TWiki> Main Web>TWikiUsers>AlexanderFedotov>AVFedotovLogA>AVFLogA012LeastSquares (revision 5)~~Edit~~~~Attach~~~~PDF~~

-- AlexanderFedotov - 2015-03-03

## Wikipedia links

## Uncorrelated measurements

where the *weight matrix* of dimension
is diagonal and defined as the *inverse of the diagonal covariance matrix for* :
i.e. .

In matrix notation (considering and as columns and respectively), one has
or
In a general case of linear transformation ,
the covariance matrice for is transformed into that for
via . Hence,
With by the definition of , that simplifies to
and
## Correlated measurements

where .
and
## An iterative solution

The subset of the parameters corresponding to the group
of indices, can be considered as an column
Let denote the set of indices
that are complementary to indices in .
asd

## Linear Least Squares |

- Least squares http://en.wikipedia.org/wiki/Least_squares
- Linear least squares http://en.wikipedia.org/wiki/Least_squares#Linear_least_squares
- Weighted least squares http://en.wikipedia.org/wiki/Least_squares#Weighted_least_squares

- Linear least squares (mathematics) http://en.wikipedia.org/wiki/Linear_least_squares_%28mathematics%29
- Weighted linear least squares http://en.wikipedia.org/wiki/Linear_least_squares_%28mathematics%29#Weighted_linear_least_squares

- Propagation of uncertainty http://en.wikipedia.org/wiki/Propagation_of_uncertainty

Let with variances be measurements of functions with the known matrix and unknown parameters .

In linear least square method, one estimates the parameter vector by minimizing over the expression

In matrix notation (considering and as columns and respectively), one has

The estimate is the solution of the system of equations

Note, that

Let be uncorrelated measurements as those in the previous section, and ( is an invertible matrix). Then are generally correlated and have the covariance matrix .

With and , one gets

Similarly,

Thus, all the formulae for the correlated measurements are similar to those for the uncorrelated , with the only complication being the replacement of a diagonal weight matrix with a non-diagonal one:

Let indices of parameters be distributed among groups
with sizes respectively.

Consider the following iterative procedure.

- Start with an vector as an initial approximation to
- Make N steps or, equivalently, iterations for , thus finding vectors by minimizing over at step :
where is the "point" where as a function of parameters , takes minimum (while the rest parameters are fixed at the values obtained in the previous iteration: ) - Repeat (3.3) infinitly, defining for (i.e. ) .

Topic revision: r5 - 2015-03-11 - AlexanderFedotov

**Webs**

- ABATBEA
- ACPP
- ADCgroup
- AEGIS
- AfricaMap
- AgileInfrastructure
- ALICE
- AliceEbyE
- AliceSPD
- AliceSSD
- AliceTOF
- AliFemto
- ALPHA
- ArdaGrid
- ASACUSA
- AthenaFCalTBAna
- Atlas
- AtlasLBNL
- AXIALPET
- CAE
- CALICE
- CDS
- CENF
- CERNSearch
- CLIC
- Cloud
- CloudServices
- CMS
- Controls
- CTA
- CvmFS
- DB
- DefaultWeb
- DESgroup
- DPHEP
- DM-LHC
- DSSGroup
- EGEE
- EgeePtf
- ELFms
- EMI
- ETICS
- FIOgroup
- FlukaTeam
- Frontier
- Gaudi
- GeneratorServices
- GuidesInfo
- HardwareLabs
- HCC
- HEPIX
- ILCBDSColl
- ILCTPC
- IMWG
- Inspire
- IPv6
- IT
- ItCommTeam
- ITCoord
- ITdeptTechForum
- ITDRP
- ITGT
- ITSDC
- LAr
- LCG
- LCGAAWorkbook
- Leade
- LHCAccess
- LHCAtHome
- LHCb
- LHCgas
- LHCONE
- LHCOPN
- LinuxSupport
- Main
- Medipix
- Messaging
- MPGD
- NA49
- NA61
- NA62
- NTOF
- Openlab
- PDBService
- Persistency
- PESgroup
- Plugins
- PSAccess
- PSBUpgrade
- R2Eproject
- RCTF
- RD42
- RFCond12
- RFLowLevel
- ROXIE
- Sandbox
- SocialActivities
- SPI
- SRMDev
- SSM
- Student
- SuperComputing
- Support
- SwfCatalogue
- TMVA
- TOTEM
- TWiki
- UNOSAT
- Virtualization
- VOBox
- WITCH
- XTCA

Welcome Guest

Copyright &© 2008-2019 by the contributing authors. All material on this collaboration platform is the property of the contributing authors.

Ideas, requests, problems regarding TWiki? Send feedback

Ideas, requests, problems regarding TWiki? Send feedback