# Linear and Ridge Regressions Computation¶

## Batch Processing¶

Linear and ridge regressions in the batch processing mode follow the general workflow described in Regression Usage Model.

### Training¶

For a description of the input and output, refer to Regression Usage Model.

The following table lists parameters of linear and ridge regressions at the training stage. Some of these parameters or their values are specific to a linear or ridge regression algorithm.

Parameter

Default Value

Description

algorithmFPType

float

The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

method

defaultDense

Available methods for linear regression training:

• defaultDense - the normal equations method

• qrDense - the method based on QR decomposition

interceptFlag

true

A flag that indicates a need to compute $$\beta_{0j}$$.

Parameter

Default Value

Description

algorithmFPType

float

The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

method

defaultDense

Default computation method used by the ridge regression. The only method supported at the training stage is the normal equations method.

ridgeParameters

A numeric table of size $$1 \times 1$$ that contains the default ridge parameter equal to $$1$$.

The numeric table of size $$1 \times k$$ ($$k$$ is the number of dependent variables) or $$1 \times 1$$. The contents of the table depend on its size:

• $$size = 1 \times k$$: values of the ridge parameters $$\lambda_j$$ for $$j = 1, \ldots, k$$.

• $$size = 1 \times 1$$: the value of the ridge parameter for each dependent variable $$\lambda_1 = \ldots = \lambda_k$$.

Note

This parameter can be an object of any class derived from NumericTable, except for PackedTriangularMatrix, PackedSymmetricMatrix, and CSRNumericTable.

interceptFlag

true

A flag that indicates a need to compute $$\beta_{0j}$$.

### Prediction¶

For a description of the input and output, refer to Regression Usage Model.

At the prediction stage, linear and ridge regressions have the following parameters:

Parameter

Default Value

Description

algorithmFPType

float

The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

method

defaultDense

Default performance-oriented computation method, the only method supported by the regression based prediction.

## Online Processing¶

You can use linear and ridge regression in the online processing mode only at the training stage.

This computation mode assumes that the data arrives in blocks $$i = 1, 2, 3, \ldots \text{nblocks}$$.

### Training¶

Linear and ridge regression training in the online processing mode follows the general workflow described in Regression Usage Model.

Linear and ridge regression training in the online processing mode accepts the input described below. Pass the Input ID as a parameter to the methods that provide input for your algorithm. For more details, see Algorithms.

Input ID

Input

data

Pointer to the $$n_i \times p$$ numeric table that represents the current, $$i$$-th, data block.

dependentVariables

Pointer to the $$n_i \times k$$ numeric table with responses associated with the current, $$i$$-th, data block.

Note

Both input tables can be an object of any class derived from NumericTable.

The following table lists parameters of linear and ridge regressions at the training stage in the online processing mode.

Parameter

Default Value

Description

algorithmFPType

float

The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

method

defaultDense

Available methods for linear regression training:

• defaultDense - the normal equations method

• qrDense - the method based on QR decomposition

interceptFlag

true

A flag that indicates a need to compute $$\beta_{0_j}$$.

Parameter

Default Value

Description

algorithmFPType

float

The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

method

defaultDense

Default computation method used by the ridge regression. The only method supported at the training stage is the normal equations method.

ridgeParameters

A numeric table of size $$1 \times 1$$ that contains the default ridge parameter equal to $$1$$.

The numeric table of size $$1 \times k$$ ($$k$$ is the number of dependent variables) or $$1 \times 1$$. The contents of the table depend on its size:

• size = $$1 \times k$$: values of the ridge parameters $$\lambda_j$$ for $$j = 1, \ldots, k$$.

• size = $$1 \times 1$$: the value of the ridge parameter for each dependent variable $$\lambda_1 = ... = \lambda_k$$.

Note

This parameter can be an object of any class derived from NumericTable, except for PackedTriangularMatrix, PackedSymmetricMatrix, and CSRNumericTable.

interceptFlag

true

A flag that indicates a need to compute $$\beta_{0_j}$$.

For a description of the output, refer to Regression Usage Model.

## Distributed Processing¶

You can use linear and ridge regression in the distributed processing mode only at the training stage.

This computation mode assumes that the data set is split in nblocks blocks across computation nodes.

### Training¶

Use the two-step computation schema for linear and ridge regression training in the distributed processing mode, as illustrated below:

#### Algorithm parameters¶

The following table lists parameters of linear and ridge regressions at the training stage in the distributed processing mode.

Parameter

Default Value

Description

computeStep

Not applicable

The parameter required to initialize the algorithm. Can be:

• step1Local - the first step, performed on local nodes

• step2Master - the second step, performed on a master node

algorithmFPType

float

The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

method

defaultDense

Available methods for linear regression training:

• defaultDense - the normal equations method

• qrDense - the method based on QR decomposition

interceptFlag

true

A flag that indicates a need to compute $$\beta_{0_j}$$.

Parameter

Default Value

Description

computeStep

Not applicable

The parameter required to initialize the algorithm. Can be:

• step1Local - the first step, performed on local nodes

• step2Master - the second step, performed on a master node

algorithmFPType

float

The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

method

defaultDense

Default computation method used by the ridge regression. The only method supported at the training stage is the normal equations method.

ridgeParameters

A numeric table of size $$1 \times 1$$ that contains the default ridge parameter equal to $$1$$.

The numeric table of size $$1 \times k$$ ($$k$$ is the number of dependent variables) or $$1 \times 1$$. The contents of the table depend on its size:

• size = $$1 \times k$$: values of the ridge parameters $$\lambda_j$$ for $$j = 1, \ldots, k$$.

• size = $$1 \times 1$$: the value of the ridge parameter for each dependent variable $$\lambda_1 = ... = \lambda_k$$.

Note

This parameter can be an object of any class derived from NumericTable, except for PackedTriangularMatrix, PackedSymmetricMatrix, and CSRNumericTable.

interceptFlag

true

A flag that indicates a need to compute $$\beta_{0_j}$$.

#### Step 1 - on Local Nodes¶

In this step, linear and ridge regression training accepts the input described below. Pass the Input ID as a parameter to the methods that provide input for your algorithm. For more details, see Algorithms.

Input ID

Input

data

Pointer to the $$n_i \times p$$ numeric table that represents the $$i$$-th data block on the local node.

dependentVariables

Pointer to the $$n_i \times k$$ numeric table with responses associated with the $$i$$-th data block.

Note

Both input tables can be an object of any class derived from NumericTable.

In this step, linear and ridge regression training calculates the result described below. Pass the Result ID as a parameter to the methods that access the results of your algorithm. For more details, see Algorithms.

Result ID

Result

partialModel

Pointer to the partial linear regression model that corresponds to the $$i$$-th data block.

The result can only be an object of the Model class.

#### Step 2 - on Master Node¶

In this step, linear and ridge regression training accepts the input described below. Pass the Input ID as a parameter to the methods that provide input for your algorithm. For more details, see Algorithms.

Input ID

Input

partialModels

A collection of partial models computed on local nodes in Step 1.

The collection contains objects of the Model class.

In this step, linear and ridge regression training calculates the result described below. Pass the Result ID as a parameter to the methods that access the results of your algorithm. For more details, see Algorithms.

Result ID

Result

model

Pointer to the linear or ridge regression model being trained.

The result can only be an object of the Model class.

## Examples¶

Note

There is no support for Java on GPU.

Batch Processing:

Online Processing:

Distributed Processing:

Batch Processing:

Batch Processing:

Online Processing:

Distributed Processing: