Usage

`javascript
var incrSGDRegression = require( '@stdlib/ml-incr-sgd-regression' );
`

#### incrSGDRegression( \[options] )

Creates an online linear regression model fitted via [stochastic gradient descent][stochastic-gradient-descent]. The module performs [L2 regularization][l2-regularization] of the model coefficients, shrinking them towards zero by penalizing the squared [euclidean norm][euclidean-norm] of the coefficients.

`javascript
var randu = require( '@stdlib/random-base-randu' );
var normal = require( '@stdlib/random-base-normal' );
var accumulator = incrSGDRegression();

var x1;
var x2;
var i;
var y;

// Update model as data comes in...
for ( i = 0; i < 100000; i++ ) {
x1 = randu();
x2 = randu();
y = (3.0 x1) + (-3.0 x2) + 2.0 + normal( 0.0, 1.0 );
accumulator( [ x1, x2 ], y );
}
`

The function accepts the following options:

- learningRate: string denoting the learning rate to use. Can be constant, pegasos or basic. Default: basic.
- loss: string denoting the loss function to use. Can be squaredError, epsilonInsensitive or huber. Default: squaredError.
- epsilon: insensitivity parameter. Default: 0.1.
- lambda: regularization parameter. Default: 1e-3.
- eta0: constant learning rate. Default: 0.02.
- intercept: boolean indicating whether to include an intercept. Default: true.

`javascript
var accumulator = incrSGDRegression({
'loss': 'squaredError',
'lambda': 1e-4
});
`

The learningRate decides how fast or slow the weights will be updated towards the optimal weights. Let i denote the current iteration of the algorithm (i.e. the number of data points having arrived). The possible learning rates are:

| Option | Definition |
| :-------------: | :---------------------: |
| basic (default) | 1000.0 / ( i + 1000.0 ) |
| constant | eta0 |
| pegasos | 1.0 / ( lambda \* i ) |

The used loss function is specified via the loss option. The available options are:

- epsilonInsensitive: Penalty is the absolute value of the error whenever the absolute error exceeds epsilon and zero otherwise.
- huber: Squared-error loss for observations with error smaller than epsilon in magnitude, linear loss otherwise. Should be used in order to decrease the influence of outliers on the model fit.
- squaredError: Squared error loss, i.e. the squared difference of the observed and fitted values.

The lambda parameter determines the amount of shrinkage inflicted on the model coefficients:

`javascript
var createRandom = require( '@stdlib/random-base-randu' ).factory;

var accumulator;
var coefs;
var opts;
var rand;
var x1;
var x2;
var i;
var y;

opts = {
'seed': 23
};
rand = createRandom( opts );

accumulator = incrSGDRegression({
'lambda': 1e-5
});

for ( i = 0; i < 100; i++ ) {
x1 = rand();
x2 = rand();
y = (3.0 x1) + (-3.0 x2) + 2.0;
accumulator( [ x1, x2 ], y );
}

coefs = accumulator.coefs;
// returns [ ~3.007, ~-3.002, ~2 ]

rand = createRandom( opts );
accumulator = incrSGDRegression({
'lambda': 1e-2
});

for ( i = 0; i < 100; i++ ) {
x1 = rand();
x2 = rand();
y = (3.0 x1) + (-3.0 x2) + 2.0;
accumulator( [ x1, x2 ], y );
}

coefs = accumulator.coefs;
// returns [ ~2.893, ~-2.409, ~1.871 ]
`

Higher values of lambda reduce the variance of the model coefficient estimates at the expense of introducing bias.

By default, the model contains an intercept term. To omit the intercept, set the corresponding option to false:

`javascript
var accumulator = incrSGDRegression({
'intercept': false
});
accumulator( [ 1.4, 0.5 ], 2.0 );

var dim = accumulator.coefs.length;
// returns 2

accumulator = incrSGDRegression();
accumulator( [ 1.4, 0.5 ], 2.0 );

dim = accumulator.coefs.length;
// returns 3
`

If intercept is true, an element equal to one is implicitly added to each x vector. Hence, this module performs regularization of the intercept term.

#### accumulator( x, y )

Updates the model coefficients in light of incoming data. y must be a numeric response value, x a numeric array of predictors. The number of predictors is decided upon first invocation of this method. All subsequent calls must supply x vectors of the same dimensionality.

`javascript
accumulator( [ 1.0, 0.0 ], 5.0 );
`

#### accumulator.predict( x )

Predicts the response for a new feature vector x, where x must be a numeric array of predictors. Given feature vector x = [x_0, x_1, ...] and model coefficients c = [c_0, c_1, ...], the prediction is equal to x_0c_0 + x_1c_1 + ... + c_intercept.

`javascript
var yhat = accumulator.predict( [ 0.5, 2.0 ] );
// returns
`

#### accumulator.coefs

Getter for the model coefficients / feature weights stored in an array. The coefficients are ordered as [c_0, c_1,..., c_intercept], where c_0 corresponds to the first feature in x and so on.

`javascript
var coefs = accumulator.coefs;
// returns
`