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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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 the mean.

e continue calculation of the previous section.

We use analyticity of $f$ and write the Taylor expansion with respect to $\tilde{X}$ : MATH where the $k$ is the vector component index. We take the expectation, MATH where we used the direct consequence of definition of MATH : MATH The above expression is approximated with the MATH where we use the notation MATH and $\delta X_{i,p}$ is the p-th component of the vector $\delta X_{i}$ .

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

MATH (Unscented conditions for mean)
Let $P_{X}$ be the covariance matrix MATH and $\sigma$ is any matrix with the property MATH

We seek the $\delta X_{i}$ of the form MATH where the MATH are columns of the matrix $\sigma$ and MATH are some scaling factors. Such choice satisfies the second requirement of ( Unscented conditions for mean ) if MATH The third requirement of ( Unscented conditions for mean ) transforms into MATH Hence, we require that MATH It is enough to leave a free scaling factor $\alpha=\alpha_{i}$ , $i=1,...,n$ . Then the recipes MATH would satisfy the second and third requirements of the ( Unscented conditions for mean ). The remaining factor $W_{0}$ is determined by the first requirement: MATH


Suppose the $X\in\U{211d} ^{n}$ is a random variable, the MATH is analytical function, the $\bar{X}$ is the mean MATH and the matrix $\sigma$ satisfies the condition MATH then the mean MATH is approximated by the expression MATH with the second order MATH if the MATH and $W_{i}\in\U{211d} $ are given by MATH for any scaling parameter $\alpha\in\U{211d} $ .

Notation. Index. Contents.

Copyright 2007