The Helmert–Wolf blocking (HWB) is a
least squares
The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...
solution method for the solution of a
sparse block
Block or blocked may refer to:
Arts, entertainment and media Broadcasting
* Block programming, the result of a programming strategy in broadcasting
* W242BX, a radio station licensed to Greenville, South Carolina, United States known as ''96.3 ...
system of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variable (math), variables.
For example,
: \begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of th ...
. It was first reported by
F. R. Helmert for use in
geodesy
Geodesy or geodetics is the science of measuring and representing the Figure of the Earth, geometry, Gravity of Earth, gravity, and Earth's rotation, spatial orientation of the Earth in Relative change, temporally varying Three-dimensional spac ...
problems in 1880;
(1910–1994) published his direct semianalytic solution in 1978.
It is based on ordinary
Gaussian elimination
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients. This method can a ...
in
matrix
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the m ...
form
or partial
minimization form.
Description
Limitations
The HWB solution is very fast to compute but it is optimal only if observational errors do not correlate between the data blocks. The
generalized canonical correlation
In statistics, the generalized canonical correlation analysis (gCCA), is a way of making sense of cross-correlation matrices between the sets of random variables when there are more than two sets. While a conventional CCA generalizes principal com ...
analysis (gCCA) is the statistical method of choice for making those harmful cross-covariances vanish. This may, however, become quite tedious depending on the nature of the problem.
Applications
The HWB method is critical to satellite geodesy and similar large problems. The HWB method can be extended to
fast Kalman filtering (FKF) by augmenting its
linear regression
In statistics, linear regression is a statistical model, model that estimates the relationship between a Scalar (mathematics), scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A mode ...
equation system to take into account information from numerical forecasts, physical constraints and other ancillary data sources that are available in realtime. Operational accuracies can then be computed reliably from the theory of minimum-norm quadratic unbiased estimation (
Minque
In statistics, the theory of minimum norm quadratic unbiased estimation (MINQUE) was developed by C. R. Rao. MINQUE is a theory alongside other estimation methods in estimation theory, such as the method of moments or maximum likelihood estimatio ...
) of
C. R. Rao.
See also
*
Block matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices.
Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix w ...
Notes
Statistical algorithms
Least squares
Geodesy
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