Description Usage Arguments Details Value Note Author(s) References See Also Examples

Performs relevant component analysis on the given data.

1 | ```
rca(x, chunks)
``` |

`x` |
matrix or data frame of original data. Each row is a feature vector of a data instance. |

`chunks` |
list of |

The RCA function takes a data set and a set of positive constraints as arguments and returns a linear transformation of the data space into better representation, alternatively, a Mahalanobis metric over the data space.

Relevant component analysis consists of three steps:

locate the test point

compute the distances between the test points

find

*k*shortest distances and the bla

The new representation is known to be optimal in an information theoretic sense under a constraint of keeping equivalent data points close to each other.

list of the RCA results:

`B` |
The RCA suggested Mahalanobis matrix. Distances between data points x1, x2 should be computed by (x2 - x1)' * B * (x2 - x1) |

`A` |
The RCA suggested transformation of the data. The data should be transformed by A * data |

`newX` |
The data after the RCA transformation (A). newData = A * data |

The three returned argument are just different forms of the same output. If one is interested in a Mahalanobis metric over the original data space, the first argument is all she/he needs. If a transformation into another space (where one can use the Euclidean metric) is preferred, the second returned argument is sufficient. Using A and B is equivalent in the following sense:

if y1 = A * x1, y2 = A * y2 then (x2 - x1)' * B * (x2 - x1) = (y2 - y1)' * (y2 - y1)

Note that any different sets of instances (chunklets), e.g. 1, 3, 7 and 4, 6, might belong to the same class and might belong to different classes.

Xiao Nan <http://www.road2stat.com>

Aharon Bar-Hillel, Tomer Hertz, Noam Shental, and Daphna Weinshall (2003).
Learning Distance Functions using Equivalence Relations.
*Proceedings of 20th International Conference on
Machine Learning (ICML2003)*.

See `dca`

for exploiting negative constrains.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | ```
## Not run:
set.seed(1234)
require(MASS) # generate synthetic Gaussian data
k = 100 # sample size of each class
n = 3 # specify how many class
N = k * n # total sample number
x1 = mvrnorm(k, mu = c(-10, 6), matrix(c(10, 4, 4, 10), ncol = 2))
x2 = mvrnorm(k, mu = c(0, 0), matrix(c(10, 4, 4, 10), ncol = 2))
x3 = mvrnorm(k, mu = c(10, -6), matrix(c(10, 4, 4, 10), ncol = 2))
x = as.data.frame(rbind(x1, x2, x3))
x$V3 = gl(n, k)
# The fully labeled data set with 3 classes
plot(x$V1, x$V2, bg = c("#E41A1C", "#377EB8", "#4DAF4A")[x$V3],
pch = c(rep(22, k), rep(21, k), rep(25, k)))
Sys.sleep(3)
# Same data unlabeled; clearly the classes' structure is less evident
plot(x$V1, x$V2)
Sys.sleep(3)
chunk1 = sample(1:100, 5)
chunk2 = sample(setdiff(1:100, chunk1), 5)
chunk3 = sample(101:200, 5)
chunk4 = sample(setdiff(101:200, chunk3), 5)
chunk5 = sample(201:300, 5)
chks = x[c(chunk1, chunk2, chunk3, chunk4, chunk5), ]
chunks = list(chunk1, chunk2, chunk3, chunk4, chunk5)
# The chunklets provided to the RCA algorithm
plot(chks$V1, chks$V2, col = rep(c("#E41A1C", "#377EB8",
"#4DAF4A", "#984EA3", "#FF7F00"), each = 5),
pch = rep(0:4, each = 5), ylim = c(-15, 15))
Sys.sleep(3)
# Whitening transformation applied to the chunklets
chkTransformed = as.matrix(chks[ , 1:2]) %*% rca(x[ , 1:2], chunks)$A
plot(chkTransformed[ , 1], chkTransformed[ , 2], col = rep(c(
"#E41A1C", "#377EB8", "#4DAF4A", "#984EA3", "#FF7F00"), each = 5),
pch = rep(0:4, each = 5), ylim = c(-15, 15))
Sys.sleep(3)
# The origin data after applying the RCA transformation
plot(rca(x[ , 1:2], chunks)$newX[, 1], rca(x[ , 1:2], chunks)$newX[, 2],
bg = c("#E41A1C", "#377EB8", "#4DAF4A")[gl(n, k)],
pch = c(rep(22, k), rep(21, k), rep(25, k)))
# The RCA suggested transformation of the data, dimensionality reduced
rca(x[ , 1:2], chunks)$A
# The RCA suggested Mahalanobis matrix
rca(x[ , 1:2], chunks)$B
## End(Not run)
``` |

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