In this paper, a new estimation method is introduced for the quantile spectrum, which uses a parametric form of the autoregressive (AR) spectrum coupled with nonparametric smoothing. The method begins with quantile periodograms which are constructed by trigonometric quantile regression at different quantile levels, to represent the serial dependence of time series at various quantiles. At each quantile level, we approximate the quantile spectrum by a function in the form of an ordinary AR spectrum. In this model, we first compute what we call the quantile autocovariance function (QACF) by the inverse Fourier transformation of the quantile periodogram at each quantile level. Then, we solve the Yule-Walker equations formed by the QACF to obtain the quantile partial autocorrelation function (QPACF) and the scale parameter. Finally, we smooth QPACF and the scale parameter across the quantile levels using a nonparametric smoother, convert the smoothed QPACF to AR coefficients, and obtain the AR spectral density function. Numerical results show that the proposed method outperforms other conventional smoothing techniques. We take advantage of the two-dimensional property of the estimators and train a convolutional neural network (CNN) to classify smoothed quantile periodogram of earthquake data and achieve a higher accuracy than a similar classifier using ordinary periodograms.

中文翻译：

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一种基于卷积神经网络的地震分类的分位数谱半参数估计方法

在本文中，为分位数谱引入了一种新的估计方法，该方法使用自回归 (AR) 谱的参数形式与非参数平滑相结合。该方法从分位数周期图开始，该分位数周期图由不同分位数水平的三角分位数回归构建，以表示时间序列在不同分位数上的序列相关性。在每个分位数级别，我们通过一个普通 AR 谱形式的函数来近似分位数谱。在这个模型中，我们首先通过每个分位数级别的分位数周期图的逆傅立叶变换来计算我们所说的分位数自协方差函数 (QACF)。然后，我们求解由 QACF 形成的 Yule-Walker 方程以获得分位数偏自相关函数 (QPACF) 和尺度参数。最后，我们使用非参数平滑器在分位数级别上平滑 QPACF 和尺度参数，将平滑后的 QPACF 转换为 AR 系数，并获得 AR 谱密度函数。数值结果表明，所提出的方法优于其他传统的平滑技术。我们利用估计器的二维特性并训练卷积神经网络 (CNN) 对地震数据的平滑分位数周期图进行分类，并获得比使用普通周期图的类似分类器更高的精度。