Principal Component Analysis (PCA) Designed to reduce redundancy in multispectral bands Spectral correlation from band to band Either enhancement prior to visual interpretation or pre-processing for classification or other analysis Compress all info originally in many bands into fewer bands AABU

descriptive image statistics The mode is the value that occurs most frequently in a distribution and is usually the highest point on the curve (histogram). It is common, however, to encounter more than one mode in a remote sensing dataset. The median is the value midway in the frequency distribution. One-half of the area below the distribution curve is to the right of the median, and one-half is to the left The mean is the arithmetic average and is defined as the sum of all brightness value (BV)oobservations divided by the number of observations.

Cont’ Min Max Variance Standard deviation Coefficient of variation (CV) Skewness Kurtosis

Measures of Distribution (Histogram) Asymmetry and Peak Sharpness Skewness is a measure of the asymmetry of a histogram and is computed using the formula: A perfectly symmetric histogram has a skewness value of zero. If a distribution has a long right tail of large values, it is positively skewed, and if it has a long left tail of small values, it is negatively skewed.

Measures of Distribution (Histogram) Asymmetry and Peak Sharpness A histogram may be symmetric but have a peak that is very sharp or one that is subdued when compared with a perfectly normal distribution. A perfectly normal distribution (histogram) has zero kurtosis. The greater the positive kurtosis value, the sharper the peak in the distribution when compared with a normal histogram. Conversely, a negative kurtosis value suggests that the peak in the histogram is less sharp than that of a normal distribution. Kurtosis is computed using the formula:

4. Multivariate Image Statistics Remote sensing research is often concerned with the measurement of how much radiant flux is reflected or emitted from an object in more than one band. It is useful to compute multivariate statistical measures such as covariance and correlation among the several bands to determine how the measurements covary. Later it will be shown that variance–covariance and correlation matrices are used in remote sensing principal components analysis (PCA), feature selection, classification and accuracy assessment.

Covariance The different remote-sensing-derived spectral measurements for each pixel often change together in some predictable fashion. If there is no relationship between the brightness value in one band and that of another for a given pixel, the values are mutually independent; that is, an increase or decrease in one band’s brightness value is not accompanied by a predictable change in another band’s brightness value. Because spectral measurements of individual pixels may not be independent, some measure of their mutual interaction is needed. This measure, called the covariance, is the joint variation of two variables about their common mean.

The Sample Covariance A'kif RS

Correlation To estimate the degree of interrelation between variables in a manner not influenced by measurement units, the correlation coefficient, is commonly used. The correlation between two bands of remotely sensed data, rkl, is the ratio of their covariance (covkl) to the product of their standard deviations (sksl); thus:

If we square the correlation coefficient (rkl), we obtain the sample coefficient of determination (r2), which expresses the proportion of the total variation in the values of “band 1” that can be accounted for or explained by a linear relationship with the values of the random variable “band 2.” Thus a correlation coefficient (rkl) of 0.70 results in an r2 value of 0.49, meaning that 49% of the total variation of the values of “band l” in the sample is accounted for by a linear relationship with values of “band k”.

BAND 1 VS BAND 2 Example: investigate relationship between band 1 and band 2 Data: sample group response data on DNs of band 1, and corresponding band 2 DNs A'kif RS

BAND 1 VS BAND 2 N Band 1 DNs (X ) Band 2 DNs (Y ) 1 45 2 5 42 3 10 33 45 2 5 42 3 10 33 4 15 31 20 29 A'kif RS

BAND 1 VS BAND 2

Band 1 vs band2 DNS Observe that as band 1 DNs goes up, corresponding Band 2 DNS goes down Variables covary inversely Covariance and Correlation quantify relationship A'kif RS

Covariance Variables that covary inversely, like band 1 and band , tend to appear on opposite sides of the group means Average product of deviation measures extent to which variables covary, the degree of linkage between them A'kif RS

Calculating Covariance Band 1 (X ) Band 2 (Y )   45 5 42 10 33 15 31 20 29 36 A'kif RS

Calculating Covariance Band 1 (X )   Band 2 (Y ) -10 -90 9 45 5 -5 -30 6 42 10 -3 33 15 -25 31 20 -70 -7 29 ∑= -215 A'kif RS

Covariance Calculation (2) Evaluation yields, A'kif RS

Table for Calculating rxy Band 1 (X ) X 2 XY Y 2 Band 2 (Y ) 2025 45 5 25 210 1764 42 10 100 330 1089 33 15 225 465 961 31 20 400 580 841 29 ∑= 50 750 1585 6680 180 A'kif RS

Computing rxy from Table A'kif RS

Computing Correlation A'kif RS

example Band 1 (Band 1 x Band 2) Band 2 130 7,410 57 165 5,775 35 100 Pixel Band 1 (green) Band 2 (red) Band 3 (ni) Band 4 (ni) (1,1) 130 57 180 205 (1,2) 165 35 215 255 (1,3) 100 25 135 195 (1,4) 50 200 220 (1,5) 145 65 235 Band 1 (Band 1 x Band 2) Band 2 130 7,410 57 165 5,775 35 100 2,500 25 135 6,750 50 145 9,425 65 675 31,860 232

Univariate statistics Band 1 Band 2 Band 3 Band 4 Mean (mk) 135 46.40 187 222 Variance (vark) 562.50 264.80 1007 570 (sk) 23.71 16.27 31.4 23.87 (mink) 100 25 195 (maxk) 165 65 215 255 Range (BVr) 40 80 60 Univariate statistics Band 1 Band 2 Band 3 Band 4 562.25 - 135 264.80 718.75 275.25 1007.50 537.50 64 663.75 570 Band 1 Band 2 Band 3 Band 4 - 0.35 0.95 0.53 0.94 0.16 0.87 covariance Covariance Correlation coefficient

Dimension rotation y y’ x’ 0.7x,0.7y -0.7x, 0.7y + x y 0.5x,0.87y

Principal Component Analysis (PCA) Designed to reduce redundancy in multispectral bands Spectral correlation from band to band Either enhancement prior to visual interpretation or pre-processing for classification or other analysis Compress all info originally in many bands into fewer bands AABU

Principal Component Analysis (PCA) - The math behind the button In the simple case of 45º axis rotation, Finding q PC1 PC2’ [ ] = [ ] [ ] [ ] [ ] DN1’ DN2’ cos q sin q -sin q cos q DN1 DN2 n11 n12 n21 n22 = cov = q = 45º Cov’=RTcovR; cov’ is the matrix having eigenvalues as diagonal elements and RT is the transpose of R. Eigenvalues can be found by diagonalizing cov. R has eigenvectors as column vectors http://www.cs.otago.ac.nz/cosc453/student_tutorials/principal_components.pdf AABU

Covariance under Affine Transformation A'kif RS

Covariance under Affine Transf (2) A'kif RS

Principal Component Analysis In the simple case of 45º axis rotation, The rotation in PCA depends on the data. In the top case, all the image data have similar DN2/DN1 ratios but different intensities, and PC1 passes through the elongated cluster. In the bottom example, vegetation causes there to be 2 mixing lines (different DN4/DN3 ratios (and the “tasseled cap” distribution such that PC1 still passes through the centroid of the data, but is a different rotation that in the top case. PC1 PC2

Tasseled Cap Transformation Transforms (rotates) the data so that the majority of the information is contained in 3 bands that are directly related to physical scene characteristics Brightness (weighted sum of all bands – principal variation in soil reflectance) Greenness (contrast between NIR and VIS bands Wetness (canopy and soil moisture)

Tasseled Cap Transformation (TCT) TCT is a fixed rotation that is designed so that the mixing line connecting shadow and sunlit green vegetation parallels one axis and shadow-soil another. It is similar to the PCT. Soil Green

Next lecture – Spectral Mixture Analysis