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The value of x that maximizes p(x : Y) is the maximum likelihood estimate (MLE) of x and coincides with the minimum of JW LS when W = R 1 For m < n, the problem will be under-determined and an in nite number of solutions will exist If m > n then the problem may be overdetermined, in which case no exact solution will exist In the latter case, minimization of the objective function in eqn (518) will result in the es timate x that has the minimum weighted 2-norm of y = [ y1 , , ym ] where yi = yi Hi x The following text will build up to the Kalman Filter by discussing sequentially the following set of questions: 1 What is the formula for the WLS estimate of x (ie, the minimum of eqn (518)) 2 Given that the WLS estimate xm has been calculated for a set of m measurements, if an additional measurement ym+1 becomes available, can xm be adjusted to e ciently produce the new WLS estimate xm+1 3 If x Rn is the state of a dynamic system, which is free to change according to xk = k xk 1 + Gk uk 1 + k 1 where k Rn n is known and k Rn is a zero mean random vector, how can the estimate of x at time k 1 and the measurement yk be combined to provide an optimal estimate of x at time k As the above discussion shows, each question is a natural extension of the one before it and as the following sections show, the solutions to these problems are intimately related The Kalman Filter is the solution to the last problem

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Even back in the 1990s it was estimated that a company with one thousand employees that does not offer elder care would lose $400,000 annually because of the absenteeism of employee caregivers It has also been estimated that 30 percent of all workers soon will have some type of elder-care responsibility Many companies have already recognized and taken steps to alleviate employee child-care problems, but comparatively few have done much to help personnel who are absent because they must care for elderly parents Passage of the federal Family and Medical Leave Act (FMLA) forced larger businesses to address the elder-care problem However, more than half of all Americans work for small companies that are exempt from the requirements of the FMLA Some companies have banded together to work out solutions, others have turned to existing community volunteer services, and some have instituted their own counseling services or forms of case management Perhaps the largest cooperative partnership is the American Business Collaboration for Quality Dependent Care, which has funded more than fifteen hundred child-care and elder-care projects For more information about this collaboration and the broad variety of solutions to dependent-care issues, visit the website at wwwabcdependentcarecom

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The WLS estimate x will be found using di erential calculus to minimize eqn (518) Eqns (B50 B53) are used in the derivation Multiplying out eqn (518) results in JW LS = 1 Y WY 2Y WH + x H WH x x 2

My next guest, prominent investor, Forbes columnist, and bestselling author Ken Fisher, stopped by to talk about what he learned from a lifetime of dueling with the stock market He will also shed some light on an indicator he created, the Price/Sales Ratio, favored by many investors

Therefore, the extreme points of the objective function are determined by the solution of JW LS x which yields x = (H WH) 1 H WY (522) = H WY + H WH = 0 x (521)