It's good to see that all model coefficients are statistically significant at a p-value of 0.001 i.e. at 99.999% confidence level. As against linear regression models, models in which the dependent variable is a count, rarely produce normally distributed residual error distributions. So we have to normalize the raw-residuals using other means.

"I think I need to visit the doctor", said sleepy-eyed Janhvi. I asked, still half-asleep, enjoying the lazy morning vibes. "Feeling quite tired since the last few days, and my appetite's gone. I better get diagnosed", she answered, while pulling my blanket away and half-kicking me out of bed. That was my cue to get ready and drive her to the doctor -- there went my lovely morning!

Here, we load the chocolate data into our program using pandas; we also drop two of the columns we won't be using in our calculation: competitorname and winpercent. Our y becomes the first column in the dataset which indicates if our specific sweet is chocolate (1) or not (0). The remaining columns are used as variables/features to predict our y and, thus, become our X. If you're confused about why we're doing with …[:, 0][:,np.newaxis] on line 5, this is to turn y into a column. We simply add a new dimension to convert the horizontal vector into a vertical column!

In the paper, a problem of forecasting promotion efficiency is raised. The authors propose a new approach, using the gradient boosting method for this task. Six performance indicators are introduced to capture the promotion effect. For each of them, within predefined groups of products, a model was trained. A description of using these models for forecasting and optimising promotion efficiency is provided. Data preparation and hyperparameters tuning processes are also described. The experiments were performed for three groups of products from a large grocery company.

Testing large covariance matrices is of fundamental importance in statistical analysis with high-dimensional data. In the past decade, three types of test statistics have been studied in the literature: quadratic form statistics, maximum form statistics, and their weighted combination. It is known that quadratic form statistics would suffer from low power against sparse alternatives and maximum form statistics would suffer from low power against dense alternatives. The weighted combination methods were introduced to enhance the power of quadratic form statistics or maximum form statistics when the weights are appropriately chosen. In this paper, we provide a new perspective to exploit the full potential of quadratic form statistics and maximum form statistics for testing high-dimensional covariance matrices. We propose a scale-invariant power enhancement test based on Fisher's method to combine the p-values of quadratic form statistics and maximum form statistics. After carefully studying the asymptotic joint distribution of quadratic form statistics and maximum form statistics, we prove that the proposed combination method retains the correct asymptotic size and boosts the power against more general alternatives. Moreover, we demonstrate the finite-sample performance in simulation studies and a real application.

In this paper, we propose a novel accelerated stochastic gradient method with momentum, which momentum is the weighted average of previous gradients. The weights decays inverse proportionally with the iteration times. Stochastic gradient descent with momentum (Sgdm) use weights that decays exponentially with the iteration times to generate an momentum term. Using exponentially decaying weights, variants of Sgdm with well designed and complicated formats have been proposed to achieve better performance. The momentum update rules of our method is as simple as that of Sgdm. We provide theoretical convergence properties analyses for our method, which show both the exponentially decay weights and our inverse proportionally decay weights can limit the variance of the moving direction of parameters to be optimized to a region. Experimental results empirically show that our method works well with practical problems and outperforms Sgdm, and it outperforms Adam in convolutional neural networks.

Network regularization is an effective tool for incorporating structural prior knowledge to learn coherent models over networks, and has yielded provably accurate estimates in applications ranging from spatial economics to neuroimaging studies. Recently, there has been an increasing interest in extending network regularization to the spatio-temporal case to accommodate the evolution of networks. However, in both static and spatio-temporal cases, missing or corrupted edge weights can compromise the ability of network regularization to discover desired solutions. To address these gaps, we propose a novel approach---{\it discrepancy-aware network regularization} (DANR)---that is robust to inadequate regularizations and effectively captures model evolution and structural changes over spatio-temporal networks. We develop a distributed and scalable algorithm based on the alternating direction method of multipliers (ADMM) to solve the proposed problem with guaranteed convergence to global optimum solutions. Experimental results on both synthetic and real-world networks demonstrate that our approach achieves improved performance on various tasks, and enables interpretation of model changes in evolving networks.

Ridge or more formally $\ell_2$ regularization shows up in many areas of statistics and machine learning. It is one of those essential devices that any good data scientist needs to master for their craft. In this brief ridge fest I have collected together some of the magic and beauty of ridge that my colleagues and I have encountered over the past 40 years in applied statistics.

In recent years, considerable progress has been made on improving the interpretability of machine learning models. This is essential, as complex deep learning models with millions of parameters produce state of the art results, but it can be nearly impossible to explain their predictions. While various explainability techniques have achieved impressive results, nearly all of them assume each data instance to be independent and identically distributed (iid). This excludes relational models, such as Statistical Relational Learning (SRL), and the recently popular Graph Neural Networks (GNNs), resulting in few options to explain them. While there does exist one work on explaining GNNs, GNN-Explainer, they assume access to the gradients of the model to learn explanations, which is restrictive in terms of its applicability across non-differentiable relational models and practicality. In this work, we develop RelEx, a model-agnostic relational explainer to explain black-box relational models with only access to the outputs of the black-box. RelEx is able to explain any relational model, including SRL models and GNNs. We compare RelEx to the state-of-the-art relational explainer, GNN-Explainer, and relational extensions of iid explanation models and show that RelEx achieves comparable or better performance, while remaining model-agnostic.

It is a common phenomenon that for high-dimensional and nonparametric statistical models, rate-optimal estimators balance squared bias and variance. Although this balancing is widely observed, little is known whether methods exist that could avoid the trade-off between bias and variance. We propose a general strategy to obtain lower bounds on the variance of any estimator with bias smaller than a prespecified bound. This shows to which extent the bias-variance trade-off is unavoidable and allows to quantify the loss of performance for methods that do not obey it. The approach is based on a number of abstract lower bounds for the variance involving the change of expectation with respect to different probability measures as well as information measures such as the Kullback-Leibler or chi-square divergence. Some of these inequalities rely on a new concept of information matrices. In a second part of the article, the abstract lower bounds are applied to several statistical models including the Gaussian white noise model, a boundary estimation problem, the Gaussian sequence model and the high-dimensional linear regression model. For these specific statistical applications, different types of bias-variance trade-offs occur that vary considerably in their strength. For the trade-off between integrated squared bias and integrated variance in the Gaussian white noise model, we propose to combine the general strategy for lower bounds with a reduction technique. This allows us to reduce the original problem to a lower bound on the bias-variance trade-off for estimators with additional symmetry properties in a simpler statistical model. To highlight possible extensions of the proposed framework, we moreover briefly discuss the trade-off between bias and mean absolute deviation.