Review of Statistics and Linear Algebra Mean: Variance:

Probabilities of Normal Distribution x f(x)  ++ +2  +3  -3  -2 --

Covariance Correlation coefficient If r>0, x and y are positively correlated; if r<0, x and y are negatively correlated. The magnitude of r reflects the strength of correlation between x and y. Q: please draw a diagram to show x and y relationships: a) strongly positively correlated b) strongly negatively correlated c) weakly positively correlated d) weakly negatively correlated

Variance covariance matrix: symmetric x 1 x 2 x 3 x n … x 1 x 2 x 3 x n …

x 1 x 2 x 3 x n … x 1 x 2 x 3 x n … Correlation Coefficients Matrix: Symmetric How would you get correlation coefficients matrix from the variance-covariance matrix?

Eigenvalues and eigenvectors The eigenvalue of a matrix A Characteristic polynomial: We will have three solutions, each of them is called a eigenvalue: 1, 2, 3

Eigenvectors Once we have the eigenvalues, we can substitute the eigenvalues into the following equation to solve for a eigenvector The solution to this linear systems is the eigenvector corresponding to the eigenvalue. Therefore, there is as many eigenvectors as eigenvalues. The eigenvectors can be thought of a basis in a n-dimensional space, meaning that each eigenvectors is like the direction of axis. What is special is that these axes are perpendicular to each other (or orthogonal to each other). All points along the vector direction in the multidimensional space are solutions to the above linear system. Usually, one only use a vector of unit length as the eigenvector.

Principal Component Analysis of Remotely Sensed Data Step 1: calculate variance-covariance matrix/correlation matrix Step 2: calculate eigenvalues and eigenvectors for the above matrix Step 3: transform the data using the eigenvectors. Pixel 1 Pixel 2 Pixel n  = nx6 6x6 nx6 n=lines  samples

Eigenvalues are the variances of principal components, the percent variance or information that a principal component represents is Because satellite data across bands are often highly correlated, usually 95% of the information can be compressed in a few bands. Eigenvectors: The coefficients for each eigenvectors are the weights that a band carry to a principal component. The information content for each component can be explained from: (1) the sign of each coefficients; (2) the magnitude of each coefficients Principal Component Transformation can (1) reduce dimensionality (2) reduce noise (3) improve visual interpretability Interpretation of PCA

PCA Example x1 x2 PC1 PC2 In a extreme case: x1 and x2 is on a straight line, we only need one dimension to represent the whole dataset.

Kauth-Thomas (KT) Transformation (or Tasseled Cap Transformation) Empirical observation of crop development 1.Soils form a line in spectral space 2.Growth of crops make the point moving away from the soil line. On bright soil, growth of crops making the scene less bright, but greener. On dark soil, growth of crops in makes the scene greener, but not as much change in the brightness. Red NIR Dark soil Bright soil Mature crop Soil line Senescence 3. As crops mature, they reach the same point in the spectral space regardless to their soil background. At this point, little soil background can be seen due to canopy closure, minimize its impact on the overall spectral signals. 4. When crops senesce and turn yellow their trajectories remain together and mover away from the green spot. The development of vegetation takes place almost totally in the same plane, while the yellowing development moves out of this plane

Based on the above observation, Kauth and Thomas (1976) that developed a linear transformation from the original 4 Landsat MSS bands to a new set of axes which are orthogonal to each other. The first axis passes along the soil line, and the second axis is perpendicular to the first one passing through the plane of vegetation development. The third axis indicates crop senescence which is perpendicular to both soil and vegetation line. A fourth axis is required to account for the remaining variation. Kauth and Thomas named the four axes as: soil-brightness green-stuff yellow-stuff non-such Only the first two components are often used. The transformation coefficients are:

KT Transformation for TM data Transformation Coefficients for TM images: The most valuable transformations are the first three components: brightness, greenness and wetness. They usually consist of more than 95% of the total information from the 6 reflective bands.

Compare KT Transformation and PCA Common: 1. Linear transformations. 2. Transformed components are orthogonal to each other. Different: 1.PCA coefficients varies from scene, KT coefficients are fixed. 2.PCA components may vary from scene, but KT components are fixed in what each component represents. 3.Interpretation of principal components is not always straightforward and sometimes can be difficult.

Vegetation Indices 1. Normalized Difference Vegetation Index (NDVI) NDVI: [-1.0, 1.0] Often, the more the leaves of vegetation present, the bigger the contrast in reflectance in the red and near-infrared spectra.

2. Perpendicular Vegetation Index (PVI) Wet soil Dry soil Full vegetation cover Partial vegetation cover Where a and b are slope and intercept of the soil line

3. Simple Ratio 4. Soil Adjusted Vegetation Index (SAVI) Where L is an adjustment factor for soil. Huete (1988) found the optimal value for L is 0.5. Huete, 1988.A soil-adjusted vegetation index (SAVI). Remote Sensing of Environment, 25:

5. Global Environmental Monitoring Index: where Pinty and Verstraete, Gemi: a non-linear index to monitor global vegetation from satellites. Vegetatio, 101:15-20.

6. Atmospherically Resistant Vegetation Index Where Developed for use with EOS-MODIS data on a global scale by Kaufman and Tanre (1992). The  value is usually takes the value of 1.0. What this does is to correct for atmospheric effect on the reflectance value for red band Kaufman and Tanre, Atmospherically Resistant Vegetation Index (ARVI) for EOS-MODIS. IEEE Trans. Geosci. Rem. Sen. 30(2):

7. Soil and Atmospheric Resistant Vegetation Index Huete and Liu, An error and sensitivity analysis of the atmospheric- and soil-correcting variants of the Normalized Difference Vegetation Index for the MODIS-EOS. IEEE Trans. Geosci. Rem. Sen. 32:

8. Enhanced Vegetation Index Where C1, C2 coefficients adjusting for atmospheric effects and L is a soil adjustment factor. They are empirically determined as C1=6.0, C2=7.5 and L=1.0. EVI has improved sensitivity to high biomass regions. Huete and Justice, 1999 MODIS vegetation index.