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One of the implicit assumptions in the derivation of the Kalman lter algorithm was that the gains and the arithmetic necessary to calculate them would take place using in nite precision real valued numbers Computer implementations of the Kalman lter use a nite accuracy subset of the rational numbers Although the precision of each computation may be quite high (especially in oating point applications), the number of calculations involved in the Kalman lter algorithm can allow the cumulative e ect of the machine arithmetic to a ect the lter performance This issue becomes increasingly important in applications with either a long duration or a high state dimension Both factors result in a greater number of calculations If the lter state is observable from the measurements and controllable from the process driving noise, then the e ects of nite machine precision are usually minor Therefore, it is common to add a small amount of process driving noise even to states which are theoretically considered to be constants Two common results of nite machine precision are loss of symmetry or loss of positive de niteness of the covariance matrix These two topics are discussed in the following subsections

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When the smoke cleared, LTCM had lost a staggering $5 billion in less than a year, leaving it with only $200 million to $300 million in equity, according to press reports The professors, who ironically had designed their computer models to minimize risk, were probably shocked at how it all unraveled so quickly Although LTCM s meltdown occurred in 1998, what happened to it is a case study in risk mismanagement In fact, a number of universities include LTCM as part of the curriculum Business students study the hedge fund s collapse and identify the mistakes, as well as the lessons that can be learned One of the lessons: No matter how brilliant the system, there are unexpected events that can force the crowd to extremes and play havoc with your models The professors learned the hard way that the stock market is not a mathematical experiment, or game, that performs as efficiently in real life as in the laboratory If you want to learn more details about the rise and fall of Long-Term Capital Management, read When Genius Failed by Roger Lowenstein

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The error covariance matrix P = E x x is by its de nition symmetric However, numeric errors can result in the numeric representation of P becoming non-symmetric Four techniques are common for maintaining the symmetry of P: 1 Use a form of the Kalman lter measurement update, such as the Joseph form, that is symmetric 2 Resymmetrize the covariance matrices at regular intervals using the equation 1 P+P P= 2

3 Because P should be symmetric, its upper and lower triangular portions are redundant Instead of computing both, only calculate the main diagonal and upper triangular portion of P, then (if necessary) copy the upper portion appropriately into the lower triangular portion This approach also substantially decreases the memory and computational requirements 4 Use a UD factorized or square root implementation of the Kalman Filter as discussed in Subsection 592

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The error covariance matrix P = E x x is by its de nition positive semi-de nite Computed values of P can lose the positive semi-de nite property When this occurs, the Kalman lter gains for the corresponding states have the wrong sign, and the state may temporarily diverge Even if the sign eventually corrects itself, subsequent performance will su er since the covariance matrix is no longer accurate In addition, the interim divergence can be arbitrarily bad Any symmetric matrix, in particular P, can be factorized as P = UDU Special purpose algorithms have been derived which propagate the factors U and D instead of P itself The factors contain the same information as P, the factorized algorithms can be shown to be theoretically equivalent to the Kalman lter, and the algorithms automatically maintain both the positive de niteness and symmetry of the covariance matrix The main drawback of square root algorithms are that they are more complex to program Also, some formulations require more computation More general square root implementations of the Kalman lter have also been derived Various algorithms are presented and compared in for example [60]

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