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Least Squares Method: Calculation of regression coefficients

Least Squares Method: Calculation of regression coefficients . Standard regression equation: T S =< T is > + c T ( X -< X >), Coefficients are estimated from matchups: c =H -1 <( X -< X >) Δ -1 ( T is -< T is >)> X =(x 1 ,x 2 ,x 3 ,…, x N ) T is a vector of regressors,

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Least Squares Method: Calculation of regression coefficients

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  1. Least Squares Method: Calculation of regression coefficients • Standard regression equation: TS=<Tis> + cT(X-<X>), • Coefficients are estimated from matchups: c=H-1<(X-<X>) Δ-1(Tis-<Tis>)> • X=(x1,x2,x3,…,xN)T is a vector of regressors, • <X> is mean X over a set of {X} • Δis covariance matrix of noise • H=<(X-<X>) Δ-1(X-<X>)T> is Hessian - covariance matrix of regressors, • H-1 is covariance matrix of coefficients

  2. Fisher’s information distance in the space of regressors ρ=[(X-<X>)TH-1(X-<X>)]1/2 • ρ characterizes to what extent a given vector of regressors X belongs to {X}, or, how well X canbe distinguished from <X> • On the other hand, H-1 is a covariance matrix of coefficients’ vector c. Therefore, ρ can be interpreted as a standard deviation of the error of TSestimated from {X} • So, ρis a natural measure of the quality of regression estimate

  3. Daytime dependencies of retrieval metrics from ρfor regression algorithms (from matchups)

  4. Daytime dependencies of retrieval metrics from ρfor incremental regression (from matchups)

  5. Nighttime dependencies of retrieval metrics from ρfor regression algorithms (from matchups)

  6. Nighttime dependencies of retrieval metrics from ρfor incremental regression (from matchups)

  7. SST-CMC, (VIIRS, 06/04/2014, Day)

  8. SST-CMC, (VIIRS, 06/04/2014, Night)

  9. Geographical distribution of ρ, (VIIRS, 06/04/2014, Day)

  10. Geographical distribution of ρ, (VIIRS, 06/04/2014, Night)

  11. Statistics of SST-CMC as functions of ρ (VIIRS, Day, 6/4-6/16/2014)

  12. Statistics of SST-CMC as functions of ρ (VIIRS, Night, 6/4-6/16/2014)

  13. Statistics of SST-CMC as functions of ρ (MetopB, Day, 6/3-6/13/2014)

  14. Statistics of SST-CMC as functions of ρ (MetopB, Night, 6/3-6/13/2014)

  15. SST-CMC, (Metop-B, 06/05/2014), all clear pixels Day Night

  16. SST-CMC, (Metop-B, 06/05/2014), ρ < 1.5 Day Night

  17. SST-CMC, (Metop-B, 06/05/2014), 1.5 < ρ <3.5 Day Night

  18. SST-CMC, (Metop-B, 06/05/2014), 3.5 < ρ <5.5 Day Night

  19. SST-CMC, (Metop-B, 06/05/2014), ρ > 5.5 Day Night

  20. VIIRS, 06/12/2014, Regression: Histograms of SST-CMC for different ρ

  21. Metop-B, 06/12/2014, Regression: Histograms of SST-CMC for different ρ

  22. VIIRS, 06/12/2014, IncR: Histograms of SST-CMC for different ρ

  23. Metop-B, 06/12/2014, IncR: Histograms of SST-CMC for different ρ

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