain a single estiate fro a continuous sequence of easureents. In this case the syste (the EDM) is in fact static but appears to be oving due to easureent errors.,

12 he Kalan filter processes the data sequence providing a "best estiate" at each easureent epoch based on all the previous easureents and their easureent precisions. he second eaple is a ore conventional navigation proble: and one that has been used in a nuber of tets (e.g., Cross 99) to deonstrate the usefulness of the Kalan filter. In this eaple the easureents are non-linear functions of the eleents of the state vector and we have shown how a linearized priary easureent odel is obtained and used. We have also shown, by assuing that position is a continuous function of tie, how the ebers of the dynaic odel are obtained the transition atri, the coefficient atri H and the syste driving noise w where the dynaic odel corrections are v = Hw. he assuption that position is a continuous function of tie ay not be correct in all navigation probles, but is adequate in our eaple. In both of our eaples, data have been generated to siulate actual easureents affected by rando errors. hese siulated easureents have norally distributed rando errors and the easureents are independent, i.e., there is no correlation between easureents at different epochs. his ay not reflect actual easureents obtained fro real situations, where there could be soe unaccounted-for systeatic errors and/or possibly unnown correlations between easureents. In such cases, careful analysis of residuals and data ay be required to identify systeatic errors (or deficiencies in the odels) and correlated data; with a possible need to odify the Kalan filter equations to allow for correlation. Bearing in ind this assuption, our eaples deonstrate how closely the Kalan filter estiates approach the "true" values, even with a liited aount of data (the second eaple) and quite noisy easureents. REFERECES Brown, R.G. and Hwang, P.Y.C., 99, Introduction to Rando Signals and Applied Kalan Filtering, nd ed, John Wiley & Sons, Inc. Cross, P.A., 987, 'Kalan filtering and its application to offshore position-fiing', he Hydrographic Journal, o. 44, pp. 9-5, April 987. Cross, P.A., 99, Advanced least squares applied to position fiing, Woring Paper o. 6, Departent of Land Surveying, University of East London, 5 pages, oveber 99. (Originally published by orth East London Polytechnic in 98) Kalan, R.E., 96, 'A new approach to linear filtering and prediction probles', ransactions of the ASME Journal of Basic Engineering, Series 8D, pp. 5-45, March 96. Kraiwsy, E.J., 975, A Synthesis of Recent Advances in the Method of Least Squares, Lecture otes o. 4, 99 reprint, Departent of Surveying Engineering, University of ew Brunswic, Fredricton, Canada. Sorenson, H.W., 97, 'Least squares estiation: fro Gauss to Kalan', IEEE Spectru, Vol. 7, pp. 6-68, July 97. Correlation is a statistical easure of the linear independence of easureents. If two easureents are independent then their correlation will be zero. Correlated easureents require special treatent in any least squares process.