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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 transformation.

uppose the vector-valued random variable is transformed by the analytical function into a vector-valued variable : The variable belongs to a certain class of random variables (such as the class of normal variables). We would like to evaluate statistics of such mean and covariance matrix in some efficient way. Consequently, we select a variable with the same statistics and use it in place of . Since such replacement is an approximate, it is acceptable to seek approximate procedure for evaluation of statistics of .

We restrict our attention to the class of normal vector-valued random variables. For this reason we evaluate the mean and covariance matrix of . Let . We treat the as a small quantity. We propose to approximate the mean with the sum where the are some deterministic vectors in the value space of , are some real numbers and the range of the index is to be determined later. We seek , such that would approximate with the second order:

Consistently, we will be seeking approximation of the covariance matrix of the random variable with the expression for some coefficients sought to deliver

We now derive the and that deliver the second order approximations.

 i. Unscented approximation of the mean.
 ii. Unscented approximation of covariance matrix.
 Notation. Index. Contents.