In this work we present a large scale comparison study for the major machine learning models for time series forecasting. Specifically, we apply the models on the monthly M3 time series competition data (around a thousand time series). There have been very few, if any, large scale comparison studies for machine learning models for the regression or the time series forecasting problems, so we hope this study would fill this gap. The models considered are multilayer perceptron, Bayesian neural networks, radial basis functions, generalized regression neural networks (also called kernel regression), K-nearest neighbor regression, CART regression trees, support vector regression, and Gaussian processes. The study reveals significant differences between the different methods. The best two methods turned out to be the multilayer perceptron and the Gaussian process regression. In addition to model comparisons, we have tested different preprocessing methods and have shown that they have different impacts on the performance.

Class imbalance occurs when the class distribution is not equal. Namely, one class is under-represented (minority class), and the other class has significantly more samples in the data (majority class). The class imbalance problem is prevalent in many real world applications. Generally, the under-represented minority class is the class of interest. The synthetic minority over-sampling technique (SMOTE) method is considered the most prominent method for handling unbalanced data. The SMOTE method generates new synthetic data patterns by performing linear interpolation between minority class samples and their K nearest neighbors. However, the SMOTE generated patterns do not necessarily conform to the original minority class distribution. This paper develops a novel theoretical analysis of the SMOTE method by deriving the probability distribution of the SMOTE generated samples. To the best of our knowledge, this is the first work deriving a mathematical formulation for the SMOTE patterns’ probability distribution. This allows us to compare the density of the generated samples with the true underlying class-conditional density, in order to assess how representative the generated samples are. The derived formula is verified by computing it on a number of densities versus densities computed and estimated empirically.

Deep neural networks (DNNs) have achieved the state of the art performance in numerous fields. However, DNNs need high computation times, and people always expect better performance in a lower computation. Therefore, we study the human somatosensory system and design a neural network (SpinalNet) to achieve higher accuracy with fewer computations. Hidden layers in traditional NNs receive inputs in the previous layer, apply activation function, and then transfer the outcomes to the next layer. In the proposed SpinalNet, each layer is split into three splits: 1) input split, 2) intermediate split, and 3) output split. Input split of each layer receives a part of the inputs. The intermediate split of each layer receives outputs of the intermediate split of the previous layer and outputs of the input split of the current layer. The number of incoming weights becomes significantly lower than traditional DNNs. The SpinalNet can also be used as the fully connected or classification layer of DNN and supports both traditional learning and transfer learning. We observe significant error reductions with lower computational costs in most of the DNNs. Traditional learning on the VGG-5 network with SpinalNet classification layers provided the state-of-the-art (SOTA) performance on QMNIST, Kuzushiji-MNIST, and EMNIST (Letters, Digits, and Balanced) datasets. Traditional learning with ImageNet pre-trained initial weights and SpinalNet classification layers provided the SOTA performance on STL-10, Fruits 360, Bird225, and Caltech-101 datasets. The scripts of the proposed SpinalNet training are available at the following link: https://github.com/dipuk0506/SpinalNet Impact Statement—Research in deep neural networks (DNNs) has gained significant attention from industries and academia due to their eye-catching performance. DNNs have enabled machines to perform myriad tasks with high accuracy that once only humans could do. Several researchers have recently proposed different types of NNs and have achieved high accuracy. The recent success of biologically inspired convolutional neural networks and the miraculous spinal architecture of humans has motivated us to develop a neural network with gradual inputs. We have achieved superior performance in several datasets. After the first online appearance of the initial version of this paper, several researchers have applied the proposed neural network in several new datasets and reported promising results. We may observe numerous novel applications of SpinalNet in upcoming years.

In many pattern classification problems, an estimate of the posterior probabilities (rather than only a classification) is required. This is usually the case when some confidence measure in the classification is needed. In this article, we propose a new posterior probability estimator. The proposed estimator considers the K-nearest neighbors. It attaches a weight to each neighbor that contributes in an additive fashion to the posterior probability estimate. The weights corresponding to the K-nearest-neighbors (which add to 1) are estimated from the data using a maximum likelihood approach. Simulation studies confirm the effectiveness of the proposed estimator.

This paper derives the value of the integral of the product of the error function and the normal probability density as a series of the Hermite polynomial and the normalized incomplete Gamma function. This expression is beneficial, and can be used for evaluating the bivariate normal integral as a series expansion. This expansion is a good alternative to the well-known tetrachoric series, when the correlation coefficient, $\rho$ , is large in absolute value.