Bayesian multiperiod forecasts for ARX models

Bayestian muliperiod forecasts for AR models with random independent exogenous variables under normal-gamma and normal-inverted Wishart prior assumptions are investigated. By suitably arranging the integration order of the model's parameters, at-density mixture approximation is analytically derived to provide an estimator of the posterior predictive density for any future observation. In particular, a suitablet-density is proposed by a convenient closed form. The precision of the discussed methods is examined by using some simulated data and one set of real data up to lead-six-ahead forecasts. It is found that the numerical results of the discussed methods are rather close. In particular, when sample sizes are sufficiently large, it is encouraging to apply a convenientt-density in practical usage. In fact, thist-density estimator asymptotically converges to the true density.This research was supported by the National Science Council, Republic of China under contract #NSC82-0208-M-008-086.     

On the cutting stock problem under stochastic demand

This paper addresses the one-dimensional cutting stock problem when demand is a random variable. The problem is formulated as a two-stage stochastic nonlinear program with recourse. The first stage decision variables are the number of objects to be cut according to a cutting pattern. The second stage decision variables are the number of holding or backordering items due to the decisions made in the first stage. The problem’s objective is to minimize the total expected cost incurred in both stages, due to waste and holding or backordering penalties. A Simplex-based method with column generation is proposed for solving a linear relaxation of the resulting optimization problem. The proposed method is evaluated by using two well-known measures of uncertainty effects in stochastic programming: the value of stochastic solution—VSS—and the expected value of perfect information—EVPI. The optimal two-stage solution is shown to be more effective than the alternative wait-and-see and expected value approaches, even under small variations in the parameters of the problem.     

Improved prediction for a multivariate normal distribution with unknown mean and variance

The prediction problem for a multivariate normal distribution is considered where both mean and variance are unknown. When the Kullback–Leibler loss is used, the Bayesian predictive density based on the right invariant prior, which turns out to be a density of a multivariate t-distribution, is the best invariant and minimax predictive density. In this paper, we introduce an improper shrinkage prior and show that the Bayesian predictive density against the shrinkage prior improves upon the best invariant predictive density when the dimension is greater than or equal to three.     

Efficiency measurement using independent component analysis and data envelopment analysis

Efficiency measurement is an important issue for any firm or organization. Efficiency measurement allows organizations to compare their performance with their competitors’ and then develop corresponding plans to improve performance. Various efficiency measurement tools, such as conventional statistical methods and non-parametric methods, have been successfully developed in the literature. Among these tools, the data envelopment analysis (DEA) approach is one of the most widely discussed. However, problems of discrimination between efficient and inefficient decision-making units also exist in the DEA context (Adler and Yazhemsky, 2010). In this paper, a two-stage approach of integrating independent component analysis (ICA) and data envelopment analysis (DEA) is proposed to overcome this issue. We suggest using ICA first to extract the input variables for generating independent components, then selecting the ICs representing the independent sources of input variables, and finally, inputting the selected ICs as new variables in the DEA model. A simulated dataset and a hospital dataset provided by the Office of Statistics in Taiwan’s Department of Health are used to demonstrate the validity of the proposed two-stage approach. The results show that the proposed method can not only separate performance differences between the DMUs but also improve the discriminatory capability of the DEA’s efficiency measurement.     

Two-stage local M-estimation of additive models

This paper studies local M-estimation of the nonparametric components of additive models.A two-stage local M-estimation procedure is proposed for estimating the additive components and their derivatives.Under very mild conditions,the proposed estimators of each additive component and its derivative are jointly asymptotically normal and share the same asymptotic distributions as they would be if the other components were known.The established asymptotic results also hold for two particular local M-estimations:the local least squares and least absolute deviation estimations.However,for general two-stage local M-estimation with continuous and nonlinear ψ-functions,its implementation is time-consuming.To reduce the computational burden,one-step approximations to the two-stage local M-estimators are developed.The one-step estimators are shown to achieve the same effciency as the fully iterative two-stage local M-estimators,which makes the two-stage local M-estimation more feasible in practice.The proposed estimators inherit the advantages and at the same time overcome the disadvantages of the local least-squares based smoothers.In addition,the practical implementation of the proposed estimation is considered in details.Simulations demonstrate the merits of the two-stage local M-estimation,and a real example illustrates the performance of the methodology.     

Forecasting retained earnings of privately held companies with PCA and L1 regression

We use proprietary data collected by SVB Analytics, an affiliate of Silicon Valley Bank, to forecast the retained earnings of privately held companies. Combining methods of principal component analysis (PCA) and L¹/quantile regression, we build multivariate linear models that feature excellent in‐sample fit and strong out‐of‐sample predictive accuracy. The combined PCA and L¹ technique effectively deals with multicollinearity and non‐normality of the data, and also performs favorably when compared against a variety of other models. Additionally, we propose a variable ranking procedure that explains which variables from the current quarter are most predictive of the next quarter's retained earnings. We fit models to the top five variables identified by the ranking procedure and thereby, discover interpretable models with excellent out‐of‐sample performance. Copyright © 2013 John Wiley & Sons, Ltd.     

The use of forecast gradients in 3DVar data assimilation

In this paper, we propose an optimization approach for data assimilation by the use of forecast gradients. The proposed objective function consists of two data-fitting terms. The first term is based on the difference between the gradients of the forecast and the analysis, and the second term is based on the difference between the observations and the analysis in observation space. The motivation for using forecast gradients is that the forecast values provide an estimation of the system state, but they may not be accurate enough. We therefore propose to construct analysis gradients driven by the forecast gradients, instead of the forecast values. The associated data-fitting term can be interpreted by using the second-order finite difference matrix as the inverse of the background error covariance matrix in the 3DVar setting. In the proposed approach, it is not necessary to estimate the background covariance matrix and to deal with its inverse in the 3DVar algorithm. The existence and uniqueness of the analysis solution of the proposed objective function are established in this paper. The solution can be calculated by using the conjugate gradient method iteratively. Experimental results based on Community Multiscale Air Quality (CMAQ) and Weather Research and Forecasting (WRF) simulations are presented. We show in our air quality data assimilation experiment that the performance of the proposed method is better than that of the 3DVar method and the En3DVar method. The average improvements over the CMAQ simulation results for single-species NO₂, O₃, SO₂, NO, and CO are 18.9%, 34.0%, 22.2%, 4.3%, and 91.9%, respectively; and for the multiple-species PM2.5 and PM10, the improvements are 61.2% and 70.1%, respectively. In addition, the performance of the proposed method in temperature data assimilation is improved by 45.1% compared with the 3DVar method.     

Optimal two-stage ordering policy with Bayesian information updating

We investigate in this paper an optimal two-stage ordering policy for seasonal products. Before the selling season, a retailer can place orders for a seasonal product from her supplier at two distinct stages satisfying the lead-time requirement. Market information is collected at the first stage and is used to update the demand forecast at the second stage by using Bayesian approach. The ordering cost at the first stage is known but the ordering cost at the second stage is uncertain. A two-stage dynamic optimization problem is formulated and an optimal policy is derived using dynamic programming. The optimal ordering policy exhibits nice structural properties and can easily be implemented by a computer program. The detailed implementation scheme is proposed. The service level and profit uncertainty level under the optimal policy are discussed. Extensive numerical analyses are carried out to study the performance of the optimal policy.     

Confident estimation for density of a biological population based on line transect sampling

Line transect sampling is a very useful method in survey of wildlife population. Confident interval estimation for density D of a biological population is proposed based on a sequential design. The survey area is occupied by the population whose size is unknown. A stopping rule is proposed by a kernel-based estimator of density function of the perpendicular data at a distance. With this stopping rule, we construct several confidence intervals for D by difference procedures. Some bias reduction techniques are used to modify the confidence intervals. These intervals provide the desired coverage probability as the bandwidth in the stopping rule approaches zero. A simulation study is also given to illustrate the performance of this proposed sequential kernel procedure.     

Computational inversion of electron micrographs using L2‐gradient flows—convergence analysis

A gradient flow‐based explicit finite element method (L2GF) for reconstructing the 3D density function from a set of 2D electron micrographs has been proposed in recently published papers. The experimental results showed that the proposed method was superior to the other classical algorithms, especially for the highly noisy data. However, convergence analysis of the L2GF method has not been conducted. In this paper, we present a complete analysis on the convergence of L2GF method for the case of using a more general form regularization term, which includes the Tikhonov‐type regularizer and modified or smoothed total variation regularizer as two special cases. We further prove that the L²‐gradient flow method is stable and robust. These results demonstrate that the iterative variational reconstruction method derived from the L²‐gradient flow approach is mathematically sound and effective and has desirable properties. Copyright © 2013 John Wiley & Sons, Ltd.     

考虑消费动机和动态竞争的电影日需求预测模型

提出并验证考虑消费动机和动态竞争的电影日需求预测模型。考虑非粉丝及粉丝型的消费动机,构建电影消费两阶段过程模型;融合该模型和Bass模型,考虑竞争导致市场潜量的动态性,考虑映前被关注度、口碑、节假日对票房的影响,提出电影日需求预测模型。利用2016~2017年上映的电影数据验证该模型,并与Bass模型对比分析。结果显示,该模型预测效果优于Bass模型。因考虑竞争导致的动态市场潜量,考虑粉丝型消费者由续集效应及改编效应导致的动态市场潜量提升,该模型能显著提高预测准确度。利用映前被关注度和电影口碑数据,该模型能实现映前及上映早期的预测。该模型可推广至存在消费动机不同、市场动态竞争的其它短生命周期体验品的需求预测,是对Bass模型的改进。     

Spectral modeling of time series with missing data

Singular spectrum analysis is a natural generalization of principal component methods for time series data. In this paper we propose an imputation method to be used with singular spectrum-based techniques which is based on a weighted combination of the forecasts and hindcasts yield by the recurrent forecast method. Despite its ease of implementation, the obtained results suggest an overall good fit of our method, being able to yield a similar adjustment ability in comparison with the alternative method, according to some measures of predictive performance.     

Some properties of a simple moving average when applied to forecasting a time series

Simple (equally weighted) moving averages are frequently used to estimate the current level of a time series, with this value being projected as a forecast for future observations. A key measure of the effectiveness of the method is the sampling error of the estimator, which this paper defines in terms of characteristics of the data. This enables the optimal length of the average for any steady state model to be established and the lead time forecast error derived. A comparison of the performance of a simple moving average (SMA) with an exponentially weighted moving average (EWMA) is made. It is shown that, for a steady state model, the variance of the forecast error is typically less than 3% higher than the appropriate EWMA. This relatively small difference may explain the inconclusive results from the empirical studies about the relative predictive performance of the two methods.     

ARMA模型在哈尔滨气温预测中的应用

将时间序列分析引入到气温时间序列预测的研究中,深入分析气温样本数据,并对其建立ARMA模型.采用最佳准则函数法确定模型的阶数,并利用自相关函数对模型的残差进行了检验.通过条件期望预测和适时修正预测方法求得预测值,与真实值的比较得到适时修正预测精确度比条件期望预测的精确度高.     

Convergence rates for two-stage confidence intervals based on U-statistics

We consider a modified two-stage procedure for constructing a fixed-width confidence interval for the mean of a U-statistic. First, we discuss a few asymptotic results with the associated rates of convergence. The main result gives the rate of convergence for the coverage probability of our proposed confidence interval which is seen to be slower than that for the purely sequential procedure.     

Two-stage complex Rosenbrock schemes for stiff systems

New two-stage Rosenbrock schemes with complex coefficients are proposed for stiff systems of differential equations. The schemes are fourth-order accurate and satisfy enhanced stability requirements. A one-parameter family of L1-stable schemes with coefficients explicitly calculated by formulas involving only fractions and radicals is constructed. A single L2-stable scheme is found in this family. The coefficients of the fourth-order accurate L4-stable scheme previously obtained by P.D Shirkov are refined. Several fourth-order schemes are constructed that are high-order accurate for linear problems and possess the limiting order of L-decay. The schemes proposed are proved to converge. A symbolic computation algorithm is developed that constructs order conditions for multistage Rosenbrock schemes with complex coefficients. This algorithm is used to design the schemes proposed and to obtain fifth-order accurate conditions.     

Choosing the optimal model parameters for Granger causality in application to time series with main timescale

The problem of determining the presence and direction of coupling between experimentally observed time series is of immediate interest in many relevant areas of knowledge. One of the approaches to its solution is the method of nonlinear Granger causality. The algorithm is based on the construction of predictive models and its effectiveness depends on the proper selection of model parameters.The most important of them for signals with a characteristic time scale fluctuations are the time lag used in the reconstruction of the state vector, and the range forecast. In this paper, we propose two criteria for evaluating performance of the method of nonlinear Granger causality, which allows one to select the lag and range forecast and achieves the best sensitivity and specificity. The sensitivity is determined by range of weakness the method can detect and specificity means the ability to avoid false positive results. Because of the proposed criteria on the example of several unidirectionally coupled reference systems were received practical advice on the selection of the following model parameters: lag and range forecast.     

Nonparametric variable selection and its application to additive models

Variable selection for multivariate nonparametric regression models usually involves parameterized approximation for nonparametric functions in the objective function. However, this parameterized approximation often increases the number of parameters significantly, leading to the “curse of dimensionality” and inaccurate estimation. In this paper, we propose a novel and easily implemented approach to do variable selection in nonparametric models without parameterized approximation, enabling selection consistency to be achieved. The proposed method is applied to do variable selection for additive models. A two-stage procedure with selection and adaptive estimation is proposed, and the properties of this method are investigated. This two-stage algorithm is adaptive to the smoothness of the underlying components, and the estimation consistency can reach a parametric rate if the underlying model is really parametric. Simulation studies are conducted to examine the performance of the proposed method. Furthermore, a real data example is analyzed for illustration.

     

An SQP algorithm for extended linear-quadratic problems in stochastic programming

Extended Linear-Quadratic Programming (ELQP) problems were introduced by Rockafellar and Wets for various models in stochastic programming and multistage optimization. Several numerical methods with linear convergence rates have been developed for solving fully quadratic ELQP problems, where the primal and dual coefficient matrices are positive definite. We present a two-stage sequential quadratic programming (SQP) method for solving ELQP problems arising in stochastic programming. The first stage algorithm realizes global convergence and the second stage algorithm realizes superlinear local convergence under a condition calledB-regularity.B-regularity is milder than the fully quadratic condition; the primal coefficient matrix need not be positive definite. Numerical tests are given to demonstrate the efficiency of the algorithm. Solution properties of the ELQP problem underB-regularity are also discussed.Supported by the Australian Research Council.     

Functional k-sample problem when data are density functions

This paper deals with the k-sample problem for functional data when the observations are density functions. We introduce test procedures based on distances between pairs of density functions (L ₁ distance and Hellinger distance, among others). A simulation study is carried out to compare the practical behaviour of the proposed tests. Theoretical derivations have been done in order to allow weighted samples in the test procedures. The paper ends with a real data example: for a collection of European regions we estimate the regional relative income densities and then we test the significance of the country effect.