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 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 1 Time Series.
 A. Time series forecasting.
 B. Updating a linear forecast.
 C. Kalman filter I.
 D. Kalman filter II.
 a. General Kalman filter problem.
 b. General Kalman filter solution.
 c. Convolution of normal distributions.
 d. Kalman filter calculation for linear model.
 e. Kalman filter in non-linear situation.
 f. Unscented transformation.
 i. Unscented approximation of the mean.
 ii. Unscented approximation of covariance matrix.
 E. Simultaneous equations.
 2 Classical statistics.
 3 Bayesian statistics.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Unscented approximation of covariance matrix. e continue calculations of the section ( Unscented transformation section ) and use the summary ( Unscented mean summary ).

Similarly to the previous section ( Unscented mean section ) we write Taylor expansions for quantities of interest:      The above matrix is approximated with the sum We set for some , then We substitute the Taylor expansions for : By comparing the expressions for and we conclude that it is enough to have the following properties of : The first five would be satisfied if The above conditions are true by the structure of and the properties and if we set . Indeed,  We next investigate the requirement where the where calculate in the previous section and the comes from the relationship Note that is -th column of , hence the satisfy hence, and the is required to have the property where is any matrix such that It is enough to set Summary

Suppose the is a random variable, the is analytical function, the is the mean , is the covariance matrix of and the matrix satisfies the condition then the covariance matrix of  is approximated by the expression with the second order if the and are given by for any scaling parameter .

 Notation. Index. Contents.