smile.regression — org.clojars.haifengl/smile 3.1.1

Regression Analysis

cart clj

(cart formula data)

(cart formula data max-depth max-nodes node-size)

Regression tree.

A classification/regression tree can be learned by splitting the training set into subsets based on an attribute value test. This process is repeated on each derived subset in a recursive manner called recursive partitioning. The recursion is completed when the subset at a node all has the same value of the target variable, or when splitting no longer adds value to the predictions.

The algorithms that are used for constructing decision trees usually work top-down by choosing a variable at each step that is the next best variable to use in splitting the set of items. 'Best' is defined by how well the variable splits the set into homogeneous subsets that have the same value of the target variable. Different algorithms use different formulae for measuring 'best'. Used by the CART algorithm, Gini impurity is a measure of how often a randomly chosen element from the set would be incorrectly labeled if it were randomly labeled according to the distribution of labels in the subset. Gini impurity can be computed by summing the probability of each item being chosen times the probability of a mistake in categorizing that item. It reaches its minimum (zero) when all cases in the node fall into a single target category. Information gain is another popular measure, used by the ID3, C4.5 and C5.0 algorithms. Information gain is based on the concept of entropy used in information theory. For categorical variables with different number of levels, however, information gain are biased in favor of those attributes with more levels. Instead, one may employ the information gain ratio, which solves the drawback of information gain.

Classification and Regression Tree techniques have a number of advantages over many of those alternative techniques.

• Simple to understand and interpret: In most cases, the interpretation of results summarized in a tree is very simple. This simplicity is useful not only for purposes of rapid classification of new observations, but can also often yield a much simpler 'model' for explaining why observations are classified or predicted in a particular manner.
• Able to handle both numerical and categorical data: Other techniques are usually specialized in analyzing datasets that have only one type of variable.
• Nonparametric and nonlinear: The final results of using tree methods for classification or regression can be summarized in a series of (usually few) logical if-then conditions (tree nodes). Therefore, there is no implicit assumption that the underlying relationships between the predictor variables and the dependent variable are linear, follow some specific non-linear link function, or that they are even monotonic in nature. Thus, tree methods are particularly well suited for data mining tasks, where there is often little a priori knowledge nor any coherent set of theories or predictions regarding which variables are related and how. In those types of data analytics, tree methods can often reveal simple relationships between just a few variables that could have easily gone unnoticed using other analytic techniques.

One major problem with classification and regression trees is their high variance. Often a small change in the data can result in a very different series of splits, making interpretation somewhat precarious. Besides, decision-tree learners can create over-complex trees that cause over-fitting. Mechanisms such as pruning are necessary to avoid this problem. Another limitation of trees is the lack of smoothness of the prediction surface.

Some techniques such as bagging, boosting, and random forest use more than one decision tree for their analysis.

formula is a symbolic description of the model to be fitted. data is the data frame of the explanatory and response variables. max-depth is the maximum depth of the tree. max-nodes is the maximum number of leaf nodes in the tree. node-size is the minimum size of leaf nodes.

gbm clj

(gbm formula data)

(gbm formula data loss ntrees max-depth max-nodes node-size shrinkage subsample)

Gradient boosted classification trees.

Generic gradient boosting at the t-th step would fit a regression tree to pseudo-residuals. Let J be the number of its leaves. The tree partitions the input space into J disjoint regions and predicts a constant value in each region. The parameter J controls the maximum allowed level of interaction between variables in the model. With J = 2 (decision stumps), no interaction between variables is allowed. With J = 3 the model may include effects of the interaction between up to two variables, and so on. Hastie et al. comment that typically 4 ≤ J ≤ 8 work well for boosting and results are fairly insensitive to the choice of in this range, J = 2 is insufficient for many applications, and J > 10 is unlikely to be required.

Fitting the training set too closely can lead to degradation of the model's generalization ability. Several so-called regularization techniques reduce this over-fitting effect by constraining the fitting procedure. One natural regularization parameter is the number of gradient boosting iterations T (i.e. the number of trees in the model when the base learner is a decision tree). Increasing T reduces the error on training set, but setting it too high may lead to over-fitting. An optimal value of T is often selected by monitoring prediction error on a separate validation data set.

Another regularization approach is the shrinkage which times a parameter η (called the 'learning rate') to update term. Empirically it has been found that using small learning rates (such as η < 0.1) yields dramatic improvements in model's generalization ability over gradient boosting without shrinking (η = 1). However, it comes at the price of increasing computational time both during training and prediction: lower learning rate requires more iterations.

Soon after the introduction of gradient boosting Friedman proposed a minor modification to the algorithm, motivated by Breiman's bagging method. Specifically, he proposed that at each iteration of the algorithm, a base learner should be fit on a subsample of the training set drawn at random without replacement. Friedman observed a substantial improvement in gradient boosting's accuracy with this modification.

Subsample size is some constant fraction f of the size of the training set. When f = 1, the algorithm is deterministic and identical to the one described above. Smaller values of f introduce randomness into the algorithm and help prevent over-fitting, acting as a kind of regularization. The algorithm also becomes faster, because regression trees have to be fit to smaller datasets at each iteration. Typically, f is set to 0.5, meaning that one half of the training set is used to build each base learner.

Also, like in bagging, sub-sampling allows one to define an out-of-bag estimate of the prediction performance improvement by evaluating predictions on those observations which were not used in the building of the next base learner. Out-of-bag estimates help avoid the need for an independent validation dataset, but often underestimate actual performance improvement and the optimal number of iterations.

Gradient tree boosting implementations often also use regularization by limiting the minimum number of observations in trees' terminal nodes. It's used in the tree building process by ignoring any splits that lead to nodes containing fewer than this number of training set instances. Imposing this limit helps to reduce variance in predictions at leaves.

formula is a symbolic description of the model to be fitted. data is the data frame of the explanatory and response variables. loss is the loss function for regression. ntrees is the number of iterations (trees). max-depth is the maximum depth of the tree. max-nodes is the maximum number of leaf nodes in the tree. node-size is the minimum size of leaf nodes. shrinkage is the shrinkage parameter in (0, 1] controls the learning rate of procedure. subsample is the sampling fraction for stochastic tree boosting.

gpr clj

(gpr x y kernel noise)

(gpr x y kernel noise normalize tol max-iter)

Gaussian process.

A Gaussian process is a stochastic process whose realizations consist of random values associated with every point in a range of times (or of space) such that each such random variable has a normal distribution. Moreover, every finite collection of those random variables has a multivariate normal distribution.

A Gaussian process can be used as a prior probability distribution over functions in Bayesian inference. Given any set of N points in the desired domain of your functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of N points with some desired kernel, and sample from that Gaussian. Inference of continuous values with a Gaussian process prior is known as Gaussian process regression.

The fitting is performed in the reproducing kernel Hilbert space with the 'kernel trick'. The loss function is squared-error. This also arises as the kriging estimate of a Gaussian random field in spatial statistics.

A significant problem with Gaussian process prediction is that it typically scales as O(n³). For large problems (e.g. n > 10,000) both storing the Gram matrix and solving the associated linear systems are prohibitive on modern workstations. An extensive range of proposals have been suggested to deal with this problem. A popular approach is the reduced-rank Approximations of the Gram Matrix, known as Nystrom approximation. Greedy approximation is another popular approach that uses an active set of training points of size m selected from the training set of size n > m. We assume that it is impossible to search for the optimal subset of size m due to combinatorics. The points in the active set could be selected randomly, but in general we might expect better performance if the points are selected greedily w.r.t. some criterion. Recently, researchers had proposed relaxing the constraint that the inducing variables must be a subset of training/test cases, turning the discrete selection problem into one of continuous optimization.

This method fits a regular Gaussian process model.

x is the training dataset. y is the response variable. kernel is the Mercer kernel. noise is the noise variance, which also works as a regularization parameter. normalize is the option to normalize the response variable. tol is the stopping tolerance for HPO. max-iter is the maximum number of iterations for HPO. No HPO if maxIter <= 0.

gpr-approx clj

(gpr-approx x y t kernel noise)

(gpr-approx x y t kernel noise normalize)

Approximate Gaussian process with a subset of regressors. x is the training dataset. y is the response variable. t is the inducing input, which are pre-selected or inducing samples acting as active set of regressors. In simple case, these can be chosen randomly from the training set or as the centers of k-means clustering. kernel is the Mercer kernel. noise is the noise variance, which also works as a regularization parameter. normalize is the option to normalize the response variable.

gpr-nystrom clj

(gpr-nystrom x y t kernel noise)

(gpr-nystrom x y t kernel noise normalize)

Approximate Gaussian process with Nystrom approximation of kernel matrix. x is the training dataset. y is the response variable. t is the inducing input, which are pre-selected or inducing samples acting as active set of regressors. In simple case, these can be chosen randomly from the training set or as the centers of k-means clustering. kernel is the Mercer kernel. noise is the noise variance, which also works as a regularization parameter. normalize is the option to normalize the response variable.

lasso clj

(lasso formula data lambda)

(lasso formula data lambda tol max-iter)

Least absolute shrinkage and selection operator.

The Lasso is a shrinkage and selection method for linear regression. It minimizes the usual sum of squared errors, with a bound on the sum of the absolute values of the coefficients (i.e. L₁-regularized). It has connections to soft-thresholding of wavelet coefficients, forward stage-wise regression, and boosting methods.

The Lasso typically yields a sparse solution, of which the parameter vector β has relatively few nonzero coefficients. In contrast, the solution of L₂-regularized least squares (i.e. ridge regression) typically has all coefficients nonzero. Because it effectively reduces the number of variables, the Lasso is useful in some contexts.

For over-determined systems (more instances than variables, commonly in machine learning), we normalize variables with mean 0 and standard deviation

1. For under-determined systems (less instances than variables, e.g. compressed sensing), we assume white noise (i.e. no intercept in the linear model) and do not perform normalization. Note that the solution is not unique in this case.

There is no analytic formula or expression for the optimal solution to the L₁-regularized least squares problems. Therefore, its solution must be computed numerically. The objective function in the L₁-regularized least squares is convex but not differentiable, so solving it is more of a computational challenge than solving the L₂-regularized least squares. The Lasso may be solved using quadratic programming or more general convex optimization methods, as well as by specific algorithms such as the least angle regression algorithm.

formula is a symbolic description of the model to be fitted. data is the data frame of the explanatory and response variables. lambda is the shrinkage/regularization parameter. tol is the tolerance for stopping iterations (relative target duality gap). max-iter is the maximum number of iterations.

lm clj

(lm formula data)

(lm formula data method stderr recursive)

Fitting linear models (ordinary least squares).

In linear regression, the model specification is that the dependent variable is a linear combination of the parameters (but need not be linear in the independent variables). The residual is the difference between the value of the dependent variable predicted by the model, and the true value of the dependent variable. Ordinary least squares obtains parameter estimates that minimize the sum of squared residuals, SSE (also denoted RSS).

The OLS estimator is consistent when the independent variables are exogenous and there is no multicollinearity, and optimal in the class of linear unbiased estimators when the errors are homoscedastic and serially uncorrelated. Under these conditions, the method of OLS provides minimum-variance mean-unbiased estimation when the errors have finite variances.

There are several different frameworks in which the linear regression model can be cast in order to make the OLS technique applicable. Each of these settings produces the same formulas and same results, the only difference is the interpretation and the assumptions which have to be imposed in order for the method to give meaningful results. The choice of the applicable framework depends mostly on the nature of data at hand, and on the inference task which has to be performed.

Least squares corresponds to the maximum likelihood criterion if the experimental errors have a normal distribution and can also be derived as a method of moments estimator.

Once a regression model has been constructed, it may be important to confirm the goodness of fit of the model and the statistical significance of the estimated parameters. Commonly used checks of goodness of fit include the R-squared, analysis of the pattern of residuals and hypothesis testing. Statistical significance can be checked by an F-test of the overall fit, followed by t-tests of individual parameters.

Interpretations of these diagnostic tests rest heavily on the model assumptions. Although examination of the residuals can be used to invalidate a model, the results of a t-test or F-test are sometimes more difficult to interpret if the model's assumptions are violated. For example, if the error term does not have a normal distribution, in small samples the estimated parameters will not follow normal distributions and complicate inference. With relatively large samples, however, a central limit theorem can be invoked such that hypothesis testing may proceed using asymptotic approximations.

formula is a symbolic description of the model to be fitted. data is the data frame of the explanatory and response variables. method is the fitting method ('qr' or 'svd'). recursive is the flag if the return model supports recursive least squares.

mlp clj

(mlp x y builders)

(mlp x y builders epochs eta alpha lambda)

Multilayer perceptron neural network.

An MLP consists of several layers of nodes, interconnected through weighted acyclic arcs from each preceding layer to the following, without lateral or feedback connections. Each node calculates a transformed weighted linear combination of its inputs (output activations from the preceding layer), with one of the weights acting as a trainable bias connected to a constant input. The transformation, called activation function, is a bounded non-decreasing (non-linear) function, such as the sigmoid functions (ranges from 0 to 1). Another popular activation function is hyperbolic tangent which is actually equivalent to the sigmoid function in shape but ranges from -1 to 1. More specialized activation functions include radial basis functions which are used in RBF networks.

The representational capabilities of a MLP are determined by the range of mappings it may implement through weight variation. Single layer perceptrons are capable of solving only linearly separable problems. With the sigmoid function as activation function, the single-layer network is identical to the logistic regression model.

The universal approximation theorem for neural networks states that every continuous function that maps intervals of real numbers to some output interval of real numbers can be approximated arbitrarily closely by a multi-layer perceptron with just one hidden layer. This result holds only for restricted classes of activation functions, which are extremely complex and NOT smooth for subtle mathematical reasons. On the other hand, smoothness is important for gradient descent learning. Besides, the proof is not constructive regarding the number of neurons required or the settings of the weights. Therefore, complex systems will have more layers of neurons with some having increased layers of input neurons and output neurons in practice.

The most popular algorithm to train MLPs is back-propagation, which is a gradient descent method. Based on chain rule, the algorithm propagates the error back through the network and adjusts the weights of each connection in order to reduce the value of the error function by some small amount. For this reason, back-propagation can only be applied on networks with differentiable activation functions.

During error back propagation, we usually times the gradient with a small number η, called learning rate, which is carefully selected to ensure that the network converges to a local minimum of the error function fast enough, without producing oscillations. One way to avoid oscillation at large η, is to make the change in weight dependent on the past weight change by adding a momentum term.

Although the back-propagation algorithm may performs gradient descent on the total error of all instances in a batch way, the learning rule is often applied to each instance separately in an online way or stochastic way. There exists empirical indication that the stochastic way results in faster convergence.

In practice, the problem of over-fitting has emerged. This arises in convoluted or over-specified systems when the capacity of the network significantly exceeds the needed free parameters. There are two general approaches for avoiding this problem: The first is to use cross-validation and similar techniques to check for the presence of over-fitting and optimally select hyper-parameters such as to minimize the generalization error. The second is to use some form of regularization, which emerges naturally in a Bayesian framework, where the regularization can be performed by selecting a larger prior probability over simpler models; but also in statistical learning theory, where the goal is to minimize over the 'empirical risk' and the 'structural risk'.

For neural networks, the input patterns usually should be scaled/standardized. Commonly, each input variable is scaled into interval [0, 1] or to have mean 0 and standard deviation 1.

For penalty functions and output units, the following natural pairings are recommended:

• linear output units and a least squares penalty function.
• a two-class cross-entropy penalty function and a logistic activation function.
• a multi-class cross-entropy penalty function and a softmax activation function.

By assigning a softmax activation function on the output layer of the neural network for categorical target variables, the outputs can be interpreted as posterior probabilities, which are very useful.

x is the training samples. y is the response variable. builders are the builders of layers from bottom to top. epochs is the the number of epochs of stochastic learning. eta is the the learning rate. alpha is the momentum factor. lambda is the weight decay for regularization.

random-forest clj

(random-forest formula data)

(random-forest formula data ntrees mtry max-depth max-nodes node-size subsample)

Random forest.

Random forest is an ensemble classifier that consists of many decision trees and outputs the majority vote of individual trees. The method combines bagging idea and the random selection of features.

Each tree is constructed using the following algorithm:

1. If the number of cases in the training set is N, randomly sample N cases with replacement from the original data. This sample will be the training set for growing the tree.
2. If there are M input variables, a number m << M is specified such that at each node, m variables are selected at random out of the M and the best split on these m is used to split the node. The value of m is held constant during the forest growing.
3. Each tree is grown to the largest extent possible. There is no pruning.

The advantages of random forest are:

• For many data sets, it produces a highly accurate classifier.
• It runs efficiently on large data sets.
• It can handle thousands of input variables without variable deletion.
• It gives estimates of what variables are important in the classification.
• It generates an internal unbiased estimate of the generalization error as the forest building progresses.
• It has an effective method for estimating missing data and maintains accuracy when a large proportion of the data are missing.

The disadvantages are

• Random forests are prone to over-fitting for some datasets. This is even more pronounced on noisy data.
• For data including categorical variables with different number of levels, random forests are biased in favor of those attributes with more levels. Therefore, the variable importance scores from random forest are not reliable for this type of data.

formula is a symbolic description of the model to be fitted. data is the data frame of the explanatory and response variables. ntrees is the number of trees. mtry is the number of random selected features to be used to determine the decision at a node of the tree. dim/3 seems to give generally good performance, where dim is the number of variables. max-depth is the maximum depth of the tree. max-nodes is the maximum number of leaf nodes in the tree. node-size is the minimum size of leaf nodes. subsample is the sampling rate for training tree. 1.0 means sampling with replacement. < 1.0 means sampling without replacement.

rbfnet clj

(rbfnet x y neurons)

(rbfnet x y neurons normalized)

Radial basis function networks.

A radial basis function network is an artificial neural network that uses radial basis functions as activation functions. It is a linear combination of radial basis functions. They are used in function approximation, time series prediction, and control.

In its basic form, radial basis function network is in the form

y(x) = &Sigma; w<sub>i</sub> &phi;(||x-c<sub>i</sub>||)

where the approximating function y(x) is represented as a sum of N radial basis functions φ, each associated with a different center c_i, and weighted by an appropriate coefficient w_i. For distance, one usually chooses Euclidean distance. The weights w_i can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights.

The centers c_i can be randomly selected from training data, or learned by some clustering method (e.g. k-means), or learned together with weight parameters undergo a supervised learning processing (e.g. error-correction learning).

The popular choices for φ comprise the Gaussian function and the so called thin plate splines. The advantage of the thin plate splines is that their conditioning is invariant under scalings. Gaussian, multi-quadric and inverse multi-quadric are infinitely smooth and and involve a scale or shape parameter, r_{<small>0</small>} > 0. Decreasing r_{<small>0</small>} tends to flatten the basis function. For a given function, the quality of approximation may strongly depend on this parameter. In particular, increasing r_{<small>0</small>} has the effect of better conditioning (the separation distance of the scaled points increases).

A variant on RBF networks is normalized radial basis function (NRBF) networks, in which we require the sum of the basis functions to be unity. NRBF arises more naturally from a Bayesian statistical perspective. However, there is no evidence that either the NRBF method is consistently superior to the RBF method, or vice versa.

SVMs with Gaussian kernel have similar structure as RBF networks with Gaussian radial basis functions. However, the SVM approach 'automatically' solves the network complexity problem since the size of the hidden layer is obtained as the result of the QP procedure. Hidden neurons and support vectors correspond to each other, so the center problems of the RBF network is also solved, as the support vectors serve as the basis function centers. It was reported that with similar number of support vectors/centers, SVM shows better generalization performance than RBF network when the training data size is relatively small. On the other hand, RBF network gives better generalization performance than SVM on large training data.

x is the training samples. y is the response variable. neurons are the radial basis functions. If normalized is true, train a normalized RBF network.

ridge clj

(ridge formula data lambda)

Ridge Regression.

When the predictor variables are highly correlated amongst themselves, the coefficients of the resulting least squares fit may be very imprecise. By allowing a small amount of bias in the estimates, more reasonable coefficients may often be obtained. Ridge regression is one method to address these issues. Often, small amounts of bias lead to dramatic reductions in the variance of the estimated model coefficients. Ridge regression is such a technique which shrinks the regression coefficients by imposing a penalty on their size. Ridge regression was originally developed to overcome the singularity of the X'X matrix. This matrix is perturbed so as to make its determinant appreciably different from 0.

Ridge regression is a kind of Tikhonov regularization, which is the most commonly used method of regularization of ill-posed problems. Another interpretation of ridge regression is available through Bayesian estimation. In this setting the belief that weight should be small is coded into a prior distribution.

formula is a symbolic description of the model to be fitted. data is the data frame of the explanatory and response variables. lambda is the shrinkage/regularization parameter.

svm clj

(svm x y kernel eps C)

(svm x y kernel eps C tol)

Support vector regression.

Like SVM for classification, the model produced by SVM depends only on a subset of the training data, because the cost function ignores any training data close to the model prediction (within a threshold).

x is the training data. y is the response variable. kernel is the kernel function. eps is the loss function error threshold. C is the soft margin penalty parameter. tol is the tolerance of convergence test.