review of statistics and linear algebra mean: variance:
Post on 12-Jan-2016
222 Views
Preview:
TRANSCRIPT
Review of Statistics and Linear Algebra
Mean:
N
iixN 1
1
Variance:
N
iixN 1
22 )(1
Probabilities of Normal Distribution
x
f(x)
+ +2 +3-3 -2 -
%7.99)]3()3[(
%5.95)]2()2[(
%3.68)]()[(
xP
xP
xP
Covariance ))((1
),cov(1
yyxxN
yx i
N
ii
Correlation coefficientyxyx
i
N
ii yx
yyxxN
r
),cov())((
1
1
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
)var(...),cov(),cov(),cov(
..............
),cov(...)var(),cov(),cov(
),cov(...),cov()var(),cov(
),cov(...),cov(),cov()var(
321
332313
232212
131211
nnnn
n
n
n
xxxxxxx
xxxxxxx
xxxxxxx
xxxxxxx
x1 x2 x3 xn…
x1
x2
x3
xn
…
1...
..............
...1
...1
...1
321
33231
22321
11312
nnn
n
n
n
rrr
rrr
rrr
rrr
x1 x2 x3 xn…
x1
x2
x3
xn
…
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
0)( xIA
333231
232221
131211
aaa
aaa
aaa
100
010
001
333231
232221
131211
aaa
aaa
aaa
Characteristic polynomial:
041
32
23
1 cccc 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
0)( 1 xIA
0
0
0
3
2
1
333231
232221
131211
x
x
x
aaa
aaa
aaa
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 matrixStep 2: calculate eigenvalues and eigenvectors for the above matrixStep 3: transform the data using the eigenvectors.
nnnn aaaa
bbbb
bbbb
bbbb
7321
73332313
72332212
71312111
...
..............
...
...
...Pixel 1Pixel 2
Pixel n
6636616
63332313
62332212
61312111
...
..............
...
...
...
vvvv
vvvv
vvvv
vvvv
nnnn PCPCPCPC
PCPCPCPC
PCPCPCPC
PCPCPCPC
6321
63332313
62332212
61312111
...
..............
...
...
...
=
nx6 6x6 nx6
n=lines samples
Eigenvalues are the variances of principal components, the percent variance or information that a principal component represents is
100%
1
n
ii
pp
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 space2. 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-suchOnly the first two components are often used.The transformation coefficients are:
81.0543.0012.0223.0
194.0039.0522.0829.0
491.0600.0562.0290.0
264.0586.0632.0433.0
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)
redNIR
redNIRNDVI
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 soilFull vegeta
tion co
ver
Partial
vegetatio
n cover
1
*2
a
baPVI redNIR
Where a and b are slope and intercept of the soil line
red
NIRSR
3. Simple Ratio
4. Soil Adjusted Vegetation Index (SAVI)
LLSAVI
redNIR
redNIR
)1(
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, 25:295-309
5. Global Environmental Monitoring Index:
red
redGEMI
1
125.0)25.01(
5.0
5.05.1)(2 22
redNIR
redNIRredNIR
where
Pinty and Verstraete, 1992. Gemi: a non-linear index to monitor global vegetation from satellites. Vegetatio, 101:15-20.
6. Atmospherically Resistant Vegetation Index
rbNIR
rbNIR
RR
RRARVI
Where )( redblueredrb RRRR
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, 1992. Atmospherically Resistant Vegetation Index (ARVI) for EOS-MODIS. IEEE Trans. Geosci. Rem. Sen. 30(2):261-270.
7. Soil and Atmospheric Resistant Vegetation Index
LRR
RRSARVI
rbNIR
rbNIR
Huete and Liu, 1994. 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. 32:897-905
8. Enhanced Vegetation Index
)1(2Re1
Re LLRCRCR
RREVI
BluedNIR
dNIR
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. http://modarch.gsfc.nasa.gov/MODIS/LAND/#vegetation-indices
top related