Sales forecasts are crucial for the E-commerce business. State-of-the-art techniques typically apply only univariate methods to make prediction for each series independently. However, due to the short nature of sales times series in E-commerce, univariate methods don't apply well. In this article, we propose a global model which outperforms state-of-the-art models on real dataset. It is achieved by using Tree Boosting Methods that exploit non-linearity and cross-series information. We also proposed a preprocessing framework to overcome the inherent difficulties in the E-commerce data. In particular, we use different schemes to limit the impact of the volatility of the data.
Other regression methods have been used, such as generalized additive methods @cite_4 , support vector machines @cite_10 and Recurrent Network @cite_11 . All this method performs only univariate forecasting and therefore run into the same problems.
@cite_10 @cite_4 @cite_11
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Classical regression methods treat covariates as a vector and estimate a corresponding vector of regression coefficients. Modern applications in medical imaging generate covariates of more complex form such as multidimensional arrays (tensors). Traditional statistical and computational methods are proving insufficient for analysis of these high-throughput data due to their ultrahigh dimensionality as well as complex structure. In this article, we propose a new family of tensor regression models that efficiently exploit the special structure of tensor covariates. Under this framework, ultrahigh dimensionality is reduced to a manageable level, resulting in efficient estimation and prediction. A fast and highly scalable estimation algorithm is proposed for maximum likelihood estimation and its associated asymptotic properties are studied. Effectiveness of the new methods is demonstrated on both synthetic and real MRI imaging data. Supplementary materials for this article are available online. Many conjugate gradient-like methods for solving linear systems @math use recursion formulas for updating residual vectors instead of computing the residuals directly. For such methods it is shown that the difference between the actual residuals and the updated approximate residual vectors generated in finite precision arithmetic depends on the machine precision @math and on the maximum norm of an iterate divided by the norm of the true solution. It is often observed numerically, and can sometimes be proved, that the norms of the updated approximate residual vectors converge to zero or, at least, become orders of magnitude smaller than the machine precision. In such cases, the actual residual norm reaches the level @math times the maximum ratio of the norm of an iterate to that of the true solution. Using exact arithmetic theory to bound the size of the iterates, we give a priori estimates of the size of the final residual for a number of algorithms. Cascaded regression has recently become the method of choice for solving non-linear least squares problems such as deformable image alignment. Given a sizeable training set, cascaded regression learns a set of generic rules that are sequentially applied to minimise the least squares problem. Despite the success of cascaded regression for problems such as face alignment and head pose estimation, there are several shortcomings arising in the strategies proposed thus far. Specifically, (a) the regressors are learnt independently, (b) the descent directions may cancel one another out and (c) handcrafted features (e.g., HoGs, SIFT etc.) are mainly used to drive the cascade, which may be sub-optimal for the task at hand. In this paper, we propose a combined and jointly trained convolutional recurrent neural network architecture that allows the training of an end-to-end to system that attempts to alleviate the aforementioned drawbacks. The recurrent module facilitates the joint optimisation of the regressors by assuming the cascades form a nonlinear dynamical system, in effect fully utilising the information between all cascade levels by introducing a memory unit that shares information across all levels. The convolutional module allows the network to extract features that are specialised for the task at hand and are experimentally shown to outperform hand-crafted features. We show that the application of the proposed architecture for the problem of face alignment results in a strong improvement over the current state-of-the-art. In multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory, if not incorrect, if the prediction vectors are not orthogonal. Proposed is an estimation procedure based on adding small positive quantities to the diagonal of X′X. Introduced is the ridge trace, a method for showing in two dimensions the effects of nonorthogonality. It is then shown how to augment X′X to obtain biased estimates with smaller mean square error. Given a collection of r ≥ 2 linear regression problems in p dimensions, suppose that the regression coefficients share partially common supports of size at most s. This set-up suggests the use of l1 l∞-regularized regression for joint estimation of the p×r matrix of regression coefficients. We analyze the high-dimensional scaling of l1 l∞-regularized quadratic programming, considering both consistency rates in l∞-norm, and how the minimal sample size n required for consistent variable selection scales with model dimension, sparsity, and overlap between the supports. We first establish bounds on the l∞-error as well sufficient conditions for exact variable selection for fixed design matrices, as well as for designs drawn randomly from general Gaussian distributions. Specializing to the case r = 2 linear regression problems with standard Gaussian designs whose supports overlap in a fraction α ∈ [0,1] of their entries, we prove that l1 l∞-regularized method undergoes a phase transition characterized by the rescaled sample size θ1,∞(n, p, s, α) = n (4 - 3 α) s log(p-(2- α) s) . An implication is that the use of l1 l∞-regularization yields improved statistical efficiency if the overlap parameter is large enough ( α >; 2 3), but has worse statistical efficiency than a naive Lasso-based approach for moderate to small overlap (α <; 2 3 ). Empirical simulations illustrate the close agreement between theory and actual behavior in practice. These results show that caution must be exercised in applying l1 l∞ block regularization: if the data does not match its structure very closely, it can impair statistical performance relative to computationally less expensive schemes. In this paper, we treat the problem of continuous pose estimation for object categories as a regression problem on the basis of only 2D training information. While regression is a natural framework for continuous problems, regression methods so far achieved inferior results with respect to 3D-based and 2D-based classification-and-refinement approaches. This may be attributed to their weakness to high intra-class variability as well as to noisy matching procedures and lack of geometrical constraints. We propose to apply regression to Fisher-encoded vectors computed from large cells by learning an array of Fisher regressors. Fisher encoding makes our algorithm flexible to variations in class appearance, while the array structure permits to indirectly introduce spatial context information in the approach. We formulate our problem as a MAP inference problem, where the likelihood function is composed of a generative term based on the prediction error generated by the ensemble of Fisher regressors as well as a discriminative term based on SVM classifiers. We test our algorithm on three publicly available datasets that envisage several difficulties, such as high intra-class variability, truncations, occlusions, and motion blur, obtaining state-of-the-art results. We consider high dimensional sparse regression with arbitrary - possibly, severe or coordinated - errors in the covariates matrix. We are interested in understanding how many corruptions we can tolerate, while identifying the correct support. To the best of our knowledge, neither standard outlier rejection techniques, nor recently developed robust regression algorithms (that focus only on corrupted response variables), nor recent algorithms for dealing with stochastic noise or erasures, can provide guarantees on support recovery. As we show, neither can the natural brute force algorithm that takes exponential time to find the subset of data and support columns, that yields the smallest regression error. We explore the power of a simple idea: replace the essential linear algebraic calculation - the inner product - with a robust counterpart that cannot be greatly affected by a controlled number of arbitrarily corrupted points: the trimmed inner product. We consider three popular algorithms in the uncorrupted setting: Thresholding Regression, Lasso, and the Dantzig selector, and show that the counterparts obtained using the trimmed inner product are provably robust. We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces, a common practical limiting difficulty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n^2). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p^2 n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this paper, we show that in the context of kernel ridge regression, for approximations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the degrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such methods, and is often seen as the implicit number of parameters of non-parametric estimators. This result enables simple algorithms that have sub-quadratic running time complexity, but provably exhibit the same predictive performance than existing algorithms, for any given problem instance, and not only for worst-case situations. We focus on the distribution regression problem: regressing to vector-valued outputs from probability measures. Many important machine learning and statistical tasks fit into this framework, including multi-instance learning and point estimation problems without analytical solution (such as hyperparameter or entropy estimation). Despite the large number of available heuristics in the literature, the inherent two-stage sampled nature of the problem makes the theoretical analysis quite challenging, since in practice only samples from sampled distributions are observable, and the estimates have to rely on similarities computed between sets of points. To the best of our knowledge, the only existing technique with consistency guarantees for distribution regression requires kernel density estimation as an intermediate step (which often performs poorly in practice), and the domain of the distributions to be compact Euclidean. In this paper, we study a simple, analytically computable, ridge regression-based alternative to distribution regression, where we embed the distributions to a reproducing kernel Hilbert space, and learn the regressor from the embeddings to the outputs. Our main contribution is to prove that this scheme is consistent in the two-stage sampled setup under mild conditions (on separable topological domains enriched with kernels): we present an exact computational-statistical efficiency trade-off analysis showing that our estimator is able to match the one-stage sampled minimax optimal rate [Caponnetto and De Vito, 2007; , 2009]. This result answers a 17-year-old open question, establishing the consistency of the classical set kernel [Haussler, 1999; Gaertner et. al, 2002] in regression. We also cover consistency for more recent kernels on distributions, including those due to [Christmann and Steinwart, 2010]. Summary. We consider the problem of selecting grouped variables (factors) for accurate prediction in regression. Such a problem arises naturally in many practical situations with the multifactor analysis-of-variance problem as the most important and well-known example. Instead of selecting factors by stepwise backward elimination, we focus on the accuracy of estimation and consider extensions of the lasso, the LARS algorithm and the non-negative garrotte for factor selection. The lasso, the LARS algorithm and the non-negative garrotte are recently proposed regression methods that can be used to select individual variables. We study and propose efficient algorithms for the extensions of these methods for factor selection and show that these extensions give superior performance to the traditional stepwise backward elimination method in factor selection problems. We study the similarities and the differences between these methods. Simulations and real examples are used to illustrate the methods. We consider high dimensional sparse regression, and develop strategies able to deal with arbitrary -- possibly, severe or coordinated -- errors in the covariance matrix @math . These may come from corrupted data, persistent experimental errors, or malicious respondents in surveys recommender systems, etc. Such non-stochastic error-in-variables problems are notoriously difficult to treat, and as we demonstrate, the problem is particularly pronounced in high-dimensional settings where the primary goal is support recovery of the sparse regressor. We develop algorithms for support recovery in sparse regression, when some number @math out of @math total covariate response pairs are arbitrarily (possibly maliciously) corrupted . We are interested in understanding how many outliers, @math , we can tolerate, while identifying the correct support. To the best of our knowledge, neither standard outlier rejection techniques, nor recently developed robust regression algorithms (that focus only on corrupted response variables), nor recent algorithms for dealing with stochastic noise or erasures, can provide guarantees on support recovery. Perhaps surprisingly, we also show that the natural brute force algorithm that searches over all subsets of @math covariate response pairs, and all subsets of possible support coordinates in order to minimize regression error, is remarkably poor, unable to correctly identify the support with even @math corrupted points, where @math is the sparsity. This is true even in the basic setting we consider, where all authentic measurements and noise are independent and sub-Gaussian. In this setting, we provide a simple algorithm -- no more computationally taxing than OMP -- that gives stronger performance guarantees, recovering the support with up to @math corrupted points, where @math is the dimension of the signal to be recovered. In recursive linear models, the multivariate normal joint distribution of all variables exhibits a dependence structure induced by a recursive (or acyclic) system of linear structural equations. These linear models have a long tradition and appear in seemingly unrelated regressions, structural equation modelling, and approaches to causal inference. They are also related to Gaussian graphical models via a classical representation known as a path diagram. Despite the models' long history, a number of problems remain open. In this paper, we address the problem of computing maximum likelihood estimates in the subclass of 'bow-free' recursive linear models. The term 'bow-free' refers to the condition that the errors for variables i and j be uncorrelated if variable i occurs in the structural equation for variable j. We introduce a new algorithm, termed Residual Iterative Conditional Fitting (RICF), that can be implemented using only least squares computations. In contrast to existing algorithms, RICF has clear convergence properties and yields exact maximum likelihood estimates after the first iteration whenever the MLE is available in closed form. Recommender problems with large and dynamic item pools are ubiquitous in web applications like content optimization, online advertising and web search. Despite the availability of rich item meta-data, excess heterogeneity at the item level often requires inclusion of item-specific "factors" (or weights) in the model. However, since estimating item factors is computationally intensive, it poses a challenge for time-sensitive recommender problems where it is important to rapidly learn factors for new items (e.g., news articles, event updates, tweets) in an online fashion. In this paper, we propose a novel method called FOBFM (Fast Online Bilinear Factor Model) to learn item-specific factors quickly through online regression. The online regression for each item can be performed independently and hence the procedure is fast, scalable and easily parallelizable. However, the convergence of these independent regressions can be slow due to high dimensionality. The central idea of our approach is to use a large amount of historical data to initialize the online models based on offline features and learn linear projections that can effectively reduce the dimensionality. We estimate the rank of our linear projections by taking recourse to online model selection based on optimizing predictive likelihood. Through extensive experiments, we show that our method significantly and uniformly outperforms other competitive methods and obtains relative lifts that are in the range of 10-15 in terms of predictive log-likelihood, 200-300 for a rank correlation metric on a proprietary My Yahoo! dataset; it obtains 9 reduction in root mean squared error over the previously best method on a benchmark MovieLens dataset using a time-based train test data split. The additive isotonic least-squares regression model has been fit using a sequential pooled adjacent violators algorithm, estimating each isotonic component in turn, and looping until convergence. However, the individual components are not, in general, estimable. The sum of the components, i.e. the expected value of the response, has a unique estimate, which can be found using a single cone projection. Estimators for the individual components are then easily obtained, which are unique if the conditions for estimability hold. Parametrically modelled covariates are easily included in the cone projection specification. The cone structure also provides information about the degrees of freedom of the fit, which can be used in inference methods, variable selection, and estimation of the model variance. Simulations show that these methods can compare favourably to standard parametric methods, even when the parametric assumptions are correct. The estimation and inference methods can be extended to other constraints such as convex regression or isotonic regression on partial orderings. Many machine learning and pattern classification methods have been applied to the diagnosis of Alzheimer’s disease (AD) and its prodromal stage, i.e., mild cognitive impairment (MCI). Recently, rather than predicting categorical variables as in classification, several pattern regression methods have also been used to estimate continuous clinical variables from brain images. However, most existing regression methods focus on estimating multiple clinical variables separately and thus cannot utilize the intrinsic useful correlation information among different clinical variables. On the other hand, in those regression methods, only a single modality of data (usually only the structural MRI) is often used, without considering the complementary information that can be provided by different modalities. In this paper, we propose a general methodology, namely Multi-Modal Multi-Task (M3T) learning, to jointly predict multiple variables from multi-modal data. Here, the variables include not only the clinical variables used for regression but also the categorical variable used for classification, with different tasks corresponding to prediction of different variables. Specifically, our method contains two key components, i.e., (1) a multi-task feature selection which selects the common subset of relevant features for multiple variables from each modality, and (2) a multi-modal support vector machine which fuses the above-selected features from all modalities to predict multiple (regression and classification) variables. To validate our method, we perform two sets of experiments on ADNI baseline MRI, FDG-PET, and cerebrospinal fluid (CSF) data from 45 AD patients, 91 MCI patients, and 50 healthy controls (HC). In the first set of experiments, we estimate two clinical variables such as Mini Mental State Examination (MMSE) and Alzheimer’s Disease Assessment Scale - Cognitive Subscale (ADAS-Cog), as well as one categorical variable (with value of ‘AD’, ‘MCI’ or ‘HC’), from the baseline MRI, FDG-PET, and CSF data. In the second set of experiments, we predict the 2-year changes of MMSE and ADAS-Cog scores and also the conversion of MCI to AD from the baseline MRI, FDG-PET, and CSF data. The results on both sets of experiments demonstrate that our proposed M3T learning scheme can achieve better performance on both regression and classification tasks than the conventional learning methods. We study parameter estimation and asymptotic inference for sparse nonlinear regression. More specifically, we assume the data are given by y = f(x > ) + , where f is nonlinear. To recover , we propose an ‘1-regularized least-squares estimator. Unlike classical linear regression, the corresponding optimization problem is nonconvex because of the nonlinearity of f. In spite of the nonconvexity, we prove that under mild conditions, every stationary point of the objective enjoys an optimal statistical rate of convergence. In addition, we provide an ecient algorithm that provably converges to a stationary point. We also access the uncertainty of the obtained estimator. Specifically, based on any stationary point of the objective, we construct valid hypothesis tests and confidence intervals for the low dimensional components of the high-dimensional parameter . Detailed numerical results are provided to back up our theory. Semi-supervised Support Vector Machines for reject inference are proposed.The method uses information of both the accepted and rejected applicants.The method deals with labelled and unlabelled classes of the outcome.The model is tested on real consumer loans with a low acceptance rate.Predictive accuracy is improved by the new model compared to traditional methods. We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces, a common practi cal limiting difficulty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n 2 ). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p 2 n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this paper, we show that in the context of kernel ridge regression, for approximations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the degrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such method s, and is often seen as the implicit number of parameters of non-parametric estimators. This result enables simple algorithms that have sub-quadratic running time complexity, but provably exhibit the same predictive performancethan existing algorithms, for any given problem instance, and not only for worst-case situations. Consider data (x1,y1),…,(xn,yn), where each xi may be vector valued, and the distribution of yi given xi is a mixture of linear regressions. This provides a generalization of mixture models which do not include covariates in the mixture formulation. This mixture of linear regressions formulation has appeared in the computer science literature under the name “Hierarchical Mixtures of Experts” model. This model has been considered from both frequentist and Bayesian viewpoints. We focus on the Bayesian formulation. Previously, estimation of the mixture of linear regression model has been done through straightforward Gibbs sampling with latent variables. This paper contributes to this field in three major areas. First, we provide a theoretical underpinning to the Bayesian implementation by demonstrating consistency of the posterior distribution. This demonstration is done by extending results in Barron, Schervish and Wasserman (Annals of Statistics 27: 536–561, 1999) on bracketing entropy to the regression setting. Second, we demonstrate through examples that straightforward Gibbs sampling may fail to effectively explore the posterior distribution and provide alternative algorithms that are more accurate. Third, we demonstrate the usefulness of the mixture of linear regressions framework in Bayesian robust regression. The methods described in the paper are applied to two examples. We investigate the problem of predicting variables of ordinal scale. This task is referred to as ordinal regression and is complementary to the standard machine learning tasks of classification and metric regression. In contrast to statistical models we present a distribution independent formulation of the problem together with uniform bounds of the risk functional. The approach presented is based on a mapping from objects to scalar utility values. Similar to support vector methods we derive a new learning algorithm for the task of ordinal regression based on large margin rank boundaries. We give experimental results for an information retrieval task: learning the order of documents with respect to an initial query. Experimental results indicate that the presented algorithm outperforms more naive approaches to ordinal regression such as support vector classification and support vector regression in the case of more than two ranks.