Locality Preserving Fisher Discriminant Analysis (LPFDA) is a supervised variant of LPP.
It can also be seemed as an improved version of LDA where the locality structure of the data
is preserved. The algorithm aims at getting a subspace projection matrix by solving a generalized
eigenvalue problem.

```
do.lpfda(
X,
label,
ndim = 2,
type = c("proportion", 0.1),
preprocess = c("center", "scale", "cscale", "whiten", "decorrelate"),
t = 10
)
```

## 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.

- 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.

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

for more details.

- t
bandwidth parameter for heat kernel in \((0,\infty)\).

## Value

a named list containing

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

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

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

## References

Zhao X, Tian X (2009).
“Locality Preserving Fisher Discriminant Analysis for Face Recognition.”
In Huang D, Jo K, Lee H, Kang H, Bevilacqua V (eds.), *Emerging Intelligent Computing Technology and Applications*, 261--269.

## Examples

```
## generate data of 3 types with clear difference
set.seed(100)
dt1 = aux.gensamples(n=20)-50
dt2 = aux.gensamples(n=20)
dt3 = aux.gensamples(n=20)+50
## merge the data and create a label correspondingly
X = rbind(dt1,dt2,dt3)
label = rep(1:3, each=20)
## try different proportion of connected edges
out1 = do.lpfda(X, label, type=c("proportion",0.10))
out2 = do.lpfda(X, label, type=c("proportion",0.25))
out3 = do.lpfda(X, label, type=c("proportion",0.50))
## visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
plot(out1$Y, pch=19, col=label, main="10% connectivity")
plot(out2$Y, pch=19, col=label, main="25% connectivity")
plot(out3$Y, pch=19, col=label, main="50% connectivity")
par(opar)
```