Mixture discriminant analysis.

```
mda(formula, data, subclasses, sub.df, tot.df, dimension, eps,
iter, weights, method, keep.fitted, trace, …)
```

formula

of the form `y~x`

it describes the response and
the predictors. The formula can be more complicated, such as
`y~log(x)+z`

etc (see `formula`

for more details).
The response should be a factor representing the response variable,
or any vector that can be coerced to such (such as a logical
variable).

data

data frame containing the variables in the formula (optional).

subclasses

Number of subclasses per class, default is 3. Can be a vector with a number for each class.

sub.df

If subclass centroid shrinking is performed, what is the effective degrees of freedom of the centroids per class. Can be a scalar, in which case the same number is used for each class, else a vector.

tot.df

The total df for all the centroids can be specified rather than separately per class.

dimension

The dimension of the reduced model. If we know our final model will be confined to a discriminant subspace (of the subclass centroids), we can specify this in advance and have the EM algorithm operate in this subspace.

eps

A numerical threshold for automatically truncating the dimension.

iter

A limit on the total number of iterations, default is 5.

weights

*NOT* observation weights! This is a special
weight structure, which for each class assigns a weight (prior
probability) to each of the observations in that class of belonging
to one of the subclasses. The default is provided by a call to
`mda.start(x, g, subclasses, trace, …)`

(by this time
`x`

and `g`

are known). See the help for
`mda.start`

. Arguments for `mda.start`

can be
provided via the `…`

argument to mda, and the
`weights`

argument need never be accessed. A previously fit
mda object can be supplied, in which case the final subclass
`responsibility`

weights are used for `weights`

. This
allows the iterations from a previous fit to be continued.

method

regression method used in optimal scaling. Default is
linear regression via the function `polyreg`

, resulting in the
usual mixture model. Other possibilities are `mars`

and
`bruto`

. For penalized mixture discriminant models
`gen.ridge`

is appropriate.

keep.fitted

a logical variable, which determines whether the
(sometimes large) component `"fitted.values"`

of the `fit`

component of the returned `mda`

object should be kept. The
default is `TRUE`

if `n * dimension < 5000`

.

trace

if `TRUE`

, iteration information is printed. Note
that the deviance reported is for the posterior class likelihood,
and not the full likelihood, which is used to drive the EM algorithm
under `mda`

. In general the latter is not available.

…

additional arguments to `mda.start`

and to
`method`

.

An object of class `c("mda", "fda")`

. The most useful extractor
is `predict`

, which can make many types of predictions from this
object. It can also be plotted, and any functions useful for fda
objects will work here too, such as `confusion`

and `coef`

.

The object has the following components:

the percent between-group variance explained by each dimension (relative to the total explained.)

optimal scaling regression sum-of-squares for each dimension (see reference).

subclass means in the discriminant space. These are also
scaled versions of the final theta's or class scores, and can be
used in a subsequent call to `mda`

(this only makes sense if
some columns of theta are omitted---see the references)

(internal) a class scoring matrix which allows
`predict`

to work properly.

dimension of discriminant space.

subclass membership priors, computed in the fit. No effort is currently spent in trying to keep these above a threshold.

class proportions for the training data.

fit object returned by `method`

.

the call that created this object (allowing it to be
`update`

-able).

confusion matrix when classifying the training data.

These are the subclass membership probabilities for each member of the training set; see the weights argument.

a pointer list which identifies which elements of certain lists belong to individual classes.

The multinomial log-likelihood of the fit. Even though
the full log-likelihood drives the iterations, we cannot in general
compute it because of the flexibility of the `method`

used.
The deviance can increase with the iterations, but generally does not.

The method functions are required to take arguments x and y where both can be matrices, and should produce a matrix of fitted.values the same size as y. They can take additional arguments weights and should all have a … for safety sake. Any arguments to method() can be passed on via the … argument of mda. The default method polyreg has a degree argument which allows polynomial regression of the required total degree. See the documentation for predict.fda for further requirements of method. The package earth is suggested for this package as well; earth is a more detailed implementation of the mars model, and works as a method argument. The function mda.start creates the starting weights; it takes additional arguments which can be passed in via the … argument to mda. See the documentation for mda.start.

``Flexible Disriminant Analysis by Optimal Scoring'' by Hastie, Tibshirani and Buja, 1994, JASA, 1255-1270.

``Penalized Discriminant Analysis'' by Hastie, Buja and Tibshirani, 1995, Annals of Statistics, 73-102

``Discriminant Analysis by Gaussian Mixtures'' by Hastie and Tibshirani, 1996, JRSS-B, 155-176.

``Elements of Statisical Learning - Data Mining, Inference and Prediction'' (2nd edition, Chapter 12) by Hastie, Tibshirani and Friedman, 2009, Springer

`predict.mda`

,
`mars`

,
`bruto`

,
`polyreg`

,
`gen.ridge`

,
`softmax`

,
`confusion`

# NOT RUN { data(iris) irisfit <- mda(Species ~ ., data = iris) irisfit ## Call: ## mda(formula = Species ~ ., data = iris) ## ## Dimension: 4 ## ## Percent Between-Group Variance Explained: ## v1 v2 v3 v4 ## 96.02 98.55 99.90 100.00 ## ## Degrees of Freedom (per dimension): 5 ## ## Training Misclassification Error: 0.02 ( N = 150 ) ## ## Deviance: 15.102 data(glass) # random sample of size 100 samp <- c(1, 3, 4, 11, 12, 13, 14, 16, 17, 18, 19, 20, 27, 28, 31, 38, 42, 46, 47, 48, 49, 52, 53, 54, 55, 57, 62, 63, 64, 65, 67, 68, 69, 70, 72, 73, 78, 79, 83, 84, 85, 87, 91, 92, 94, 99, 100, 106, 107, 108, 111, 112, 113, 115, 118, 121, 123, 124, 125, 126, 129, 131, 133, 136, 139, 142, 143, 145, 147, 152, 153, 156, 159, 160, 161, 164, 165, 166, 168, 169, 171, 172, 173, 174, 175, 177, 178, 181, 182, 185, 188, 189, 192, 195, 197, 203, 205, 211, 212, 214) glass.train <- glass[samp,] glass.test <- glass[-samp,] glass.mda <- mda(Type ~ ., data = glass.train) predict(glass.mda, glass.test, type="post") # abbreviations are allowed confusion(glass.mda,glass.test) # }