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 the mean.

e continue calculation of the previous section.

We use analyticity of and write the Taylor expansion with respect to : where the is the vector component index. We take the expectation, where we used the direct consequence of definition of : The above expression is approximated with the where we use the notation and is the p-th component of the vector .

Hence, the second order of approximation would be delivered if Therefore, it is enough to satisfy the following conditions

 (Unscented conditions for mean)
Let be the covariance matrix and is any matrix with the property

We seek the of the form where the are columns of the matrix and are some scaling factors. Such choice satisfies the second requirement of ( Unscented conditions for mean ) if The third requirement of ( Unscented conditions for mean ) transforms into Hence, we require that It is enough to leave a free scaling factor , . Then the recipes would satisfy the second and third requirements of the ( Unscented conditions for mean ). The remaining factor is determined by the first requirement:

Summary

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

 Notation. Index. Contents.