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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 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: MATH MATH MATH MATH MATH MATH The above matrix is approximated with the sum MATH We set MATH for some $c$ , then MATH We substitute the Taylor expansions for $f\left( X\right) $ : MATH By comparing the expressions for $P_{kq}$ and $P_{kq}^{\ast}$ we conclude that it is enough to have the following properties of $\kappa$ : MATH The first five would be satisfied if MATH The above conditions are true by the structure of $\kappa_{ij}$ and the properties MATH and if we set $c=0$ . Indeed, MATH MATH We next investigate the requirement MATH where the $\delta X_{i,s}$ where calculate in the previous section MATH and the $\sigma$ comes from the relationship MATH Note that $\sigma_{i}$ is $i$ -th column of $\sigma$ , hence the $\delta X$ satisfy MATH hence, MATH and the $\kappa$ is required to have the property MATH where $\theta_{ij}$ is any matrix such that MATH It is enough to set MATH


Suppose the $X\in\U{211d} ^{n}$ is a random variable, the MATH is analytical function, the $\bar{X}$ is the mean MATH , $P^{X}$ is the covariance matrix of $X$ and the matrix $\sigma$ satisfies the condition MATH then the covariance matrix of $Y$ 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