# Principal Components Analysis

A practical introduction in Principal Components Analysis (or Transformation) that aims to:

- highlight the importance of the values returned by PCA
- address numerical accuracy issues with respect to the default implementation of PCA in GRASS through the
*i.pca*module.

*This page is still not complete*

# Principal Components Analysis

Principal Components Analysis (PCA) is a dimensionality reduction technique used extensively in Remote Sensing studies (e.g. in change detection studies, image enhancement tasks and more). PCA is in fact a linear transformation applied on (usually) highly correlated multidimensional (e.g. multispectral) data. The input dimensions are transformed in a new coordinate system in which the produced dimensions (called principal components) contain, in decreasing order, the greatest variance related with unchanged landscape features.

PCA has two algebraic solutions:

**Eigenvectors of Covariance**(or Correlation) of a given data matrix**Singular Value Decomposition**of a given data matrix

The **SVD** method is used for numerical accuracy [R Documentation]

## Background

The basic steps of the transformation are:

**organizing a dataset in a matrix****data centering**(that is: subtracting the dimensions means from themself so each of the dimensions in the dataset has zero mean)- calculate
- the
**covariance matrix**(**non-standardised PCA**) or - the
**correlation matrix**(**standartised PCA**, also known as scaling)

- the
- calculate either
- the
**eigenvectors**and**eigenvalues**of the covariance (or the correlation) matrix or - the
**SVD**of the data matrix

- the
**sort variances in decreasing order**(decreasing eigenvalues; this is default in eigenvalue analysis)**project original dataset**(PC's or PC scores: eigenvector * input-data) to get*signals*

**Why is data centering performed?**

Data centering reduces the square mean error of approximating the input data [*A.A. Miranda, Y.-A. Le Borgne, and G. Bontempi. New Routes from Minimal Approximation Error to Principal Components, Volume 27, Number 3 / June, 2008, Neural Processing Letters, Springer*].

**Why is data scaling performed?**

Scaling normalises(=standartises) the variables to have unit variance before the analysis takes place. This normalisation prevents certain features to dominate the analysis because of their large numerical values [*Duda and Hart (1973), Eklundh and Singh (1993)*]. However "if the variables are measured on comparable scales, unstandartised data may be appropriate". [*Maindonald, John; Braun, John: Data Analysis and Graphics Using R. 2. Aufl. 2007, Cambridge University Press*]

## Solutions to PCA

The **Eigenvector** solution to PCA involves:

- calculation of
- the covariance matrix of the given multidimensional dataset (non-standardised PCA)
**or** - the correlation matrix of the given multidimensional dataset (standardised PCA)

- the covariance matrix of the given multidimensional dataset (non-standardised PCA)
- calculation of the eigenvalues and eigenvectors of the covariance (or correlation) matrix
- transformation of the input dataset using the eigenvalues as weighting coefficients

## Terminology

**eigenvalues** represent either the variance of the original data contained in each principal component (in case they were computed from the covariance matrix), or the amount of correlation captured by the respective principle components (in case they were computed from the correlation matrix).

**eigenvectors**

- act as weighting coefficients
- represent the contribution of each original dimension the principal components
- tell how the principle components relate to the the data variables (dimensions).

Suppose an eigenvector is (0, 0, 1, 0), this indicates that the corresponding principle component is equal to dimension 3 and perpendicular to dimensions 1, 2 and 4. If it is (0.7, 0.7, 0, 0) the PC is equally determined by dimension 1 and 2 (it averages them), and not determined by dimensions 3 and 4. If it is (0.7, 0, -0.7, 0) it is the difference between dimension 1 and 3. Eigenvectors are normalized, meaning that the sum of its squared elements equals 1.

**loadings** The individual numbers in an eigenvector are called loadings.

**scores** When the original data are projected onto the eigenvectors, the resulting new variables are called the principle components; the new data values are called PC scores.

## Performing PCA with GRASS

**m.eigensystem**

The *m.eigensystem* module implements the eigenvector solution to PCA. The respective function in R is *princomp()*. A comparison of their results confirms their almost identical performance. Specifically,

- the standard deviations (sdev) reported by
*princomp()*are (almost) identical with the variances (eigenvalues) reported by*m.eigensystem*.

*princomp()*scales (also referred as normalization) the eigenvectors and so does*m.eigensystem*. The scaled(=normalised) eigenvectors produced by m.eigensystem are marked with the capital letter N.

**i.pca**

The *i.pca* module performs PCA based on the SVD solution with data centering but without scaling. A comparison of the results derived by *i.pca* and R's *prcomp()* function confirms this. Specifically, *i.pca* yields the same eigenvectors as R's *prcomp()* function does with the following options:

prcomp(x, center=TRUE, scale=FALSE)

where x is a numeric or complex matrix (or data frame) which provides the data for the principal components analysis (R Documentation).

## Useful details

**princomp()**

If one computes principle components with R, the loadings are printed by default such that loadings close to 0 (between -.1 and .1, this can be controlled) are left out. This can be overriden (see the help page of function loadings). The reason for this is that for large loading tables, the real information is in loadings not close to 0; reading large loading tables is much easier when only important loadings are printed.

# Examples using SPOT imagery

Get SPOT images from the Spearfish dataset

## Eigenvectors solution

### Based on the covariance matrix

#### Using **m.eigensystem**

**m.eigensystem**

Note: download m.eigensystem from GRASS_AddOns#m.eigensystem GRASS-Addons

Command in one line

(echo 3; r.covar spot.ms.1,spot.ms.2,spot.ms.3) | m.eigensystem

Output

r.covar: complete ... 100% E 1159.7452017844 .0000000000 88.38 V .6910021591 .0000000000 V .7205280412 .0000000000 V .4805108400 .0000000000 N.6236808478.0000000000 N.6503301526.0000000000 N.4336967751.0000000000 W 21.2394712045 .0000000000 W 22.1470141296 .0000000000 W 14.7695575384 .0000000000 E 5.9705414972 .0000000000 .45 V .7119385973 .0000000000 V -.6358200627 .0000000000 V -.0703936743 .0000000000 N .7438340890 .0000000000 N -.6643053754 .0000000000 N -.0735473745 .0000000000 W 1.8175356507 .0000000000 W -1.6232096923 .0000000000 W -.1797107407 .0000000000 E 146.5031967184 .0000000000 11.16 V .2265837636 .0000000000 V .3474697082 .0000000000 V -.8468727535 .0000000000 N .2402770238 .0000000000 N .3684685345 .0000000000 N -.8980522763 .0000000000 W 2.9082771721 .0000000000 W 4.4598880523 .0000000000 W -10.8698904856 .0000000000

*Note that the output is not sorted by relative importance.*

The **red-bold** numbers are the normalised eigen vectors. Compare the above results with R's *princomp()* function below.

#### Using R's **princomp()** function

**princomp()**

Launch R from within GRASS

R

Load spgrass6() and projection information in R

library(spgrass6) G <- gmeta6()

Read spot.ms bands

spot.ms <- readRAST6(c('spot.ms.1', 'spot.ms.2', 'spot.ms.3'))

Work around NA's

spot.ms.nas <- which(is.na(spot.ms@data$spot.ms.1) & is.na(spot.ms@data$spot.ms.2) & is.na(spot.ms@data$spot.ms.3)) spot.ms.values <- which(!is.na(spot.ms@data$spot.ms.1) & !is.na(spot.ms@data$spot.ms.2) & !is.na(spot.ms@data$spot.ms.3)) spot.ms.nonas <- spot.ms.values@data[spot.ms.values, ]

A better option is to use R's *complete.cases()* function. See *?complete.cases* within R.

Command

princomp(spot.ms.nonas)

Another possible way is to use the function na.omit() and outputs are exactly the same

princomp(na.omit(spot.ms))

Output

Call: princomp(x = modis) Standard deviations: Comp.1 Comp.2 Comp.3 34.055018 12.103846 2.443468 3 variables and 1231860 observations.

Get loadings

princomp(spot.ms.nonas)$loadings Loadings: Comp.1 Comp.2 Comp.3 spot.ms.1-0.6240.240 0.744 spot.ms.2-0.650-0.368 -0.664 spot.ms.3-0.4340.898 Comp.1 Comp.2 Comp.3 SS loadings 1.000 1.000 1.000 Proportion Var 0.333 0.333 0.333 Cumulative Var 0.333 0.667 1.000

The **red-bold** eigen vectors of the first principal component, ease of the comparison with the results derived from GRASS' *m.eigensystem* module above.

Note the missing eigen value in row 3, column 3. It is omitted on purpose. You will appreciate how useful this is when you compute 20 PC's from 120 dimensions. It will be printed when you explicitly ask for showing loadings between -0.1 and 0.1, by

print(princomp(spot.ms.nonas)$loadings, cutoff=0)

### Based on the correlation matrix

#### Using **m.eigensystem**

**m.eigensystem**

Command in one line

(echo 3; r.covar -r spot.ms.1,spot.ms.2,spot.ms.3) | m.eigensystem

Output

r.covar: complete ... 100% E 2.5897901435 .0000000000 86.33 V -.7080795162 .0000000000 V -.6979341819 .0000000000 V -.6128387525 .0000000000 N -.6062688915 .0000000000 N -.5975822957 .0000000000 N -.5247222419 .0000000000 W -.9756579133 .0000000000 W -.9616787268 .0000000000 W -.8444263177 .0000000000 E .0123265666 .0000000000 .41 V -.6690685456 .0000000000 V .6302711261 .0000000000 V .0552608155 .0000000000 N -.7265842171 .0000000000 N .6844516242 .0000000000 N .0600112449 .0000000000 W -.0806690651 .0000000000 W .0759912909 .0000000000 W .0066627528 .0000000000 E .3978832898 .0000000000 13.26 V .3005969725 .0000000000 V .3883277727 .0000000000 V -.7895613377 .0000000000 N.3232853332.0000000000 N.4176378502.0000000000 N-.8491555920.0000000000 W .2039218921 .0000000000 W .2634375639 .0000000000 W -.5356302845 .0000000000

*Note that the output is not sorted by relative importance. In this example the second principal component (accounts for 13.26% of the original variance) can be created by using numbers (i.e. the W lines) from the third group of eigen vectors. To compare with princomp()'s results look at column Comp.2 below*.

#### Using R's **princomp()** function

**princomp()**

Perform PCA

princomp(spot.ms.nonas, cor=TRUE)

Output

Call: princomp(x = spot.ms.nonas, cor = TRUE) Standard deviations: Comp.1 Comp.2 Comp.3 1.6092826 0.6307795 0.1110256 3 variables and 1231860 observations.

Get loadings

princomp(spot.ms.nonas, cor=TRUE)$loadings

Output

Loadings: Comp.1 Comp.2 Comp.3 spot.ms.1 -0.606-0.3230.727 spot.ms.2 -0.598-0.418-0.684 spot.ms.3 -0.5250.849Comp.1 Comp.2 Comp.3 SS loadings 1.000 1.000 1.000 Proportion Var 0.333 0.333 0.333 Cumulative Var 0.333 0.667 1.000

## SVD solution

#### Using *i.pca*

Command

```
i.pca input=spot.ms.1,spot.ms.2,spot.ms.3 out=pca.spot.ms rescale=0,0
```

Output

Eigen values, (vectors), and [percent importance]: PC11170.12(-0.6251,-0.6536,-0.4268)[88.07%] PC2 152.49 ( 0.2328, 0.3658,-0.9011)[11.48%] PC3 6.01 ( 0.7450,-0.6626,-0.0765) [0.45%]

#### Using R's *prcomp()* function

The following example replicates *i.pca'*s solution using the same data **with data centering and without scaling** (options *center=TRUE* and *scale=FALSE*). Note that these settings are the defaults.

Command

prcomp(spot.ms.nonas)

Output

Standard deviations: [1] 34.055032 12.103851 2.443469 Rotation: PC1 PC2 PC3 spot.ms.1 -0.6236808 0.2402770 -0.74383409 spot.ms.2 -0.6503302 0.3684685 0.66430538 spot.ms.3 -0.4336968 -0.8980523 0.07354738

In this example *i.pca'*s performance seems to be identical to R's *prcomp()* function which means that data centering is applied prior to the actual PCA.

The three SPOT bands used in this example have the following ranges:

- spot.ms.1

min=24 max=254

- spot.ms.2

min=14 max=254

- spot.ms.3

min=12 max=254

Nevertheless, as shown in the examples using MODIS surface reflectance products (read below) in which the range of the bands varies significantly, *i.pca'*s results do not match the results of R's *prcomp()* function with the parameter *center* set to *TRUE*. Instead, the results derived from *i.pca* are almost identical when the parameter *center* is set to *FALSE*.

The following example performs PCA **without data centering and scaling** (options *center=FALSE* and *scale=FALSE*).

Command

prcomp(spot.ms.nonas, center=FALSE, scale=FALSE)

Output

Standard deviations: [1]101.0092715.79861 2.83775 Rotation: PC1 PC2 PC3 spot.ms.1 -0.5605487 -0.3652694 0.7432116 spot.ms.2 -0.4726613 -0.5958032 -0.6493149 spot.ms.3 -0.6799827 0.7152600 -0.1613280

+++

The following example performs PCA **with data centering and scaling** (options *center=TRUE* and *scale=TRUE*)

Command

prcomp(spot.ms.nonas, center=TRUE, scale=TRUE)

Output

Standard deviations: [1] 1.6092826 0.6307795 0.1110256 Rotation: PC1 PC2 PC3 spot.ms.1 -0.6062688 0.3232856 -0.72658417 spot.ms.2 -0.5975823 0.4176378 0.68445170 spot.ms.3 -0.5247224 -0.8491555 0.06001103

# Examples using MODIS surface reflectance products

## Eigenvectors solution

### Based on the covariance matrix

#### Using **m.eigensystem**

**m.eigensystem**

Command in one line

(echo 3; r.covar b02,b06,b07) | m.eigensystem

Output

- The E line is the eigen value. (Real part, imaginary part, percent importance)
- The
`V`lines are the eigen vectors associated with E. - The N lines are the
`V`vectors normalized to have a magnitude of 1.**These are the scaled eigen vectors that correspond to princomp()'s results presented in the following section.** - The W lines are the N vector multiplied by the square root of the magnitude of the eigen value(E).
**Generally referred to as "the factor loadings", also called "component loadings".**

r.covar: complete ... 100% E 778244.0258462029 .0000000000 79.20 V .5006581842 .0000000000 V .8256483300 .0000000000 V .6155834548 .0000000000 N.4372107421.0000000000 N.7210155161.0000000000 N.5375717557.0000000000 W 385.6991853500 .0000000000 W 636.0664787886 .0000000000 W 474.2358050886 .0000000000 E 192494.5769628266 .0000000000 19.59 V -.8689798010 .0000000000 V .0996340298 .0000000000 V .5731134848 .0000000000 N -.8309940700 .0000000000 N .0952787255 .0000000000 N .5480609638 .0000000000 W -364.5920328433 .0000000000 W 41.8027823088 .0000000000 W 240.4573848757 .0000000000 E 11876.4548199713 .0000000000 1.21 V .2872248982 .0000000000 V -.5731591248 .0000000000 V .5351449518 .0000000000 N .3439413070 .0000000000 N -.6863370819 .0000000000 N .6408165005 .0000000000 W 37.4824307850 .0000000000 W -74.7964308085 .0000000000 W 69.8356366100 .0000000000

In general, the solution to the eigen system results in complex
numbers (with both real and imaginary parts). However, in the example
above, since the input matrix is symmetric (i.e., inverting the rows and columns
gives the same matrix) the eigen system has only real values (i.e., the
imaginary part is zero).
This fact makes it possible to use eigen vectors to perform principle component
transformation of data sets. The covariance or correlation
matrix of any data set is symmetric
and thus has only real eigen values and vectors.

The **red-bold** eigen vectors of the first principal component, ease of the comparison with the results derived from R's *princomp()* function that follows.

~~Using the W vector, new maps can be created:~~. The new maps (coordinate system) system is formed by the normalized eigenvectors (N) of the variance–covariance (or correlation) matrix [*Tso, B. & P.M. Mather. Classification methods for remotely sensed data. 2001. Taylor & Francis, London ; New York*].

r.mapcalc'pc.1 = .4372107421*map.1 +.7210155161 *map.2 + .5375717557*map.3'r.mapcalc'pc.2 = -.8309940700 *map.1 + .0952787255*map.2 + .5480609638*map.3'r.mapcalc'pc.3 = .3439413070*map.1 - .6863370819*map.2 + .6408165005*map.3'

Visualize results:

d.mon x0 d.rast pc.1 d.rast pc.2 d.rast pc.3

#### Using R's **princomp()** function

**princomp()**

Command

princomp(modis)

Output

Call: princomp(x = modis) Standard deviations: Comp.1 Comp.2 Comp.3 857.5737 436.0922 108.5083 3 variables and 350596 observations.

Get loadings

(princomp(modis))$loadings Loadings: Comp.1 Comp.2 Comp.3 b02-0.4180.839 0.348 b06-0.725-0.684 b07-0.547-0.539 0.641

Comp.1 Comp.2 Comp.3 SS loadings 1.000 1.000 1.000 Proportion Var 0.333 0.333 0.333 Cumulative Var 0.333 0.667 1.000

The **red-bold** eigen vectors of the first principal component, ease of the comparison with the results derived from GRASS' *m.eigensystem* module above.

### Based on the correlation matrix

#### Using **m.eigensystem**

**m.eigensystem**

Command in one line

(echo 3; r.covar -r MOD07_b02,MOD07_b06,MOD07_b07)|m.eigensystem

Output

r.covar: complete ... 100% E 2.2915877718 .0000000000 76.39 V -.5755655569 .0000000000 V -.7660355041 .0000000000 V -.6809380186 .0000000000 N -.4896413269 .0000000000 N -.6516766616 .0000000000 N -.5792830912 .0000000000 W -.7412186091 .0000000000 W -.9865075560 .0000000000 W -.8769182329 .0000000000 E .6740687010 .0000000000 22.47 V .8667178982 .0000000000 V -.1116525720 .0000000000 V -.6069908335 .0000000000 N.8145815825.0000000000 N-.1049362531.0000000000 N-.5704780699.0000000000 W .6687852213 .0000000000 W -.0861544341 .0000000000 W -.4683721194 .0000000000 E .0343435272 .0000000000 1.14 V .2486404469 .0000000000 V -.6006166822 .0000000000 V .4655120098 .0000000000 N .3109794470 .0000000000 N -.7512029762 .0000000000 N .5822249325 .0000000000 W .0576307320 .0000000000 W -.1392129859 .0000000000 W .1078979635 .0000000000

The **red-bold** eigen vectors of the second principal component, ease of the comparison with the results derived from R's *princomp()* function that follows.

#### Using R's **princomp()** function

**princomp()**

Command

princomp(mod07, cor=TRUE)

Output

Call: princomp(x = mod07, cor = TRUE)

Standard deviations: Comp.1 Comp.2 Comp.3 1.5030740 0.8397807 0.1885121 3 variables and 350596 observations.

Get loadings

(princomp(mod07, cor=TRUE))$loadings

Output

Loadings: Comp.1 Comp.2 Comp.3 MOD2007_242_500_sur_refl_b02 -0.4810.8200.310 MOD2007_242_500_sur_refl_b06 -0.656-0.102-0.748 MOD2007_242_500_sur_refl_b07 -0.582-0.5630.587 Comp.1 Comp.2 Comp.3 SS loadings 1.000 1.000 1.000 Proportion Var 0.333 0.333 0.333 Cumulative Var 0.333 0.667 1.000

The **red-bold** eigen vectors of the second component, ease of the comparison with the results derived from GRASS' *m.eigensystem* module above.

Comments
**Add comments here...**

## SVD

#### Using *i.pca*

Command

i.pca input=b2,b6,b7 output=pca.b267

Output

Eigen values, (vectors), and [percent importance]: PC16307563.04( -0.64, -0.65, -0.42 ) [ 98.71% ] PC2 78023.63 ( -0.71, 0.28, 0.64 ) [ 1.22% ] PC3 4504.60 ( -0.30, 0.71, -0.64 ) [ 0.07% ]

#### Using R's *prcomp()* function

The following example shows that *i.pca'* does not perform data centering in this specific case:

Command

prcomp(mod07, center=FALSE, scale=FALSE)

Output

Standard deviations: [1]4288.3788476.8904 114.3971 Rotation: PC1 PC2 PC3 MOD2007_242_500_sur_refl_b02 -0.6353238 0.7124070 -0.2980602 MOD2007_242_500_sur_refl_b06 -0.6485551 -0.2826985 0.7067234 MOD2007_242_500_sur_refl_b07 -0.4192135 -0.6423066 -0.6416403

- Performing data centering manually in grass (using
*r.mapcalc*) and repeating the i.pca gives the same results as prcomp(x, center=TRUE, scale=FALSE).**Example to be added.** - The eigenvector matrices match although
*prcomp()*reports loadings (=eigenvectors) column-wise and*i.pca*row-wise. - The eigenvalues do
**not**match. To exemplify, the standard deviation for PC1 reported by*prcomp()*is**4288.3788**and the variance reported by*i.pca*is 6307563.04 [ sqrt(6307563.04) =**2511.486**]

**More examples to be added**

# References

Jon Shlens, "Tutorial on Principal Component Analysis, Dec 2005," [1] (accessed on March, 2009).

## e-mails in GRASS-user mailing list

There are many posts concerning the functionality of *i.pca*. Most of them are questioning the non-reporting of eigenvalues (an issue recently fixed).

**Posts in grass-user mailing list**

**More sources to be added**