# Minimum Variance with Constrain

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### Contents:

- Overview
- Inputs
- Outputs
- Bugs/Caveats
- History
- References
- Other MPE/ISSI PlugIns

### Overview

This PlugIn performs a minimum variance with an additional constrain. When enabled, the minimum variance is constrained so that the normal component of the input vector time series (or another input vector) is zero. This contrain may give a more stable result of the variance analysis.

The input may also be a 3x3 covariance matrix or 3x3 covariance matrix series. If this is the case, the eigenvalues and eigenvectors of this matrix is calculated. The time series output is then a empty dummy, and the Covariance Matrix output is a copy of the input. Constraining have no effect, and no plots are generated.

### Input data

Required inputs are :
- Input TS or Matrix - vector or 3x3 matrix time series or 3x3 matrix without timetags
- Constrain type - integer
- time interval
- name of output for hodograms

Constrain can be one of the following :
0 - standard covariance matrix (see Blue ISSI book, ch 8)
1 - constrained (same as setting constrain == 1) (see Blue ISSI book, ch 8)
2 - use a Siscoe type variance matrix (see Siscoe et al, JGR no 73, 1968)
3 - use a Weimer type variance matrix (see Weimer, JGR, vol 109, dec 2004)
4 - use Mij = < Vi > < Vj > (Weimer for large N. For demonstration only - this gives nonsense)

Use the methods 2,3 with care; Method 4 is for demonstration purposes, and should not be used. Both method 2 and 4 produce ill-posed covariance matrices, and give one or two negative eigenvalues.

### Outputs

Output quantities are:
- MinVar_TS - input data vector rotated into a MVA coordinate system
- rotation matrix - 3x3 rotation matrix between input- and output vector
- covariance matrix from the minimum variance analysis
- projection matrix from constrained MVA - invalid for uconstrained MVA

If the input is a time series, a hodogram plot is generated. The boundary normal estimate is marked with green.

### Bugs/Caveats

When constrained, or using the Weimer methods, the minimum eigenvalue == 0. The corresponding eigenvector describes the average direction of the input vector. The eigenvector corresponding to the smallest non-zero eigenvalue serves as an estimate of the boundary normal. Keep this in mind when interpreting the hodogram.

Constraining has no effect if the input is a covariance matrix or matrix input. Also, no plot is generated in this case.

### History

2002-05-14 - initial version
2002-08-25 - covariance as output to WL
2004-01-07 - QSAS 2.x version
2004-01-15 - allow matrix input
2005-05-02 - output projection matrix (used for constrain)
2005-08-08 - added possibility to select covariance matrix type
2005-08-19 - corrected minor bug in printout of Bn for matrix type 2

2005-11-09 - removed redundant link with lapack

### References

Siscoe et al, JGR, 1968 : Description of reduced covariance matrix
Weimer, JGR, vol 109, A12, 2004 : Description of Weimer type covariance matrix
Bargatze et al, JGR, A07, vol 110, 2005 : Discussion of Weimers matrix
ISSI book, 'Multi Spacecraft Analysis', chapter 8

SEH, 19 Aug, 2005