The Zeros of Orthogonal Polynomials and Markov–Bernstein Inequalities for Jacobi-Exponential Weights on (−1,1)

Let Ux=∏i=1rx−tipi, 0<p<∞, −1=tr<tr−1<⋯<t2<t1=1, r≥2, pi>−1/p, i=1,2,…,r, and W=e−Qx where Q:−1,1⟶0,∞. We give the estimates of the zeros of orthogonal polynomials for the Jacobi-Exponential weight WU on −1,1. In addition, Markov–Bernstein inequalities for the weight WU are also obtained.

The Exponential-Weight Mean-Variance Model: A novel computational model for the Balloon Analogue Risk Task

The Balloon Analogue Risk Task (BART) is a popular task used to measure risk-taking behavior. To identify cognitive processes associated with choice behavior on the BART, a few computational models have been proposed. However, the extant models are either too simplistic or fail to show good parameter recovery performance. Here, we propose a novel computational model, the exponential-weight mean-variance (EWMV) model, which addresses the limitations of existing models. By using multiple model comparison methods, including post hoc model fits criterion and parameter recovery, we showed that the EWMV model outperforms the existing models. In addition, we applied the EWMV model to BART data from healthy controls and substance-using populations (patients with past opiate and stimulant dependence). The results suggest that (1) the EWMV model addresses the limitations of existing models and (2) heroin-dependent individuals show reduced risk preference than other groups in the BART.

Short Term Residential Load Forecasting: An Improved Optimal Nonlinear Auto Regressive (NARX) Method with Exponential Weight Decay Function

The advancement in electrical load forecasting techniques with new algorithms offers reliable solutions to operators for operational cost reduction, optimum use of available resources, effective power management, and a reliable planning process. The focus is to develop a comprehensive understanding regarding the forecast accuracy generated by employing a state of the art optimal autoregressive neural network (NARX) for multiple, nonlinear, dynamic, and exogenous time varying input vectors. Other classical computational methods such as a bagged regression tree (BRT), an autoregressive and moving average with external inputs (ARMAX), and a conventional feedforward artificial neural network are implemented for comparative error assessment. The training of the applied method is realized in a closed loop by feeding back the predicted results obtained from the open loop model, which made the implemented model more robust when compared with conventional forecasting approaches. The recurrent nature of the applied model reduces its dependency on the external data and a produced mean absolute percentage error (MAPE) below 1%. Subsequently, more precision in handling daily grid operations with an average improvement of 16%–20% in comparison with existing computational techniques is achieved. The network is further improved by proposing a lightning search algorithm (LSA) for optimized NARX network parameters and an exponential weight decay (EWD) technique to control the input error weights.