We develop in this paper a framework of empirical gain maximization (EGM) to address the robust regression problem where heavy-tailed noise or outliers may present in the response variable. The idea of EGM is to approximate the density function of the noise distribution instead of approximating the truth function directly as usual. Unlike the classical maximum likelihood estimation that encourages equal importance of all observations and could be problematic in the presence of abnormal observations, EGM schemes can be interpreted from a minimum distance estimation viewpoint and allow the ignorance of those observations. Furthermore, it is shown that several well-known robust nonconvex regression paradigms, such as Tukey regression and truncated least square regression, can be reformulated into this new framework. We then develop a learning theory for EGM, by means of which a unified analysis can be conducted for these well-established but not fully-understood regression approaches. Resulting from the new framework, a novel interpretation of existing bounded nonconvex loss functions can be concluded. Within this new framework, the two seemingly irrelevant terminologies, the well-known Tukey's biweight loss for robust regression and the triweight kernel for nonparametric smoothing, are closely related. More precisely, it is shown that the Tukey's biweight loss can be derived from the triweight kernel. Similarly, other frequently employed bounded nonconvex loss functions in machine learning such as the truncated square loss, the Geman-McClure loss, and the exponential squared loss can also be reformulated from certain smoothing kernels in statistics. In addition, the new framework enables us to devise new bounded nonconvex loss functions for robust learning.

Hybrid system identification is a key tool to achieve reliable models of Cyber-Physical Systems from data. PieceWise Affine models guarantees universal approximation, local linearity and equivalence to other classes of hybrid system. Still, PWA identification is a challenging problem, requiring the concurrent solution of regression and classification tasks. In this work, we focus on the identification of PieceWise Auto Regressive with eXogenous input models with arbitrary regions (NPWARX), thus not restricted to polyhedral domains, and characterized by discontinuous maps. To this end, we propose a method based on a probabilistic mixture model, where the discrete state is represented through a multinomial distribution conditioned by the input regressors. The architecture is conceived following the Mixture of Expert concept, developed within the machine learning field. To achieve nonlinear partitioning, we parametrize the discriminant function using a neural network. Then, the parameters of both the ARX submodels and the classifier are concurrently estimated by maximizing the likelihood of the overall model using Expectation Maximization. The proposed method is demonstrated on a nonlinear piece-wise problem with discontinuous maps.

In this article, we present a novel approach to multivariate probabilistic forecasting. Our approach is based on an extension of single-output quantile regression (QR) to multivariate-targets, called quantile surfaces (QS). QS uses a simple yet compelling idea of indexing observations of a probabilistic forecast through direction and vector length to estimate a central tendency. We extend the single-output QR technique to multivariate probabilistic targets. QS efficiently models dependencies in multivariate target variables and represents probability distributions through discrete quantile levels. Therefore, we present a novel two-stage process. In the first stage, we perform a deterministic point forecast (i.e., central tendency estimation). Subsequently, we model the prediction uncertainty using QS involving neural networks called quantile surface regression neural networks (QSNN). Additionally, we introduce new methods for efficient and straightforward evaluation of the reliability and sharpness of the issued probabilistic QS predictions. We complement this by the directional extension of the Continuous Ranked Probability Score (CRPS) score. Finally, we evaluate our novel approach on synthetic data and two currently researched real-world challenges in two different domains: First, probabilistic forecasting for renewable energy power generation, second, short-term cyclists trajectory forecasting for autonomously driving vehicles. Especially for the latter, our empirical results show that even a simple one-layer QSNN outperforms traditional parametric multivariate forecasting techniques, thus improving the state-of-the-art performance.

In many fields of study, we only observe lower bounds on the true response value of some experiments. When fitting a regression model to predict the distribution of the outcomes, we cannot simply drop these right-censored observations, but need to properly model them. In this work, we focus on the concept of censored data in the light of model-based optimization where prematurely terminating evaluations (and thus generating right-censored data) is a key factor for efficiency, e.g., when searching for an algorithm configuration that minimizes runtime of the algorithm at hand. Neural networks (NNs) have been demonstrated to work well at the core of model-based optimization procedures and here we extend them to handle these censored observations. We propose (i)~a loss function based on the Tobit model to incorporate censored samples into training and (ii) use an ensemble of networks to model the posterior distribution. To nevertheless be efficient in terms of optimization-overhead, we propose to use Thompson sampling s.t. we only need to train a single NN in each iteration. Our experiments show that our trained regression models achieve a better predictive quality than several baselines and that our approach achieves new state-of-the-art performance for model-based optimization on two optimization problems: minimizing the solution time of a SAT solver and the time-to-accuracy of neural networks.

Linear Regression and Logistic Regression for beginners NEW Created by Start-Tech Academy English [Auto] Students also bought Seven to Heaven - HTML5, CSS3 and jQuery Course The complete gRPC course [Protobuf Golang Java] Spanish: The Most Useful Phrases 300 The Complete Java Masterclass: Learn Java From Scratch C Programming for Beginners - Master the C Fundamentals Preview this course GET COUPON CODE Description You're looking for a complete Linear Regression and Logistic Regression course that teaches you everything you need to create a Linear or Logistic Regression model in Python, right? You've found the right Linear Regression course! After completing this course you will be able to: Identify the business problem which can be solved using linear and logistic regression technique of Machine Learning. Create a linear regression and logistic regression model in Python and analyze its result. Confidently model and solve regression and classification problems A Verifiable Certificate of Completion is presented to all students who undertake this Machine learning basics course.

A major requirement for credit scoring models is to provide a maximally accurate risk prediction. Additionally, regulators demand these models to be transparent and auditable. Thus, in credit scoring, very simple predictive models such as logistic regression or decision trees are still widely used and the superior predictive power of modern machine learning algorithms cannot be fully leveraged. Significant potential is therefore missed, leading to higher reserves or more credit defaults. This paper works out different dimensions that have to be considered for making credit scoring models understandable and presents a framework for making ``black box'' machine learning models transparent, auditable and explainable. Following this framework, we present an overview of techniques, demonstrate how they can be applied in credit scoring and how results compare to the interpretability of score cards. A real world case study shows that a comparable degree of interpretability can be achieved while machine learning techniques keep their ability to improve predictive power.

Classification is a major tool of statistics and machine learning. A classification method first processes a training set of objects with given classes (labels), with the goal of afterward assigning new objects to one of these classes. When running the resulting prediction method on the training data or on test data, it can happen that an object is predicted to lie in a class that differs from its given label. This is sometimes called label bias, and raises the question whether the object was mislabeled. Our goal is to visualize aspects of the data classification to obtain insight. The proposed display reflects to what extent each object's label is (dis)similar to its prediction, how far each object lies from the other objects in its class, and whether some objects lie far from all classes. The display is constructed for discriminant analysis, the k-nearest neighbor classifier, support vector machines, logistic regression, and majority voting. It is illustrated on several benchmark datasets containing images and texts.

Simple Linear regression is a simple yet powerful supervised learning technique. The aim of linear regression is to identify how the input variable(explanatory variable) influences the output variable(response variable). Simple Linear regression performs the task to predict a dependent variable value (y) based on a given independent variable (x). So, this regression technique finds out a linear relationship between x (input) and y(output). Hence, the name is Linear Regression. In the figure above, X (input) is the work experience and Y (output) is the salary of a person.

This introduction aims to provide readers an intuitive understanding of Gaussian processes regression. Gaussian processes regression (GPR) models have been widely used in machine learning applications because their representation flexibility and inherently uncertainty measures over predictions. The paper starts with explaining mathematical basics that Gaussian processes built on including multivariate normal distribution, kernels, non-parametric models, joint and conditional probability. The Gaussian processes regression is then described in an accessible way by balancing showing unnecessary mathematical derivation steps and missing key conclusive results. An illustrative implementation of a standard Gaussian processes regression algorithm is provided. Beyond the standard Gaussian processes regression, existing software packages to implement state-of-the-art Gaussian processes algorithms are reviewed. Lastly, more advanced Gaussian processes regression models are specified. The paper is written in an accessible way, thus undergraduate science and engineering background will find no difficulties in following the content.

Travel time is a crucial measure in transportation. Accurate travel time prediction is also fundamental for operation and advanced information systems. A variety of solutions exist for short-term travel time predictions such as solutions that utilize real-time GPS data and optimization methods to track the path of a vehicle. However, reliable long-term predictions remain challenging. We show in this paper the applicability and usefulness of travel time i.e. delivery time prediction for postal services. We investigate several methods such as linear regression models and tree based ensembles such as random forest, bagging, and boosting, that allow to predict delivery time by conducting extensive experiments and considering many usability scenarios. Results reveal that travel time prediction can help mitigate high delays in postal services. We show that some boosting algorithms, such as light gradient boosting and catboost, have a higher performance in terms of accuracy and runtime efficiency than other baselines such as linear regression models, bagging regressor and random forest.