On Optimality of Bayesian Wavelet Estimators

Abstract.  We investigate the asymptotic optimality of several Bayesian wavelet estimators, namely, posterior mean, posterior median and Bayes Factor, where the prior imposed on wavelet coefficients is a mixture of a mass function at zero and a Gaussian density. We show that in terms of the mean squared error, for the properly chosen hyperparameters of the prior, all the three resulting Bayesian wavelet estimators achieve optimal minimax rates within any prescribed Besov space     for p  ≥ 2. For 1 ≤  p  < 2, the Bayes Factor is still optimal for (2 s +2)/(2 s +1) ≤  p  < 2 and always outperforms the posterior mean and the posterior median that can achieve only the best possible rates for linear estimators in this case.     

局部Haar条件下的变形非线性L逼近

本文讨论了局部Ｈａａｒ条件下变形的非线性Ｌ逼近，得到了包括交错定理在内的特征定理、唯一性和强唯一性定理。     

Real nonparametric regression using complex wavelets

Summary.  Wavelet shrinkage is an effective nonparametric regression technique, especially when the underlying curve has irregular features such as spikes or discontinuities. The basic idea is simple: take the discrete wavelet transform of data consisting of a signal corrupted by noise; shrink or remove the wavelet coefficients to remove the noise; then invert the discrete wavelet transform to form an estimate of the true underlying curve. Various researchers have proposed increasingly sophisticated methods of doing this by using real-valued wavelets. Complex-valued wavelets exist but are rarely used. We propose two new complex-valued wavelet shrinkage techniques: one based on multiwavelet style shrinkage and the other using Bayesian methods. Extensive simulations show that our methods almost always give significantly more accurate estimates than methods based on real-valued wavelets. Further, our multiwavelet style shrinkage method is both simpler and dramatically faster than its competitors. To understand the excellent performance of this method we present a new risk bound on its hard thresholded coefficients.     

Two preprocessing algorithms for climate time series

We propose two preprocessing algorithms suitable for climate time series. The first algorithm detects outliers based on an autoregressive cost update mechanism. The second one is based on the wavelet transform, a method from pattern recognition. In order to benchmark the algorithms'' performance we compare them to existing methods based on a synthetic data set. Eventually, for exemplary purposes, the proposed methods are applied to a data set of high-frequent temperature measurements from Novi Sad, Serbia. The results show that both methods together form a powerful tool for signal preprocessing: In case of solitary outliers the autoregressive cost update mechanism prevails, whereas the wavelet-based mechanism is the method of choice in the presence of multiple consecutive outliers.     

Uniform Convergence of Compactly Supported Wavelet Expansions of Gaussian Random Processes

New results on uniform convergence in probability for expansions of Gaussian random processes using compactly supported wavelets are given. The main result is valid for general classes of non stationary processes. An application of the obtained results to stationary processes is also presented. It is shown that the convergence rate of the expansions is exponential.     

基于LSTM神经网络的金融时间序列预测

本文提出将小波分析与纳入时间序列依赖特征的长短期记忆（LSTM）神经网络相结合，构建金融时间序列数据预测模型，以克服现有模型对金融时间序列数据非平稳、非线性、序列相关等复杂特征以及数据间非线性交互关系无法反映的缺陷。同时，以道琼斯工业指数日收盘价为例，探究LSTM神经网络对实际金融时间序列数据的预测能力，比较其与多层感知机、支持向量机、K近邻、GARCH四种模型的预测效果。实证结果表明LSTM神经网络具有更高的预测精度，能够有效预测金融时间序列数据的长短期动态变化趋势，说明了其对金融时间序列数据预测的适用性与有效性。此外，对金融时间序列数据进行小波分解与重构，可有效提高LSTM预测模型的泛化能力，以及对长短期动态趋势的预测精度。     

基于Haar小波变换的无失真图像压缩

提出了一种利用Ｈａａｒ小波进行图像无失真压缩的算法。对线性预测后的图像进行Ｈａａｒ小波分解，将各子带小波系数根据大小分解成两部分，其位置信息分别通过自适应算术编码进行了有效的压缩。试验结果表明，该算法实现简单，达到了很好的压缩效果。     

局部Haar条件下的Chebyshev同时逼近(英文)

本文在局部 Haar 条件下,讨论了紧空间上非线性 Dunham 型 Chebyshev 同时逼近,得到了同时逼近为局部最佳的充分必要条件;给出了与非线性同时逼近问题等价的相关线性同时逼近问题;证明了极小集的存在性,在这个极小集上的局部最佳逼近为局部最佳的;获得了区间上非线性同时逼近的交错定理;证明了非线性同时逼近的α阶(α≥1)的强唯一常数等于其相关线性同时逼近的α阶强唯一常数。     

Analyzing time–frequency relationship between oil price and exchange rate in Pakistan through wavelets

This study analyzed the time–frequency relationship between oil price and exchange rate for Pakistan by using measures of continuous wavelet such as wavelet power, cross-wavelet power, and cross-wavelet coherency (WTC). The results of cross-wavelet analysis indicated that covariance between oil price and exchange rate is unable to give clear-cut results, but both variables have been in phase and out phase (i.e. they are anti-cyclical and cyclical in nature) in some or other durations. However, results of squared wavelet coherence disclose that both variables are out of phase and real exchange rate was leading during the entire period studied, corresponding to the 10–15 months’ scale. These results are the unique contribution of the present study, which would have not been drawn if one would have utilized any other time series or frequency domain-based approach. This finding provides evidence of anti-cyclical relationship between oil price and real effective exchange rate; however, in most of the period studied, real exchange rate was leading and passing anti-cycle effects on oil price shocks which is the major contribution of the study.