# FeatureSelectionNCARegression

Feature selection for regression using neighborhood component analysis (NCA)

## Description

`FeatureSelectionNCARegression`

contains the data,
fitting information, feature weights, and other model parameters of a
neighborhood component analysis (NCA) model. `fsrnca`

learns the feature
weights using a diagonal adaptation of NCA and returns an instance of
`FeatureSelectionNCARegression`

object. The function
achieves feature selection by regularizing the feature weights.

## Creation

Create a `FeatureSelectionNCARegression`

object using `fsrnca`

.

## Properties

### NCA Properties

`ModelParameters`

— Model parameters

structure

This property is read-only.

Model parameters used for training the model, specified as a structure.

You can access the fields of `ModelParameters`

using dot
notation.

For example, for a FeatureSelectionNCARegression object named `mdl`

, you can access the
`LossFunction`

value using
`mdl.ModelParameters.LossFunction`

.

**Data Types: **`struct`

`Lambda`

— Regularization parameter

scalar

This property is read-only.

Regularization parameter used for training this model, specified as a scalar. For
*n* observations, the best `Lambda`

value that
minimizes the generalization error of the NCA model is expected to be a multiple of
1/*n*.

**Data Types: **`double`

`FitMethod`

— Name of the fitting method used to fit this model

`'exact'`

| `'none'`

| `'average'`

This property is read-only.

Name of the fitting method used to fit this model, specified as one of the following:

`'exact'`

— Perform fitting using all of the data.`'none'`

— No fitting. Use this option to evaluate the generalization error of the NCA model using the initial feature weights supplied in the call to`fsrnca`

.`'average'`

— The software divides the data into partitions (subsets), fits each partition using the`exact`

method, and returns the average of the feature weights. You can specify the number of partitions using the`NumPartitions`

name-value argument.

`Solver`

— Name of the solver used to fit this model

`'lbfgs'`

| `'sgd'`

| `'minibatch-lbfgs'`

This property is read-only.

Name of the solver used to fit this model, specified as one of the following:

`'lbfgs'`

— Limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm`'sgd'`

— Stochastic gradient descent (SGD) algorithm`'minibatch-lbfgs'`

— stochastic gradient descent with LBFGS algorithm applied to mini-batches

`GradientTolerance`

— Relative convergence tolerance on gradient norm

positive scalar

This property is read-only.

Relative convergence tolerance on the gradient norm for the `'lbfgs'`

and `'minibatch-lbfgs'`

solvers, specified as a positive scalar
value.

**Data Types: **`double`

`IterationLimit`

— Maximum number of iterations for optimization

positive integer

This property is read-only.

Maximum number of iterations for optimization, specified as a positive integer value.

**Data Types: **`double`

`PassLimit`

— Maximum number of passes

positive integer

This property is read-only.

Maximum number of passes for `'sgd'`

and
`'minibatch-lbfgs'`

solvers, specified as a positive integer.
Every pass processes all
of the observations in the data.

**Data Types: **`double`

`InitialLearningRate`

— Initial learning rate

positive real scalar

This property is read-only.

Initial learning rate for
`'sgd'`

and
`'minibatch-lbfgs'`

solvers,
specified as a positive real scalar.
The
learning rate decays over iterations starting at
the value specified for
`InitialLearningRate`

.

Use the
`NumTuningIterations`

and
`TuningSubsetSize`

to control
the automatic tuning of initial learning rate in
the call to `fsrnca`

.

**Data Types: **`double`

`Verbose`

— Verbosity level indicator

nonnegative integer

This property is read-only.

Verbosity level indicator, specified as a nonnegative integer. Possible values are:

0 — No convergence summary

1 — Convergence summary, including norm of gradient and objective function value

>1 — More convergence information, depending on the fitting algorithm. When you use the

`'minibatch-lbfgs'`

solver and verbosity level > 1, the convergence information includes the iteration log from intermediate mini-batch LBFGS fits.

**Data Types: **`double`

`InitialFeatureWeights`

— Initial feature weights

*p*-by-1 vector of positive real scalars

This property is read-only.

Initial feature weights, specified as a *p*-by-1 vector of positive
real scalars, where *p* is the number of predictors in
`X`

.

**Data Types: **`double`

`FeatureWeights`

— Feature weights

numeric vector | numeric matrix

This property is read-only.

Feature weights, specified as a
*p*-by-1 numeric vector or a
*p*-by-*m*
numeric matrix, where *p* is the
number of predictor variables after dummy
variables are created for categorical variables
(for more details, see
`ExpandedPredictorNames`

).

If `FitMethod`

is
`'average'`

, then
`FeatureWeights`

is a
*p*-by-*m*
matrix. *m* is the number of
partitions specified via the
`NumPartitions`

name-value
argument in the call to
`fsrnca`

.

The absolute value of
`FeatureWeights(k)`

is a measure
of the importance of predictor
`k`

. A
`FeatureWeights(k)`

value that is
close to 0 indicates that predictor
`k`

does not influence the
response in `Y`

.

**Data Types: **`double`

`FitInfo`

— Fit information

structure

This property is read-only.

Fit information, specified as a structure with the following fields.

Field Name | Meaning |
---|---|

`Iteration` | Iteration index |

`Objective` | Regularized objective function for minimization |

`UnregularizedObjective` | Unregularized objective function for minimization |

`Gradient` | Gradient of regularized objective function for minimization |

For classification,

`UnregularizedObjective`

represents the negative of the leave-one-out accuracy of the NCA classifier on the training data.For regression,

`UnregularizedObjective`

represents the leave-one-out loss between the true response and the predicted response when using the NCA regression model.For the

`'lbfgs'`

solver,`Gradient`

is the final gradient. For the`'sgd'`

and`'minibatch-lbfgs'`

solvers,`Gradient`

is the final mini-batch gradient.If

`FitMethod`

is`'average'`

, then`FitInfo`

is an*m*-by-1 structure array, where*m*is the number of partitions specified via the`NumPartitions`

name-value argument.

You can access the fields of `FitInfo`

using dot notation. For
example, for a FeatureSelectionNCARegressionobject named `mdl`

, you can access the
`Objective`

field using
`mdl.FitInfo.Objective`

.

**Data Types: **`struct`

### Other Regression Properties

`NumObservations`

— Number of observations in the training data

scalar

This property is read-only.

Number of observations in the training data (`X`

and
`Y`

) after removing `NaN`

or
`Inf`

values, specified as a scalar.

**Data Types: **`double`

`Mu`

— Predictor means

*p*-by-1 vector | `[]`

This property is read-only.

Predictor means, specified as a *p*-by-1 vector for standardized
training data. In this case, the `predict`

method centers predictor
matrix `X`

by subtracting the respective element of
`Mu`

from every column.

If data is not standardized during training, then `Mu`

is
empty.

**Data Types: **`double`

`Sigma`

— Predictor standard deviations

*p*-by-1 vector | `[]`

This property is read-only.

Predictor standard deviations, specified as a *p*-by-1 vector for
standardized training data. In this case, the `predict`

method scales
predictor matrix `X`

by dividing every column by the respective
element of `Sigma`

after centering the data using
`Mu`

.

If data is not standardized during training, then `Sigma`

is
empty.

**Data Types: **`double`

`X`

— Predictor values

*n*-by-*p* matrix

This property is read-only.

Predictor values used to train this model, specified as an
*n*-by-*p* matrix. *n* is the
number of observations and *p* is the number of predictor variables in
the training data.

**Data Types: **`double`

`Y`

— Response values

numeric vector of size *n*

This property is read-only.

Response values used to train this model, specified as a numeric vector of size
*n*, where n is the number of
observations.

**Data Types: **`double`

`W`

— Observation weights

numeric vector of size *n*

This property is read-only.

Observation weights used to train this model, specified as a numeric vector of size
*n*. The sum of observation weights is
*n*.

**Data Types: **`double`

`CategoricalPredictors`

— Categorical predictor indices

vector of positive integers | `[]`

This property is read-only.

Categorical predictor indices, specified as a vector of positive integers.
`CategoricalPredictors`

contains index values indicating that the
corresponding predictors are categorical. The index values are between 1 and
*p*, where *p* is the number of predictors used to
train the model. If none of the predictors are categorical, then this property is empty
(`[]`

).

**Data Types: **`single`

| `double`

`ResponseName`

— Response variable name

character vector

This property is read-only.

Response variable name, specified as a character vector.

**Data Types: **`char`

`PredictorNames`

— Predictor variable names

cell array of unique character vectors

This property is read-only.

Predictor variable names in order of their appearance in the predictor data,
specified as a cell array of unique character vectors. The length of
`PredictorNames`

is equal to the number of
variables in the training data `X`

used as predictor
variables.

**Data Types: **`cell`

`ExpandedPredictorNames`

— Expanded predictor names

cell array of unique character vectors

This property is read-only.

Expanded predictor names, specified as a cell array of unique character vectors.

If the model uses encoding for categorical variables, then
`ExpandedPredictorNames`

includes the names that describe the
expanded variables. Otherwise, `ExpandedPredictorNames`

is the same as
`PredictorNames`

.

**Data Types: **`cell`

## Object Functions

`loss` | Evaluate accuracy of learned feature weights on test data |

`predict` | Predict responses using neighborhood component analysis (NCA) regression model |

`refit` | Refit neighborhood component analysis (NCA) model for regression |

`selectFeatures` | Select important features for NCA classification or regression |

## Examples

### Explore `FeatureSelectionNCARegression`

Object

Load the sample data.

`load imports-85`

The first 15 columns contain the continuous predictor variables, whereas the 16th column contains the response variable, which is the price of a car. Define the variables for the neighborhood component analysis model.

Predictors = X(:,1:15); Y = X(:,16);

Fit a neighborhood component analysis (NCA) model for regression to detect the relevant features.

mdl = fsrnca(Predictors,Y);

The returned NCA model, `mdl`

, is a `FeatureSelectionNCARegression`

object. This object stores information about the training data, model, and optimization. You can access the object properties, such as the feature weights, using dot notation.

Plot the feature weights.

plot(mdl.FeatureWeights,"o") xlabel("Feature Index") ylabel("Feature Weight") grid on

The weights of the irrelevant features are zero. The `Verbose=1`

option in the call to `fsrnca`

displays the optimization information on the command line. You can also visualize the optimization process by plotting the objective function versus the iteration number.

plot(mdl.FitInfo.Iteration,mdl.FitInfo.Objective,"o-") grid on xlabel("Iteration Number") ylabel("Objective")

The `ModelParameters`

property is a `struct`

that contains more information about the model. You can access the fields of this property using dot notation. For example, see if the data was standardized or not.

mdl.ModelParameters.Standardize

`ans = `*logical*
0

`0`

means that the data was not standardized before fitting the NCA model. You can standardize the predictors when they are on very different scales using the `Standardize=true`

name-value argument in the call to `fsrnca`

.

## Version History

**Introduced in R2016b**

## See Also

`predict`

| `fsrnca`

| `refit`

| `loss`

| `selectFeatures`

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