Chapter 28

Cononical Correction Regression Analysis used for Temperature

Retrieval

This method attempts to generate an optimum statistical relationship between a number of input variables (one or more of which are generally the parameters of interest) and a number of output variables (usually the observed spectral vectors).

2n

1 ii=1

ε = - y y

Lacking probability distributions, we can best estimate y given x by minimizing the squared error between a linear estimate of y expressed as and y according to:y

(1)

=Y Xβ

=y xβ

Ordinary Least Squares (OLS) regression theory tells us that the optimal linear combination of X is given by:

More fundamentally, the least square prediction of Y is given by

or

(2)

(3)

(4)

-1T T=β X X X Y

xx

T

xx= X X

-=

n

x ux

note that if we mean center X (i.e. subtract the mean value (u) and scale by the number of observations (n) then XTX is the covariance matrix i.e.

i.e. x is redefined as:

(5)

(6)

xx= A ΛA

=U XA

we can decompose the covariance matrix into its principle components (aka eigen vectors). The Eigen vector matrix can be expressed as

The X data are projected (transformed) onto the principle components space according to:

(7)

(8)

Cononical Correlation Analysis (CCA)

PCR assumes the principle components of X will yield good predictions of Y based only on the covariance of X. In cononical correlating analysis (CCA), the joint covariance of X and Y is considered. In this process, an optimal orthogonal space is produced where the projections of both X and Y are maximally correlated.

-1 -1XX XY YY YX =Σ Σ Σ Σ A ΨA

-1 -1YY YX XX XY =Σ Σ Σ Σ B ΨB

=U XA

=V YB

The cononical correlations are the eigen values of:

where Ψ is the k x k diagonal matrix of squared cononical correlations, k is the min (p, q), A is the matrix of column eigen vectors used to transform X and B is the matrix of column eigen vectors used to transform Y, i.e.

(9)

(10)

(11)

(12)

Hernandez-Baquero and Schott point out three useful properties:

1. The cononical variables are orthogonal2. The cononical correlations are the maximum linear correlations between the data sets

3. and XX =AΣ A I YY =BΣ B I

To find the estimates of Y (i.e. ) from the predicted values

-1T Tcc= =V UB U U U U V

V =V YB

T TYY YY= = =Y VB Σ YBB Σ YI Y

Y V

If we regress the cononical variables using Equation 2, we obtain

let

and use property 3 above, such that

(13)

(14)

(15)

Canonical Correlation Analysisy1

y2

y3

yq

x1

x2

x3

xp

v1

v2

vr

u1

u2

ur

.

.

.

.

.

.

.

.

.

.

.

.

weights weights

loadings loadings

From a practical standpoint, we still need in CCR to invert the covariance matrices as expressed in Equations 9 and 10. In general, these matrices may be singular (i.e. of reduced rank) and their Inverse will not exist. In these cases, we can use the singular value decomposition (SVD) to define the rank and reconstruct the matrices using the SVD approach discussed elsewhere in these notes. The reconstructed matrices are more amenable to inversions.

Examples

In the first case, the Y ensemble was made up of n temperature and water vapor profiles at q altitudes to form a n x 2q matrix (i.e. each new row was temperatures at q altitudes followed by q corresponding water vapor values

The X data were generated to simulate nighttime Modis Airborne Simulator (MAS) collection that took place over Death Valley at 21km

The X example for this first test was made up of p = 9 element spectral radiance vectors (i.e. 9 spectral bands) forming a ( n x p ) matrix.

Locations of Radiosonde Data

IGARSS: Hernandez Table 1

CCA Implementation

MODTRAN

Radiosonde

d

u

L

L

)(

)(

zRH

zT

)(

)(

)(

zRH

zp

zT

obsL

CCA

rr VU

VU

VU

22

11

CCUV ˆsT

sT

IGARSS: Hernandez Table 2

s uL = L - L /

CCR was used to predict the τ(λ), (Luλ) , (Lpλ) and vectors and solve for the surface leaving radiance according to

(16)

A second experiment was conducted with 10 bands of LWIR data from the MASTER sensor flown over 2 water and 1 land target on 3 different missions shown in Figure 3 from Hernandez-Baquero and Schott (SPIE).

Aero Sense: Hernandez Figure 3

Aero Sense: Hernandez Table 2

Brightness temperatures were used because they are more linear with temperature and CCR is a linear process

Aero Sense: Hernandez Table 3