1.1. Generalized Linear Models¶

1.1.1. Ordinary Least Squares¶

LinearRegression fits a linear model with coefficients \(w = (w_1, ..., w_p)\) to minimize the residual sum of squares between the observed targets in the dataset, and the targets 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()
>>> reg.coef_
array([0.5, 0.5])

The coefficient estimates for Ordinary Least Squares rely on the independence of the features. When features 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 least-squares estimate becomes highly sensitive to random errors in the observed target, producing a large variance. This situation of multicollinearity can arise, for example, when data are collected without an experimental design.

1.1.2. Ridge Regression¶

Ridge regression addresses some of the problems of Ordinary Least Squares by imposing a penalty on the size of the coefficients. The ridge coefficients minimize a penalized residual sum of squares:

The complexity parameter \(\alpha \geq 0\) 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)
>>> reg.coef_
array([0.34545455, 0.34545455])
>>> reg.intercept_
0.13636...

>>> import numpy as np
>>> from sklearn import linear_model
>>> reg = linear_model.RidgeCV(alphas=np.logspace(-6, 6, 13))
>>> reg.fit([[0, 0], [0, 0], [1, 1]], [0, .1, 1])
RidgeCV(alphas=array([1.e-06, 1.e-05, 1.e-04, 1.e-03, 1.e-02, 1.e-01, 1.e+00, 1.e+01,
      1.e+02, 1.e+03, 1.e+04, 1.e+05, 1.e+06]))
>>> reg.alpha_
0.01

1.1.3. Lasso¶

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 non-zero coefficients, effectively reducing the number of features upon which the given solution is dependent. For this reason Lasso and its variants are fundamental to the field of compressed sensing. Under certain conditions, it can recover the exact set of non-zero coefficients (see Compressive sensing: tomography reconstruction with L1 prior (Lasso)).

Mathematically, it consists of a linear model with an added regularization term. The objective function to minimize is:

The lasso estimate thus solves the minimization of the least-squares penalty with \(\alpha ||w||_1\) added, where \(\alpha\) is a constant and \(||w||_1\) is the \(\ell_1\)-norm of the coefficient 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)
>>> reg.predict([[1, 1]])
array([0.8])

The function lasso_path is useful for lower-level tasks, as it computes the coefficients along the full path of possible values.

The following two references explain the iterations used in the coordinate descent solver of scikit-learn, as well as the duality gap computation used for convergence control.

1.1.3.1. Setting regularization parameter¶

The alpha parameter controls the degree of sparsity of the estimated coefficients.

1.1.3.1.1. Using cross-validation¶

scikit-learn exposes objects that set the Lasso alpha parameter by cross-validation: LassoCV and LassoLarsCV. LassoLarsCV is based on the Least Angle Regression algorithm explained below.

For high-dimensional datasets with many collinear features, 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.

1.1.3.1.2. Information-criteria based model selection¶

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 k-fold cross-validation. 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).

Examples:

• Lasso model selection: Cross-Validation / AIC / BIC

1.1.3.1.3. Comparison with the regularization parameter of SVM¶

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.

1.1.4. Multi-task Lasso¶

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 non-zero entries in the coefficient matrix W obtained with a simple Lasso or a MultiTaskLasso. The Lasso estimates yield scattered non-zeros while the non-zeros of the MultiTaskLasso are full columns.

Fitting a time-series model, imposing that any active feature be active at all times.

Mathematically, it consists of a linear model trained with a mixed \(\ell_1\) \(\ell_2\)-norm for regularization. The objective function to minimize is:

where \(\text{Fro}\) 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.

1.1.5. Elastic-Net¶

ElasticNet is a linear regression model trained with both \(\ell_1\) and \(\ell_2\)-norm regularization of the coefficients. This combination allows for learning a sparse model where few of the weights are non-zero like Lasso, while still maintaining the regularization properties of Ridge. We control the convex combination of \(\ell_1\) and \(\ell_2\) using the l1_ratio parameter.

Elastic-net is useful when there are multiple features which are correlated with one another. Lasso is likely to pick one of these at random, while elastic-net is likely to pick both.

A practical advantage of trading-off between Lasso and Ridge is that it allows Elastic-Net 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 cross-validation.

The following two references explain the iterations used in the coordinate descent solver of scikit-learn, as well as the duality gap computation used for convergence control.

1.1.6. Multi-task Elastic-Net¶

The MultiTaskElasticNet is an elastic-net 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 \(\ell_1\) \(\ell_2\)-norm and \(\ell_2\)-norm for regularization. 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 cross-validation.

1.1.7. Least Angle Regression¶

Least-angle regression (LARS) is a regression algorithm for high-dimensional 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 feature most correlated with the target. When there are multiple features having equal correlation, instead of continuing along the same feature, it proceeds in a direction equiangular between the features.

The advantages of LARS are:

• It is numerically efficient in contexts where the number of features is significantly greater than the number of samples.

• It is computationally just as fast as forward selection and has the same order of complexity as ordinary least squares.

• It produces a full piecewise linear solution path, which is useful in cross-validation or similar attempts to tune the model.

• If two features are almost equally correlated with the target, then their coefficients should increase at approximately the same rate. The algorithm thus behaves as intuition would expect, and also is more stable.

• It is easily modified to produce solutions for other estimators, like the Lasso.

The disadvantages of the LARS method include:

• Because LARS is based upon an iterative refitting of the residuals, it would appear to be especially sensitive to the effects of noise. This problem is discussed in detail by Weisberg in the discussion section of the Efron et al. (2004) Annals of Statistics article.

The LARS model can be used using estimator Lars, or its low-level implementation lars_path or lars_path_gram.

1.1.8. LARS Lasso¶

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)
>>> reg.coef_
array([0.717157..., 0.        ])

The Lars algorithm provides the full path of the coefficients along the regularization parameter almost for free, thus a common operation is to retrieve the path with one of the functions lars_path or lars_path_gram.

1.1.9. Orthogonal Matching Pursuit (OMP)¶

OrthogonalMatchingPursuit and orthogonal_mp implements the OMP algorithm for approximating the fit of a linear model with constraints imposed on the number of non-zero coefficients (ie. the \(\ell_0\) pseudo-norm).

Being a forward feature selection method like Least Angle Regression, orthogonal matching pursuit can approximate the optimum solution vector with a fixed number of non-zero elements:

Alternatively, orthogonal matching pursuit can target a specific error instead of a specific number of non-zero 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.

1.1.10. Bayesian Regression¶

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 coefficients \(w\) 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 \(X w\):

where \(\alpha\) is again treated as a random variable that is to be estimated from the data.

The advantages of Bayesian Regression are:

• It adapts to the data at hand.

• It can be used to include regularization parameters in the estimation procedure.

The disadvantages of Bayesian regression include:

• Inference of the model can be time consuming.

1.1.10.1. Bayesian Ridge Regression¶

BayesianRidge estimates a probabilistic model of the regression problem as described above. The prior for the coefficient \(w\) is given by a spherical Gaussian:

\[p(w|\lambda) = \mathcal{N}(w|0,\lambda^{-1}\mathbf{I}_{p})\]

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 regularization parameters \(\alpha\) and \(\lambda\) being estimated by maximizing the log marginal likelihood. The scikit-learn implementation is based on the algorithm described in Appendix A of (Tipping, 2001) where the update of the parameters \(\alpha\) and \(\lambda\) is done as suggested in (MacKay, 1992). The initial value of the maximization procedure can be set with the hyperparameters alpha_init and lambda_init.

There are four more hyperparameters, \(\alpha_1\), \(\alpha_2\), \(\lambda_1\) and \(\lambda_2\) of the gamma prior distributions over \(\alpha\) and \(\lambda\). These are usually chosen to be non-informative. 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()

After being fitted, the model can then be used to predict new values:

>>> reg.predict([[1, 0.]])
array([0.50000013])

The coefficients \(w\) of the model can be accessed:

>>> 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 ill-posed problems.

Examples:

• Bayesian Ridge Regression

• Curve Fitting with Bayesian Ridge Regression

References:

• Section 3.3 in Christopher M. Bishop: Pattern Recognition and Machine Learning, 2006

• David J. C. MacKay, Bayesian Interpolation, 1992.

• Michael E. Tipping, Sparse Bayesian Learning and the Relevance Vector Machine, 2001.

1.1.10.2. Automatic Relevance Determination - ARD¶

ARDRegression is very similar to Bayesian Ridge Regression, but can lead to sparser coefficients \(w\) 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 axis-parallel, elliptical Gaussian distribution.

This means each coefficient \(w_{i}\) is drawn from a Gaussian distribution, centered on zero and with a precision \(\lambda_{i}\):

\[p(w|\lambda) = \mathcal{N}(w|0,A^{-1})\]

with \(\text{diag}(A) = \lambda = \{\lambda_{1},...,\lambda_{p}\}\).

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.

Examples:

• Automatic Relevance Determination Regression (ARD)

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

1.1.11. Logistic regression¶

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, maximum-entropy classification (MaxEnt) or the log-linear classifier. In this model, the probabilities describing the possible outcomes of a single trial are modeled using a logistic function.

Logistic regression is implemented in LogisticRegression. This implementation can fit binary, One-vs-Rest, or multinomial logistic regression with optional \(\ell_1\), \(\ell_2\) or Elastic-Net regularization.

As an optimization problem, binary class \(\ell_2\) penalized logistic regression minimizes the following cost function:

Similarly, \(\ell_1\) regularized logistic regression solves the following optimization problem:

Elastic-Net regularization is a combination of \(\ell_1\) and \(\ell_2\), and minimizes the following cost function:

where \(\rho\) controls the strength of \(\ell_1\) regularization vs. \(\ell_2\) regularization (it corresponds to the l1_ratio parameter).

Note that, in this notation, it’s assumed that the target \(y_i\) takes values in the set \({-1, 1}\) at trial \(i\). We can also see that Elastic-Net is equivalent to \(\ell_1\) when \(\rho = 1\) and equivalent to \(\ell_2\) when \(\rho=0\).

The solvers implemented in the class LogisticRegression are “liblinear”, “newton-cg”, “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 scikit-learn. However, the CD algorithm implemented in liblinear cannot learn a true multinomial (multiclass) model; instead, the optimization problem is decomposed in a “one-vs-rest” 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 \(\ell_1\) regularization 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 “newton-cg” solvers only support \(\ell_2\) regularization or no regularization, 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 “one-vs-rest” setting.

The “sag” solver uses 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 non-smooth penalty="l1". This is therefore the solver of choice for sparse multinomial logistic regression. It is also the only solver that supports penalty="elasticnet".

The “lbfgs” is an optimization algorithm that approximates the Broyden–Fletcher–Goldfarb–Shanno algorithm 8, which belongs to quasi-Newton methods. The “lbfgs” solver is recommended for use for small data-sets but for larger datasets its performance suffers. 9

The following table summarizes the penalties supported by each solver:

The “lbfgs” solver is used by default for its robustness. For large datasets the “saga” solver is usually faster. For large dataset, you may also consider using SGDClassifier with ‘log’ loss, which might be even faster but requires more tuning.

LogisticRegressionCV implements Logistic Regression with built-in cross-validation support, to find the optimal C and l1_ratio parameters according to the scoring attribute. The “newton-cg”, “sag”, “saga” and “lbfgs” solvers are found to be faster for high-dimensional dense data, due to warm-starting (see Glossary).

1.1.12. Stochastic Gradient Descent - SGD¶

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/out-of-core 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).

1.1.13. Perceptron¶

The Perceptron is another simple classification algorithm suitable for large scale learning. By default:

• It does not require a learning rate.

• It is not regularized (penalized).

• It updates its model only on mistakes.

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.

1.1.14. Passive Aggressive Algorithms¶

The passive-aggressive algorithms are a family of algorithms for large-scale 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' (PA-I) or loss='squared_hinge' (PA-II). For regression, PassiveAggressiveRegressor can be used with loss='epsilon_insensitive' (PA-I) or loss='squared_epsilon_insensitive' (PA-II).

1.1.15. Robustness regression: outliers and modeling errors¶

Robust regression aims to fit a regression model in the presence of corrupt data: either outliers, or error in the model.

1.1.15.2. RANSAC: RANdom SAmple Consensus¶

RANSAC (RANdom SAmple Consensus) fits a model from random subsets of inliers from the complete data set.

RANSAC is a non-deterministic 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 non-linear regression problems and is especially popular in the field 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.

1.1.15.2.1. Details of the algorithm¶

Each iteration performs the following steps:

1. Select min_samples random samples from the original data and check whether the set of data is valid (see is_data_valid).

2. Fit a model to the random subset (base_estimator.fit) and check whether the estimated model is valid (see is_model_valid).

3. Classify all data as inliers or outliers by calculating the residuals to the estimated model (base_estimator.predict(X) - y) - all data samples with absolute residuals smaller than the residual_threshold are considered as inliers.

4. Save fitted model as best model if number of inlier samples is maximal. In case the current estimated model has the same number of inliers, it is only considered as the best model if it has better score.

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 sub-samples. 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.

Examples:

• Robust linear model estimation using RANSAC

• Robust linear estimator fitting

References:

• https://en.wikipedia.org/wiki/RANSAC

• “Random Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartography” Martin A. Fischler and Robert C. Bolles - SRI International (1981)

• “Performance Evaluation of RANSAC Family” Sunglok Choi, Taemin Kim and Wonpil Yu - BMVC (2009)

1.1.15.3. Theil-Sen estimator: generalized-median-based estimator¶

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 loses its robustness properties and becomes no better than an ordinary least squares in high dimension.

Examples:

• Theil-Sen Regression

• Robust linear estimator fitting

References:

• https://en.wikipedia.org/wiki/Theil%E2%80%93Sen_estimator

1.1.15.3.1. Theoretical considerations¶

TheilSenRegressor is comparable to the Ordinary Least Squares (OLS) in terms of asymptotic efficiency and as an unbiased estimator. In contrast to OLS, Theil-Sen is a non-parametric method which means it makes no assumption about the underlying distribution of the data. Since Theil-Sen is a median-based estimator, it is more robust against corrupted data aka outliers. In univariate setting, Theil-Sen 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 scikit-learn follows a generalization to a multivariate linear regression model 10 using the spatial median which is a generalization of the median to multiple dimensions 11.

In terms of time and space complexity, Theil-Sen scales according to

\[\binom{n_{\text{samples}}}{n_{\text{subsamples}}}\]

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:

• Theil-Sen Regression

References:

10

Xin Dang, Hanxiang Peng, Xueqin Wang and Heping Zhang: Theil-Sen Estimators in a Multiple Linear Regression Model.

11

20. Kärkkäinen and S. Äyrämö: On Computation of Spatial Median for Robust Data Mining.

1.1.15.4. Huber Regression¶

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

\[\min_{w, \sigma} {\sum_{i=1}^n\left(\sigma + H_{\epsilon}\left(\frac{X_{i}w - y_{i}}{\sigma}\right)\sigma\right) + \alpha {||w||_2}^2}\]

where

\[\begin{split}H_{\epsilon}(z) = \begin{cases} z^2, & \text {if } |z| < \epsilon, \\ 2\epsilon|z| - \epsilon^2, & \text{otherwise} \end{cases}\end{split}\]

It is advised to set the parameter epsilon to 1.35 to achieve 95% statistical efficiency.

1.1.16. Polynomial regression: extending linear models with basis functions¶

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 two-dimensional data:

If we want to fit a paraboloid to the data instead of a plane, we can combine the features in second-order 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 set of features

With this re-labeling of the data, our problem can be written

We see that the resulting polynomial regression is in the same class of linear models we considered above (i.e. the model is linear in \(w\)) and can be solved by the same techniques. By considering linear fits within a higher-dimensional 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 one-dimensional data, using polynomial features of varying degrees:

This figure is created using the PolynomialFeatures transformer, which 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 order-3 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 so-called 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