On a General Class of Count Time Series Models

本研究研究的时间记录数据是有价值的，例如，每年加利福尼亚野火的数量，每年阿拉斯加北极熊的目击，每月流感死亡人数，每年北大西洋强烈飓风等。这些数据在时间上可能是相关的，这意味着今天观察到的计数可能受到刚刚过去发生的计数的影响（与之相关）。这项工作发展了时间序列方法，考虑到适合计数的特定类型的分布，例如泊松分布、几何分布、二项分布等。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和负相关，这是当前统计计数时间序列模型所缺乏的特征。例如，每年太平洋和大西洋的飓风数量可以用泊松型分布很好地描述，但它们是负相关的：当大西洋飓风数量高时，太平洋的飓风数量往往较低（反之亦然）。为了得出准确的气候变化结论，考虑这种相关性是很重要的。在技术层面上，一个离散时间更新过程被用作二进制（0 - 1）序列的一般模型。通过选择一个特殊的初始更新寿命长度，使更新过程平稳。然后用不同的方法叠加固定二进位序列的副本，以建立所寻求的边际分布。用泊松、几何和二项边缘分布构造平稳序列是很容易的。这些方法很容易产生具有负相关和/或长记忆的平稳计数序列，这些方面是经典的整数自回归移动平均计数技术无法实现的。推理问题，协变量和多元序列也被考虑。

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.