Local Fisher Discriminant Analysis (LFDA) is a linear dimension reduction method for
supervised case, i.e., labels are given. It reflects *local* information to overcome
undesired results of traditional Fisher Discriminant Analysis which results in a poor mapping
when samples in a single class form form several separate clusters.

```
do.lfda(
X,
label,
ndim = 2,
preprocess = c("center", "scale", "cscale", "decorrelate", "whiten"),
type = c("proportion", 0.1),
symmetric = c("union", "intersect", "asymmetric"),
localscaling = TRUE
)
```

## Arguments

- X
an \((n\times p)\) matrix or data frame whose rows are observations
and columns represent independent variables.

- label
a length-\(n\) vector of data class labels.

- ndim
an integer-valued target dimension.

- preprocess
an additional option for preprocessing the data.
Default is "center". See also `aux.preprocess`

for more details.

- type
a vector of neighborhood graph construction. Following types are supported;
`c("knn",k)`

, `c("enn",radius)`

, and `c("proportion",ratio)`

.
Default is `c("proportion",0.1)`

, connecting about 1/10 of nearest data points
among all data points. See also `aux.graphnbd`

for more details.

- symmetric
one of `"intersect"`

, `"union"`

or `"asymmetric"`

is supported. Default is `"union"`

. See also `aux.graphnbd`

for more details.

- localscaling
`TRUE`

to use local scaling method for construction affinity matrix, `FALSE`

for binary affinity.

## Value

a named list containing

- Y
an \((n\times ndim)\) matrix whose rows are embedded observations.

- projection
a \((p\times ndim)\) whose columns are basis for projection.

- trfinfo
a list containing information for out-of-sample prediction.

## References

Sugiyama M (2006).
“Local Fisher Discriminant Analysis for Supervised Dimensionality Reduction.”
In *Proceedings of the 23rd International Conference on Machine Learning*, 905--912.

Zelnik-manor L, Perona P (2005).
“Self-Tuning Spectral Clustering.”
In Saul LK, Weiss Y, Bottou L (eds.), *Advances in Neural Information Processing Systems 17*, 1601--1608.
MIT Press.

## Examples

```
## generate 3 different groups of data X and label vector
x1 = matrix(rnorm(4*10), nrow=10)-20
x2 = matrix(rnorm(4*10), nrow=10)
x3 = matrix(rnorm(4*10), nrow=10)+20
X = rbind(x1, x2, x3)
label = rep(1:3, each=10)
## try different affinity matrices
out1 = do.lfda(X, label)
out2 = do.lfda(X, label, localscaling=FALSE)
## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2))
plot(out1$Y, col=label, main="binary affinity matrix")
plot(out2$Y, col=label, main="local scaling affinity")
par(opar)
```