The following are a set of methods intended for regression in which the target value is expected to be a linear combination of the input variables. In mathematical notion, if \(\hat{y}\) is the predicted value.
Across the module, we designate the vector \(w = (w_1, ..., w_p)\) as coef_
and \(w_0\) as intercept_
.
To perform classification with generalized linear models, see Logistic regression.
LinearRegression
fits a linear model with coefficients \(w = (w_1, ..., w_p)\) to minimize the residual sum of squares between the observed responses in the dataset, and the responses predicted by the linear approximation. Mathematically it solves a problem of the form:
LinearRegression
will take in its fit
method arrays X, y and will store the coefficients \(w\) of the linear model in its coef_
member:
>>> from sklearn import linear_model >>> reg = linear_model.LinearRegression() >>> reg.fit ([[0, 0], [1, 1], [2, 2]], [0, 1, 2]) ... LinearRegression(copy_X=True, fit_intercept=True, n_jobs=None, normalize=False) >>> reg.coef_ array([0.5, 0.5])
However, coefficient estimates for Ordinary Least Squares rely on the independence of the model terms. When terms are correlated and the columns of the design matrix \(X\) have an approximate linear dependence, the design matrix becomes close to singular and as a result, the leastsquares estimate becomes highly sensitive to random errors in the observed response, producing a large variance. This situation of multicollinearity can arise, for example, when data are collected without an experimental design.
Examples:
This method computes the least squares solution using a singular value decomposition of X. If X is a matrix of size (n, p) this method has a cost of \(O(n p^2)\), assuming that \(n \geq p\).
Ridge
regression addresses some of the problems of Ordinary Least Squares by imposing a penalty on the size of coefficients. The ridge coefficients minimize a penalized residual sum of squares,
Here, \(\alpha \geq 0\) is a complexity parameter that controls the amount of shrinkage: the larger the value of \(\alpha\), the greater the amount of shrinkage and thus the coefficients become more robust to collinearity.
As with other linear models, Ridge
will take in its fit
method arrays X, y and will store the coefficients \(w\) of the linear model in its coef_
member:
>>> from sklearn import linear_model >>> reg = linear_model.Ridge (alpha = .5) >>> reg.fit ([[0, 0], [0, 0], [1, 1]], [0, .1, 1]) Ridge(alpha=0.5, copy_X=True, fit_intercept=True, max_iter=None, normalize=False, random_state=None, solver='auto', tol=0.001) >>> reg.coef_ array([0.34545455, 0.34545455]) >>> reg.intercept_ 0.13636...
Examples:
This method has the same order of complexity than an Ordinary Least Squares.
RidgeCV
implements ridge regression with builtin crossvalidation of the alpha parameter. The object works in the same way as GridSearchCV except that it defaults to Generalized CrossValidation (GCV), an efficient form of leaveoneout crossvalidation:
>>> from sklearn import linear_model >>> reg = linear_model.RidgeCV(alphas=[0.1, 1.0, 10.0], cv=3) >>> reg.fit([[0, 0], [0, 0], [1, 1]], [0, .1, 1]) RidgeCV(alphas=[0.1, 1.0, 10.0], cv=3, fit_intercept=True, scoring=None, normalize=False) >>> reg.alpha_ 0.1
References
The Lasso
is a linear model that estimates sparse coefficients. It is useful in some contexts due to its tendency to prefer solutions with fewer parameter values, effectively reducing the number of variables upon which the given solution is dependent. For this reason, the Lasso and its variants are fundamental to the field of compressed sensing. Under certain conditions, it can recover the exact set of nonzero weights (see Compressive sensing: tomography reconstruction with L1 prior (Lasso)).
Mathematically, it consists of a linear model trained with \(\ell_1\) prior as regularizer. The objective function to minimize is:
The lasso estimate thus solves the minimization of the leastsquares penalty with \(\alpha w_1\) added, where \(\alpha\) is a constant and \(w_1\) is the \(\ell_1\)norm of the parameter vector.
The implementation in the class Lasso
uses coordinate descent as the algorithm to fit the coefficients. See Least Angle Regression for another implementation:
>>> from sklearn import linear_model >>> reg = linear_model.Lasso(alpha = 0.1) >>> reg.fit([[0, 0], [1, 1]], [0, 1]) Lasso(alpha=0.1, copy_X=True, fit_intercept=True, max_iter=1000, normalize=False, positive=False, precompute=False, random_state=None, selection='cyclic', tol=0.0001, warm_start=False) >>> reg.predict([[1, 1]]) array([0.8])
Also useful for lowerlevel tasks is the function lasso_path
that computes the coefficients along the full path of possible values.
Examples:
Note
Feature selection with Lasso
As the Lasso regression yields sparse models, it can thus be used to perform feature selection, as detailed in L1based feature selection.
The following two references explain the iterations used in the coordinate descent solver of scikitlearn, as well as the duality gap computation used for convergence control.
References
The alpha
parameter controls the degree of sparsity of the coefficients estimated.
scikitlearn exposes objects that set the Lasso alpha
parameter by crossvalidation: LassoCV
and LassoLarsCV
. LassoLarsCV
is based on the Least Angle Regression algorithm explained below.
For highdimensional datasets with many collinear regressors, LassoCV
is most often preferable. However, LassoLarsCV
has the advantage of exploring more relevant values of alpha
parameter, and if the number of samples is very small compared to the number of features, it is often faster than LassoCV
.
Alternatively, the estimator LassoLarsIC
proposes to use the Akaike information criterion (AIC) and the Bayes Information criterion (BIC). It is a computationally cheaper alternative to find the optimal value of alpha as the regularization path is computed only once instead of k+1 times when using kfold crossvalidation. However, such criteria needs a proper estimation of the degrees of freedom of the solution, are derived for large samples (asymptotic results) and assume the model is correct, i.e. that the data are actually generated by this model. They also tend to break when the problem is badly conditioned (more features than samples).
The equivalence between alpha
and the regularization parameter of SVM, C
is given by alpha = 1 / C
or alpha = 1 / (n_samples * C)
, depending on the estimator and the exact objective function optimized by the model.
The MultiTaskLasso
is a linear model that estimates sparse coefficients for multiple regression problems jointly: y
is a 2D array, of shape (n_samples, n_tasks)
. The constraint is that the selected features are the same for all the regression problems, also called tasks.
The following figure compares the location of the nonzeros in W obtained with a simple Lasso or a MultiTaskLasso. The Lasso estimates yields scattered nonzeros while the nonzeros of the MultiTaskLasso are full columns.
Fitting a timeseries model, imposing that any active feature be active at all times.
Mathematically, it consists of a linear model trained with a mixed \(\ell_1\) \(\ell_2\) prior as regularizer. The objective function to minimize is:
where \(Fro\) indicates the Frobenius norm:
and \(\ell_1\) \(\ell_2\) reads:
The implementation in the class MultiTaskLasso
uses coordinate descent as the algorithm to fit the coefficients.
ElasticNet
is a linear regression model trained with L1 and L2 prior as regularizer. This combination allows for learning a sparse model where few of the weights are nonzero like Lasso
, while still maintaining the regularization properties of Ridge
. We control the convex combination of L1 and L2 using the l1_ratio
parameter.
Elasticnet is useful when there are multiple features which are correlated with one another. Lasso is likely to pick one of these at random, while elasticnet is likely to pick both.
A practical advantage of tradingoff between Lasso and Ridge is it allows ElasticNet to inherit some of Ridge’s stability under rotation.
The objective function to minimize is in this case
The class ElasticNetCV
can be used to set the parameters alpha
(\(\alpha\)) and l1_ratio
(\(\rho\)) by crossvalidation.
The following two references explain the iterations used in the coordinate descent solver of scikitlearn, as well as the duality gap computation used for convergence control.
References
The MultiTaskElasticNet
is an elasticnet model that estimates sparse coefficients for multiple regression problems jointly: Y
is a 2D array, of shape (n_samples, n_tasks)
. The constraint is that the selected features are the same for all the regression problems, also called tasks.
Mathematically, it consists of a linear model trained with a mixed \(\ell_1\) \(\ell_2\) prior and \(\ell_2\) prior as regularizer. The objective function to minimize is:
The implementation in the class MultiTaskElasticNet
uses coordinate descent as the algorithm to fit the coefficients.
The class MultiTaskElasticNetCV
can be used to set the parameters alpha
(\(\alpha\)) and l1_ratio
(\(\rho\)) by crossvalidation.
Leastangle regression (LARS) is a regression algorithm for highdimensional data, developed by Bradley Efron, Trevor Hastie, Iain Johnstone and Robert Tibshirani. LARS is similar to forward stepwise regression. At each step, it finds the predictor most correlated with the response. When there are multiple predictors having equal correlation, instead of continuing along the same predictor, it proceeds in a direction equiangular between the predictors.
The advantages of LARS are:
The disadvantages of the LARS method include:
The LARS model can be used using estimator Lars
, or its lowlevel implementation lars_path
.
LassoLars
is a lasso model implemented using the LARS algorithm, and unlike the implementation based on coordinate_descent, this yields the exact solution, which is piecewise linear as a function of the norm of its coefficients.
>>> from sklearn import linear_model >>> reg = linear_model.LassoLars(alpha=.1) >>> reg.fit([[0, 0], [1, 1]], [0, 1]) LassoLars(alpha=0.1, copy_X=True, eps=..., fit_intercept=True, fit_path=True, max_iter=500, normalize=True, positive=False, precompute='auto', verbose=False) >>> reg.coef_ array([0.717157..., 0. ])
Examples:
The Lars algorithm provides the full path of the coefficients along the regularization parameter almost for free, thus a common operation consist of retrieving the path with function lars_path
The algorithm is similar to forward stepwise regression, but instead of including variables at each step, the estimated parameters are increased in a direction equiangular to each one’s correlations with the residual.
Instead of giving a vector result, the LARS solution consists of a curve denoting the solution for each value of the L1 norm of the parameter vector. The full coefficients path is stored in the array coef_path_
, which has size (n_features, max_features+1). The first column is always zero.
References:
OrthogonalMatchingPursuit
and orthogonal_mp
implements the OMP algorithm for approximating the fit of a linear model with constraints imposed on the number of nonzero coefficients (ie. the L _{0} pseudonorm).
Being a forward feature selection method like Least Angle Regression, orthogonal matching pursuit can approximate the optimum solution vector with a fixed number of nonzero elements:
Alternatively, orthogonal matching pursuit can target a specific error instead of a specific number of nonzero coefficients. This can be expressed as:
OMP is based on a greedy algorithm that includes at each step the atom most highly correlated with the current residual. It is similar to the simpler matching pursuit (MP) method, but better in that at each iteration, the residual is recomputed using an orthogonal projection on the space of the previously chosen dictionary elements.
Examples:
References:
Bayesian regression techniques can be used to include regularization parameters in the estimation procedure: the regularization parameter is not set in a hard sense but tuned to the data at hand.
This can be done by introducing uninformative priors over the hyper parameters of the model. The \(\ell_{2}\) regularization used in Ridge Regression is equivalent to finding a maximum a posteriori estimation under a Gaussian prior over the parameters \(w\) with precision \(\lambda^{1}\). Instead of setting lambda
manually, it is possible to treat it as a random variable to be estimated from the data.
To obtain a fully probabilistic model, the output \(y\) is assumed to be Gaussian distributed around \(X w\):
Alpha is again treated as a random variable that is to be estimated from the data.
The advantages of Bayesian Regression are:
The disadvantages of Bayesian regression include:
References
Bayesian learning for neural networks
by Radford M. NealBayesianRidge
estimates a probabilistic model of the regression problem as described above. The prior for the parameter \(w\) is given by a spherical Gaussian:
The priors over \(\alpha\) and \(\lambda\) are chosen to be gamma distributions, the conjugate prior for the precision of the Gaussian.
The resulting model is called Bayesian Ridge Regression, and is similar to the classical Ridge
. The parameters \(w\), \(\alpha\) and \(\lambda\) are estimated jointly during the fit of the model. The remaining hyperparameters are the parameters of the gamma priors over \(\alpha\) and \(\lambda\). These are usually chosen to be noninformative. The parameters are estimated by maximizing the marginal log likelihood.
By default \(\alpha_1 = \alpha_2 = \lambda_1 = \lambda_2 = 10^{6}\).
Bayesian Ridge Regression is used for regression:
>>> from sklearn import linear_model >>> X = [[0., 0.], [1., 1.], [2., 2.], [3., 3.]] >>> Y = [0., 1., 2., 3.] >>> reg = linear_model.BayesianRidge() >>> reg.fit(X, Y) BayesianRidge(alpha_1=1e06, alpha_2=1e06, compute_score=False, copy_X=True, fit_intercept=True, lambda_1=1e06, lambda_2=1e06, n_iter=300, normalize=False, tol=0.001, verbose=False)
After being fitted, the model can then be used to predict new values:
>>> reg.predict ([[1, 0.]]) array([0.50000013])
The weights \(w\) of the model can be access:
>>> reg.coef_ array([0.49999993, 0.49999993])
Due to the Bayesian framework, the weights found are slightly different to the ones found by Ordinary Least Squares. However, Bayesian Ridge Regression is more robust to illposed problem.
Examples:
References
ARDRegression
is very similar to Bayesian Ridge Regression, but can lead to sparser weights \(w\) [1] [2]. ARDRegression
poses a different prior over \(w\), by dropping the assumption of the Gaussian being spherical.
Instead, the distribution over \(w\) is assumed to be an axisparallel, elliptical Gaussian distribution.
This means each weight \(w_{i}\) is drawn from a Gaussian distribution, centered on zero and with a precision \(\lambda_{i}\):
with \(diag \; (A) = \lambda = \{\lambda_{1},...,\lambda_{p}\}\).
In contrast to Bayesian Ridge Regression, each coordinate of \(w_{i}\) has its own standard deviation \(\lambda_i\). The prior over all \(\lambda_i\) is chosen to be the same gamma distribution given by hyperparameters \(\lambda_1\) and \(\lambda_2\).
ARD is also known in the literature as Sparse Bayesian Learning and Relevance Vector Machine [3] [4].
References:
[1]  Christopher M. Bishop: Pattern Recognition and Machine Learning, Chapter 7.2.1 
[2]  David Wipf and Srikantan Nagarajan: A new view of automatic relevance determination 
[3]  Michael E. Tipping: Sparse Bayesian Learning and the Relevance Vector Machine 
[4]  Tristan Fletcher: Relevance Vector Machines explained 
Logistic regression, despite its name, is a linear model for classification rather than regression. Logistic regression is also known in the literature as logit regression, maximumentropy classification (MaxEnt) or the loglinear classifier. In this model, the probabilities describing the possible outcomes of a single trial are modeled using a logistic function.
The implementation of logistic regression in scikitlearn can be accessed from class LogisticRegression
. This implementation can fit binary, Onevs Rest, or multinomial logistic regression with optional L2 or L1 regularization.
As an optimization problem, binary class L2 penalized logistic regression minimizes the following cost function:
Similarly, L1 regularized logistic regression solves the following optimization problem
Note that, in this notation, it’s assumed that the observation \(y_i\) takes values in the set \({1, 1}\) at trial \(i\).
The solvers implemented in the class LogisticRegression
are “liblinear”, “newtoncg”, “lbfgs”, “sag” and “saga”:
The solver “liblinear” uses a coordinate descent (CD) algorithm, and relies on the excellent C++ LIBLINEAR library, which is shipped with scikitlearn. However, the CD algorithm implemented in liblinear cannot learn a true multinomial (multiclass) model; instead, the optimization problem is decomposed in a “onevsrest” fashion so separate binary classifiers are trained for all classes. This happens under the hood, so LogisticRegression
instances using this solver behave as multiclass classifiers. For L1 penalization sklearn.svm.l1_min_c
allows to calculate the lower bound for C in order to get a non “null” (all feature weights to zero) model.
The “lbfgs”, “sag” and “newtoncg” solvers only support L2 penalization and are found to converge faster for some high dimensional data. Setting multi_class
to “multinomial” with these solvers learns a true multinomial logistic regression model [5], which means that its probability estimates should be better calibrated than the default “onevsrest” setting.
The “sag” solver uses a Stochastic Average Gradient descent [6]. It is faster than other solvers for large datasets, when both the number of samples and the number of features are large.
The “saga” solver [7] is a variant of “sag” that also supports the nonsmooth penalty=”l1”
option. This is therefore the solver of choice for sparse multinomial logistic regression.
In a nutshell, the following table summarizes the solvers characteristics:
solver  ‘liblinear’  ‘lbfgs’  ‘newtoncg’  ‘sag’  ‘saga’ 

Multinomial + L2 penalty  no  yes  yes  yes  yes 
OVR + L2 penalty  yes  yes  yes  yes  yes 
Multinomial + L1 penalty  no  no  no  no  yes 
OVR + L1 penalty  yes  no  no  no  yes 
Penalize the intercept (bad) yes no no no no Faster for large datasets no no no yes yes Robust to unscaled datasets yes yes yes no no ============================ =========== ======= =========== ===== ======
The “saga” solver is often the best choice. The “liblinear” solver is used by default for historical reasons.
For large dataset, you may also consider using SGDClassifier
with ‘log’ loss.
Examples:
Differences from liblinear:
There might be a difference in the scores obtained between LogisticRegression
with solver=liblinear
or LinearSVC
and the external liblinear library directly, when fit_intercept=False
and the fit coef_
(or) the data to be predicted are zeroes. This is because for the sample(s) with decision_function
zero, LogisticRegression
and LinearSVC
predict the negative class, while liblinear predicts the positive class. Note that a model with fit_intercept=False
and having many samples with decision_function
zero, is likely to be a underfit, bad model and you are advised to set fit_intercept=True
and increase the intercept_scaling.
Note
Feature selection with sparse logistic regression
A logistic regression with L1 penalty yields sparse models, and can thus be used to perform feature selection, as detailed in L1based feature selection.
LogisticRegressionCV
implements Logistic Regression with builtin crossvalidation to find out the optimal C parameter. “newtoncg”, “sag”, “saga” and “lbfgs” solvers are found to be faster for highdimensional dense data, due to warmstarting. For the multiclass case, if multi_class
option is set to “ovr”, an optimal C is obtained for each class and if the multi_class
option is set to “multinomial”, an optimal C is obtained by minimizing the crossentropy loss.
References:
[5]  Christopher M. Bishop: Pattern Recognition and Machine Learning, Chapter 4.3.4 
[6]  Mark Schmidt, Nicolas Le Roux, and Francis Bach: Minimizing Finite Sums with the Stochastic Average Gradient. 
[7]  Aaron Defazio, Francis Bach, Simon LacosteJulien: SAGA: A Fast Incremental Gradient Method With Support for NonStrongly Convex Composite Objectives. 
Stochastic gradient descent is a simple yet very efficient approach to fit linear models. It is particularly useful when the number of samples (and the number of features) is very large. The partial_fit
method allows online/outofcore learning.
The classes SGDClassifier
and SGDRegressor
provide functionality to fit linear models for classification and regression using different (convex) loss functions and different penalties. E.g., with loss="log"
, SGDClassifier
fits a logistic regression model, while with loss="hinge"
it fits a linear support vector machine (SVM).
References
The Perceptron
is another simple classification algorithm suitable for large scale learning. By default:
The last characteristic implies that the Perceptron is slightly faster to train than SGD with the hinge loss and that the resulting models are sparser.
The passiveaggressive algorithms are a family of algorithms for largescale learning. They are similar to the Perceptron in that they do not require a learning rate. However, contrary to the Perceptron, they include a regularization parameter C
.
For classification, PassiveAggressiveClassifier
can be used with loss='hinge'
(PAI) or loss='squared_hinge'
(PAII). For regression, PassiveAggressiveRegressor
can be used with loss='epsilon_insensitive'
(PAI) or loss='squared_epsilon_insensitive'
(PAII).
References:
Robust regression is interested in fitting a regression model in the presence of corrupt data: either outliers, or error in the model.
There are different things to keep in mind when dealing with data corrupted by outliers:
Outliers in X or in y?
Outliers in the y direction  Outliers in the X direction 

Fraction of outliers versus amplitude of error
The number of outlying points matters, but also how much they are outliers.
Small outliers  Large outliers 

An important notion of robust fitting is that of breakdown point: the fraction of data that can be outlying for the fit to start missing the inlying data.
Note that in general, robust fitting in highdimensional setting (large n_features
) is very hard. The robust models here will probably not work in these settings.
Tradeoffs: which estimator?
Scikitlearn provides 3 robust regression estimators: RANSAC, Theil Sen and HuberRegressor
n_samples
>> n_features
. This is because RANSAC and Theil Sen fit on smaller subsets of the data. However, both Theil Sen and RANSAC are unlikely to be as robust as HuberRegressor for the default parameters.When in doubt, use RANSAC
RANSAC (RANdom SAmple Consensus) fits a model from random subsets of inliers from the complete data set.
RANSAC is a nondeterministic algorithm producing only a reasonable result with a certain probability, which is dependent on the number of iterations (see max_trials
parameter). It is typically used for linear and nonlinear regression problems and is especially popular in the fields of photogrammetric computer vision.
The algorithm splits the complete input sample data into a set of inliers, which may be subject to noise, and outliers, which are e.g. caused by erroneous measurements or invalid hypotheses about the data. The resulting model is then estimated only from the determined inliers.
Each iteration performs the following steps:
min_samples
random samples from the original data and check whether the set of data is valid (see is_data_valid
).base_estimator.fit
) and check whether the estimated model is valid (see is_model_valid
).base_estimator.predict(X)  y
)  all data samples with absolute residuals smaller than the residual_threshold
are considered as inliers.These steps are performed either a maximum number of times (max_trials
) or until one of the special stop criteria are met (see stop_n_inliers
and stop_score
). The final model is estimated using all inlier samples (consensus set) of the previously determined best model.
The is_data_valid
and is_model_valid
functions allow to identify and reject degenerate combinations of random subsamples. If the estimated model is not needed for identifying degenerate cases, is_data_valid
should be used as it is called prior to fitting the model and thus leading to better computational performance.
References:
The TheilSenRegressor
estimator uses a generalization of the median in multiple dimensions. It is thus robust to multivariate outliers. Note however that the robustness of the estimator decreases quickly with the dimensionality of the problem. It looses its robustness properties and becomes no better than an ordinary least squares in high dimension.
TheilSenRegressor
is comparable to the Ordinary Least Squares (OLS) in terms of asymptotic efficiency and as an unbiased estimator. In contrast to OLS, TheilSen is a nonparametric method which means it makes no assumption about the underlying distribution of the data. Since TheilSen is a medianbased estimator, it is more robust against corrupted data aka outliers. In univariate setting, TheilSen has a breakdown point of about 29.3% in case of a simple linear regression which means that it can tolerate arbitrary corrupted data of up to 29.3%.
The implementation of TheilSenRegressor
in scikitlearn follows a generalization to a multivariate linear regression model [8] using the spatial median which is a generalization of the median to multiple dimensions [9].
In terms of time and space complexity, TheilSen scales according to
which makes it infeasible to be applied exhaustively to problems with a large number of samples and features. Therefore, the magnitude of a subpopulation can be chosen to limit the time and space complexity by considering only a random subset of all possible combinations.
Examples:
References:
[8]  Xin Dang, Hanxiang Peng, Xueqin Wang and Heping Zhang: TheilSen Estimators in a Multiple Linear Regression Model. 
[9] 

The HuberRegressor
is different to Ridge
because it applies a linear loss to samples that are classified as outliers. A sample is classified as an inlier if the absolute error of that sample is lesser than a certain threshold. It differs from TheilSenRegressor
and RANSACRegressor
because it does not ignore the effect of the outliers but gives a lesser weight to them.
The loss function that HuberRegressor
minimizes is given by
where
It is advised to set the parameter epsilon
to 1.35 to achieve 95% statistical efficiency.
The HuberRegressor
differs from using SGDRegressor
with loss set to huber
in the following ways.
HuberRegressor
is scaling invariant. Once epsilon
is set, scaling X
and y
down or up by different values would produce the same robustness to outliers as before. as compared to SGDRegressor
where epsilon
has to be set again when X
and y
are scaled.HuberRegressor
should be more efficient to use on data with small number of samples while SGDRegressor
needs a number of passes on the training data to produce the same robustness.References:
Also, this estimator is different from the R implementation of Robust Regression (http://www.ats.ucla.edu/stat/r/dae/rreg.htm) because the R implementation does a weighted least squares implementation with weights given to each sample on the basis of how much the residual is greater than a certain threshold.
One common pattern within machine learning is to use linear models trained on nonlinear functions of the data. This approach maintains the generally fast performance of linear methods, while allowing them to fit a much wider range of data.
For example, a simple linear regression can be extended by constructing polynomial features from the coefficients. In the standard linear regression case, you might have a model that looks like this for twodimensional data:
If we want to fit a paraboloid to the data instead of a plane, we can combine the features in secondorder polynomials, so that the model looks like this:
The (sometimes surprising) observation is that this is still a linear model: to see this, imagine creating a new variable
With this relabeling of the data, our problem can be written
We see that the resulting polynomial regression is in the same class of linear models we’d considered above (i.e. the model is linear in \(w\)) and can be solved by the same techniques. By considering linear fits within a higherdimensional space built with these basis functions, the model has the flexibility to fit a much broader range of data.
Here is an example of applying this idea to onedimensional data, using polynomial features of varying degrees:
This figure is created using the PolynomialFeatures
preprocessor. This preprocessor transforms an input data matrix into a new data matrix of a given degree. It can be used as follows:
>>> from sklearn.preprocessing import PolynomialFeatures >>> import numpy as np >>> X = np.arange(6).reshape(3, 2) >>> X array([[0, 1], [2, 3], [4, 5]]) >>> poly = PolynomialFeatures(degree=2) >>> poly.fit_transform(X) array([[ 1., 0., 1., 0., 0., 1.], [ 1., 2., 3., 4., 6., 9.], [ 1., 4., 5., 16., 20., 25.]])
The features of X
have been transformed from \([x_1, x_2]\) to \([1, x_1, x_2, x_1^2, x_1 x_2, x_2^2]\), and can now be used within any linear model.
This sort of preprocessing can be streamlined with the Pipeline tools. A single object representing a simple polynomial regression can be created and used as follows:
>>> from sklearn.preprocessing import PolynomialFeatures >>> from sklearn.linear_model import LinearRegression >>> from sklearn.pipeline import Pipeline >>> import numpy as np >>> model = Pipeline([('poly', PolynomialFeatures(degree=3)), ... ('linear', LinearRegression(fit_intercept=False))]) >>> # fit to an order3 polynomial data >>> x = np.arange(5) >>> y = 3  2 * x + x ** 2  x ** 3 >>> model = model.fit(x[:, np.newaxis], y) >>> model.named_steps['linear'].coef_ array([ 3., 2., 1., 1.])
The linear model trained on polynomial features is able to exactly recover the input polynomial coefficients.
In some cases it’s not necessary to include higher powers of any single feature, but only the socalled interaction features that multiply together at most \(d\) distinct features. These can be gotten from PolynomialFeatures
with the setting interaction_only=True
.
For example, when dealing with boolean features, \(x_i^n = x_i\) for all \(n\) and is therefore useless; but \(x_i x_j\) represents the conjunction of two booleans. This way, we can solve the XOR problem with a linear classifier:
>>> from sklearn.linear_model import Perceptron >>> from sklearn.preprocessing import PolynomialFeatures >>> import numpy as np >>> X = np.array([[0, 0], [0, 1], [1, 0], [1, 1]]) >>> y = X[:, 0] ^ X[:, 1] >>> y array([0, 1, 1, 0]) >>> X = PolynomialFeatures(interaction_only=True).fit_transform(X).astype(int) >>> X array([[1, 0, 0, 0], [1, 0, 1, 0], [1, 1, 0, 0], [1, 1, 1, 1]]) >>> clf = Perceptron(fit_intercept=False, max_iter=10, tol=None, ... shuffle=False).fit(X, y)
And the classifier “predictions” are perfect:
>>> clf.predict(X) array([0, 1, 1, 0]) >>> clf.score(X, y) 1.0
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Licensed under the 3clause BSD License.
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