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Example 52 Consider the problem of estimating the deterministic vector x based on observations Y related to x according to Y = Hx + where is zero mean Gaussian white noise with R = var{ } = 2 Im with = 01 At least two criteria are available for judging the quality of the estimate First, the diagonal of P contains that variance of the estimate of each component of x The variance of the estimation error vector x is quanti ed by trace(P) Second, the residual measurement can be computed as Y = Y H x These two criteria have di erent properties The matrix P depends only on H and R, but not the actual measurements nor the value of x Alternatively, Y is directly a ected by the data The variance of Y and the variance of Y are distinct: var( Y) = HPH + R while var(Y) = HPH Consider the set of m = 3 measurements Y = [ 120529, with 06964 H = 06330 00885 30431, 01228 07544 02432 61943] 07071 01736 09659 (532)

3rd power, or cube, of the length of any edge Therefore, the length of any edge is equal to the 1/3 power, or cube root, of the volume

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The third power is often called the cube We can say, 4 cubed equals 64 Now if we go with the reciprocal power and work backwards, we get 641/3 = 4 This can be read as, The cube root of 64 equals 4 The 3rd power is called the cube and the 1/3 power is called the cube root because of the relationship between the edges and the interior volume of a geometric cube For any perfect cube, the volume is equal to the 3rd power of the length of any edge (Fig 8-2) Going the other way, the length of any edge is equal to the 1/3 power of the volume The figure also shows the radical notation for the cube root The fact that the radical refers to the cube root, rather than the square root, is indicated by the small numeral 3 in the upper-left part of the radical symbol

According to eqns (527-528), 00160 00091 00031 68333 00208 00016 and x = 77014 P = 00091 00031 00016 00075 89782 The diagonal of the covariance matrix P indicates that the standard deviation of x is [013, 014, 009] Since m = 3 = dim(x) and H is nonsingular, Y = 0; however, this is only a result of the number of measurements being equal to the number of unknowns It does not imply that x = 0 nor that Y = Y Evaluation of HPH shows that the standard deviation in the estimation of each component of Y is 01 With three measurements and three unknowns, while an estimate of x is now available, there has been absolutely no reduction in the error in prediction of Y As the number of measurements m increases, Y will become nonzero, while the variance of both x and Y will tend to decrease Because Y is directly a ected by the data, it can be useful in comparison with the prediction of eqn (532), for detecting anomalies in the data

1st Significant Figure 2nd Significant Figure Multiplier Tolerance 01 F 10% CSGNetworkCom 6/4/92

Higher roots When p is a positive integer equal to 4 or more, people write or talk about the numerical powers and roots directly That s because geometric hypercubes having 4 dimensions or more are not commonly named A 4-dimensional hypercube is technically called a tesseract, but you should expect incredulous stares from your listeners if you say 2 tesseracted is 16 or The tesseract root of 81 is 3 Here are some examples of higher powers and roots You can check the larger ones on your calculator if you like

Recursive Least Squares (RLS)

24 = 16 34 = 81 so 161/4 = 2 so 811/4 = 4

For the linear measurement model of eqn (517) with the set of mea surements Y = [ 1 , , ym ] , using equations (527 528), it is possible to y calculate the estimate x that is the optimal in the WLS, maximum likelihood, and mean-squared senses The question of this section is how to e ciently produce the new optimal estimate of x if an additional measurement ym+1 becomes available To be computationally e cient, the estimate xm+1 should not start from scratch (ie, require all elements of Y), but from the best previously available estimate xm Therefore, based on intuition, the objective is to develop a recursive estimation equation of the form y (533) xm+1 = xm + K( m+1 ym+1 ) where ym+1 = Hm+1 xm and K is a vector gain that is to be determined To achieve this objective, we start from the prior information The vector xm is known with E xm var( m ) x = x = var( xm ) = Pm

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