# Shrinkage-mean-based Diagonal Quadratic Discriminant Analysis (SmDQDA) from Tong, Chen, and Zhao (2012)

### Description

Given a set of training data, this function builds the Shrinkage-mean-based
Diagonal Quadratic Discriminant Analysis (SmDQDA) classifier from Tong, Chen,
and Zhao (2012). The SmDQDA classifier incorporates a Lindley-type shrunken
mean estimator into the DQDA classifier from Dudoit et al. (2002). For more
about the DQDA classifier, see `dqda`

.

The SmDQDA classifier is a modification to QDA, where the off-diagonal elements of the pooled sample covariance matrix are set to zero.

### Usage

1 2 3 4 5 6 7 8 9 10 |

### Arguments

`x` |
matrix containing the training data. The rows are the sample observations, and the columns are the features. |

`...` |
additional arguments |

`y` |
vector of class labels for each training observation |

`prior` |
vector with prior probabilities for each class. If NULL (default), then equal probabilities are used. See details. |

`formula` |
A formula of the form |

`data` |
data frame from which variables specified in |

`object` |
trained SmDQDA object |

`newdata` |
matrix of observations to predict. Each row corresponds to a new observation. |

### Details

The DQDA classifier is a modification to the well-known QDA classifier, where the off-diagonal elements of each class covariance matrix are assumed to be zero – the features are assumed to be uncorrelated. Under multivariate normality, the assumption uncorrelated features is equivalent to the assumption of independent features. The feature-independence assumption is a notable attribute of the Naive Bayes classifier family. The benefit of these classifiers is that they are fast and have much fewer parameters to estimate, especially when the number of features is quite large.

The matrix of training observations are given in `x`

. The rows of `x`

contain the sample observations, and the columns contain the features for each
training observation.

The vector of class labels given in `y`

are coerced to a `factor`

.
The length of `y`

should match the number of rows in `x`

.

An error is thrown if a given class has less than 2 observations because the variance for each feature within a class cannot be estimated with less than 2 observations.

The vector, `prior`

, contains the *a priori* class membership for
each class. If `prior`

is NULL (default), the class membership
probabilities are estimated as the sample proportion of observations belonging
to each class. Otherwise, `prior`

should be a vector with the same length
as the number of classes in `y`

. The `prior`

probabilties should be
nonnegative and sum to one.

### Value

`smdqda`

object that contains the trained SmDQDA classifier

list predicted class memberships of each row in newdata

### References

Tong, T., Chen, L., and Zhao, H. (2012), "Improved Mean Estimation and Its Application to Diagonal Discriminant Analysis," Bioinformatics, 28, 4, 531-537. http://bioinformatics.oxfordjournals.org/content/28/4/531.long

Dudoit, S., Fridlyand, J., & Speed, T. P. (2002). "Comparison of Discrimination Methods for the Classification of Tumors Using Gene Expression Data," Journal of the American Statistical Association, 97, 457, 77-87.

Dudoit, S., Fridlyand, J., & Speed, T. P. (2002). "Comparison of Discrimination Methods for the Classification of Tumors Using Gene Expression Data," Journal of the American Statistical Association, 97, 457, 77-87.

### Examples

1 2 3 4 5 6 7 8 | ```
n <- nrow(iris)
train <- sample(seq_len(n), n / 2)
smdqda_out <- smdqda(Species ~ ., data = iris[train, ])
predicted <- predict(smdqda_out, iris[-train, -5])$class
smdqda_out2 <- smdqda(x = iris[train, -5], y = iris[train, 5])
predicted2 <- predict(smdqda_out2, iris[-train, -5])$class
all.equal(predicted, predicted2)
``` |