On a General Class of Count Time Series Models

这项研究研究了时间上记录的具有计数价值的数据，例如加州野火的年度数量、阿拉斯加北极熊的年度目击次数、每月的流感死亡人数、每年的北大西洋严重飓风等。这些数据可能在时间上相关，这意味着今天观察到的计数可能会受到刚刚发生的计数的影响(与过去发生的计数相关)。这项工作开发了时间序列方法，该方法考虑了适合于计数的特定类型的分布，例如泊松、几何、二项式等。使用正确的分布有助于准确的预测和推断。这里开发的模型允许正相关性和负相关性，这是当前统计计数时间序列模型所缺少的一个特征。例如，太平洋和大西洋飓风的年度计数很好地用泊松分布来描述，但却是负相关的：当大西洋飓风计数高时，太平洋飓风计数往往低(反之亦然)。在技术层面上，离散时间更新过程被用作二进制(0-1)序列的一般模型。通过选择特殊的初始更新寿命长度来使更新过程处于静止状态。然后，以各种方式叠加固定的二进制序列的副本，以建立所寻找的边缘分布。构造具有泊松分布、几何分布和二项分布的平稳序列是很容易的。这些方法很容易产生具有负相关性和/或长记忆性的平稳计数序列-这些方面是经典的整数自回归移动平均计数技术所不能实现的。推理问题、协变量和多变量序列也被考虑。

This research studies data recorded in time that is count-valued, e.g., the annual number of Californian wildfires, yearly Alaskan polar bear sightings, monthly flu deaths, annual North Atlantic severe hurricanes, etc. The data may be correlated in time, implying that counts observed today may be influenced by (correlated with) counts occurring in the immediate past. This work develops time series methods that take into account the particular type of distribution appropriate for the counts, e.g., Poisson, geometric, binomial, etc.. Using the correct distribution facilitates accurate forecasts and inferences. The models developed here allow both positive and negative correlations, a feature absent from current statistical count time series models. For example, annual Pacific and Atlantic hurricane counts are well described by a Poisson-type distribution, but are negatively correlated: when Atlantic hurricane counts are high, Pacific counts tend to be low (and vice versa). It is important to account for such correlations to make accurate climate change conclusions.On technical levels, a discrete-time renewal process is used as a general model for a binary (zero-one) series. The renewal process is rendered stationary by selecting a special initial renewal life length. Copies of the stationary binary sequence are then superimposed in various ways to build the marginal distribution sought. It is easy to construct stationary series with Poisson, geometric, and binomial marginal distributions. The methods easily generate stationary count series with negative correlations and/or long-memory --- aspects not achievable from classical integer autoregressive moving-average count techniques. Inference issues, covariates, and multivariate series are also considered.