### 统计代写|贝叶斯网络代写Bayesian network代考|CS47100

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|贝叶斯网络代写Bayesian network代考|What is Spatial Time Series Data

Spatial time series data is a major category of spatio-temporal data that involves variations across the space as well as time. In other words, it can be defined as data to which labels have been assigned to indicate where and when these were collected. Moreover, in case of spatial time series data, the space is fixed, but the measurement value changes over a series of time (refer Fig. 1.1). These are also termed as geo-referenced time series. Time series of precipitation data collected over various locations in a space, earth surface temperature data etc. are some examples in this regard. The recent advancements in satellite remote sensing and spatially enabled sensors technology are the primary sources of this kind of data. In a naive case, it is possible to remember only the most recent value in the temporal evolution of some spatial phenomena in a fixed location. These are called geo-referenced variables.

Given spatial time series data over a set of explanatory variables and a dependent variable (or target variable), the spatial time series prediction is the process of learning a model that can predict the dependent variable from the explanatory variables [17]. Considering ‘day’ as the smallest unit of time (for example), the overall prediction problem can be formally stated as follows:

• Given, the historical daily time series dataset over $n$ variables in $V=$ $\left{V_{1}, V_{2}, \ldots, V_{n}\right}$, corresponding to a set of $K$ locations $L o c^{k}=$ $\left{\operatorname{Loc}{1}^{k}, \operatorname{Loc}{2}^{k}, \ldots, L o c_{K}^{k}\right}$ in a spatial region $R$ for previous $t$ years: $\left{y_{1}, y_{2}, \ldots, y_{t}\right}$. Also given, the spatial attribute information $S A=\left{S A_{1}^{l o c}, S A_{2}^{l o c}, \ldots, S A_{r}^{l o c}\right}$ regarding each location $l o c \in L o c^{k}$. The problem is to determine the daily state/conditions of the variables in $V$ for any location $x \in\left(\operatorname{Loc}^{k} \cup L o c^{u}\right)$ for future $i$ years $\left{y_{(t+1)}, y_{(t+2)}, \ldots, y_{(t+i)}\right}$, when the spatial attributes of $x$ are observed as

$\left{S A_{1}^{x}, S A_{2}^{x}, \ldots, S A_{r}^{x}\right}$. Here, $L o c^{u}$ is a set of $z$ new locations $\left{\operatorname{Loc}{1}^{u}, \operatorname{Loc}{2}^{u}, \ldots, L o c_{z}^{u}\right}$, such that $L o c_{j}^{u} \notin \operatorname{Loc}^{k}$, for $j=1$ to $z$, and $i$ is a positive integer, i.e. $i \in{1,2,3, \ldots}$.
A sample of spatial time series data corresponding to the domain of meteorology is depicted in the Fig. 1.2, where $V={$ Temperature $(T)$, Humidity $(H)$, Rainfall $(R)}, \quad S A={\operatorname{Latitude}(L), \operatorname{Longitude}(G)$, Elevation $(E)}, \quad$ Loc ${ }^{k}={$ Location 1, Location2, Location3 $}, y_{1}=2011, y_{2}=2012, y_{3}=2013$, and $y_{4}=2014$.

## 统计代写|贝叶斯网络代写Bayesian network代考|Spatial Time Series Prediction and Research Challenges

The major challenges in spatial time series prediction arise mainly because of the nature of the spatio-temporal data itself. Some of these are discussed below:

• First of all, unlike the traditional data, the spatio-temporal data follow the “first law of geography”, i.e. the data that are close in space and time tend to be more similar than data far apart. For example, the weather of a day is more similar to that of the previous day. Likewise, the land surface temperature of one location is more likely to be same as that of its nearby locations. This property is commonly known as autocorrelation $[7,8]$ and dictates that spatio-temporal data cannot be modeled as statistically independent data [19].
• Secondly, the spatio-temporal phenomena are not concrete “objects”. Rather, these are continuous, evolving patterns over space and time. These kinds of spatio-temporal evolutionary processes can be well captured by differential equations used in existing physics-driven approaches. However, differential equations are costly to solve and have several well-known limitations. Therefore, providing an alternative means of modeling such ST processes becomes a challenging task.
• Thirdly, the spatial/spatio-temporal data sometimes also show inter-dependency with the co-located variables. Therefore, instead of only dealing with the target variable, considering the effects of other influencing variables may improve the results of spatio-temporal data mining. A proper modeling of such spatio-temporal inter-relationships among the variables is also a critical issue.
• Further, in most of the cases, the spatio-temporal data are relatively abundant in either space, or time, but not in both [16]. For example, the satellite remote sensing imagery is significantly profuse in space, providing detailed view of large areas. However, these are relatively scarce with respect to time. On the other hand, data from fixed sensors are plentifully available over time, though these provide relatively little detail in space due to limitation in the number of spatially distributed sensors.
• Finally, the recent advancement in satellite and remote sensing technology has led to explosive growth in spatial and spatio-temporal data. Extracting useful and interesting information or patterns from these huge amount of data is also an added challenge in this regard.

In the subsequent chapters of this monograph we attempt to discuss a number of enhanced Bayesian network models in the context of the above-mentioned challenges.

## 统计代写|贝叶斯网络代写Bayesian network代考|State of the Art

In spite of the fact that the spatial relationships are powerful and informative, while predicting ST data, most of the earlier researches focused only on the temporal aspects without taking into account the spatial dependencies. The various traditional statistical time series prediction models formed the base structure of these techniques. Among these conventional statistical techniques, the Exponentially Weighted Moving Average(EWMA), Auto-regressive Moving Average(ARMA), Auto-regressive Integrated Moving Average (ARIMA), Vector Auto-regressive Moving Average (VARMA), and Generalized Auto-regressive Conditional Heteroskedastic (GARCH) models have been widely used especially for prediction of financial, economic, and meteorological time series data [18].

However, two of the major limitations of applying traditional statistical techniques for ST prediction are: firstly, most of these techniques suffer from linear and/or univariate nature and backward looking problem; and secondly, none of these takes the spatial aspects of the associated ST data into account. Therefore, a number of spatially-enhanced prediction techniques have been proposed in recent days to overcome the limitations of traditional techniques. The Space-Time Auto-regressive Moving Average (STARMA) model, Space-Time ARIMA (STARIMA) model, Spatio-temporal Kriging (ST Kriging), Bayesian Hierarchical model, Dynamic Spatio-temporal Models (DSMs) etc. are most commonly used statistical ST prediction techniques. Extensive application of these techniques can be found in traffic flow management, and atmospheric data analysis.

## 统计代写|贝叶斯网络代写Bayesian network代考|What is Spatial Time Series Data

• 给定，历史每日时间序列数据集超过n变量在= \left{V_{1}, V_{2}, \ldots, V_{n}\right}\left{V_{1}, V_{2}, \ldots, V_{n}\right}, 对应一组ķ地点大号○Cķ=$\left{\operatorname{Loc} {1}^{k}, \operatorname{Loc} {2}^{k}, \ldots, L o c_{K}^{k}\right}一世n一个sp一个吨一世一个lr和G一世○nRF○rpr和在一世○在s吨是和一个rs:\left{y_{1}, y_{2}, \ldots, y_{t}\right}.一个ls○G一世在和n,吨H和sp一个吨一世一个l一个吨吨r一世b在吨和一世nF○r米一个吨一世○nSA=\left{S A_{1}^{loc}, S A_{2}^{loc}, \ldots, S A_{r}^{loc}\right}r和G一个rd一世nG和一个CHl○C一个吨一世○nloc \in L oc^{k}.吨H和pr○bl和米一世s吨○d和吨和r米一世n和吨H和d一个一世l是s吨一个吨和/C○nd一世吨一世○ns○F吨H和在一个r一世一个bl和s一世n在F○r一个n是l○C一个吨一世○nx \in\left(\operatorname{Loc}^{k} \cup L oc^{u}\right)F○rF在吨在r和一世是和一个rs\left{y_{(t+1)}, y_{(t+2)}, \ldots, y_{(t+i)}\right},在H和n吨H和sp一个吨一世一个l一个吨吨r一世b在吨和s○Fx$ 被观察为

\left{S A_{1}^{x}, S A_{2}^{x}, \ldots, S A_{r}^{x}\right}\left{S A_{1}^{x}, S A_{2}^{x}, \ldots, S A_{r}^{x}\right}. 这里，大号○C在是一组和新地点\left{\operatorname{Loc}{1}^{u}, \operatorname{Loc}{2}^{u}, \ldots, L o c_{z}^{u}\right}\left{\operatorname{Loc}{1}^{u}, \operatorname{Loc}{2}^{u}, \ldots, L o c_{z}^{u}\right}, 这样大号○Cj在∉地方ķ， 为了j=1至和， 和一世是一个正整数，即一世∈1,2,3,….

## 统计代写|贝叶斯网络代写Bayesian network代考|Spatial Time Series Prediction and Research Challenges

• 首先，与传统数据不同，时空数据遵循“地理学第一定律”，即时空相近的数据往往比相距远的数据更相似。例如，一天的天气与前一天的天气更相似。同样，一个地点的地表温度更可能与其附近地点的地表温度相同。此属性通常称为自相关[7,8]并规定时空数据不能被建模为统计上独立的数据[19]。
• 其次，时空现象不是具体的“对象”。相反，这些是在空间和时间上连续的、不断演变的模式。现有物理驱动方法中使用的微分方程可以很好地捕捉到这些类型的时空演化过程。然而，微分方程的求解成本很高，并且有几个众所周知的局限性。因此，为此类 ST 流程建模提供替代方法成为一项具有挑战性的任务。
• 第三，空间/时空数据有时也显示出与同位变量的相互依赖关系。因此，不仅仅处理目标变量，考虑其他影响变量的影响，可能会改善时空数据挖掘的结果。对变量之间的这种时空相互关系进行适当建模也是一个关键问题。
• 此外，在大多数情况下，时空数据在空间或时间上都相对丰富，但在两者中都不是[16]。例如，卫星遥感图像在太空中非常丰富，可以提供大面积的详细视图。然而，这些相对于时间而言是相对稀缺的。另一方面，随着时间的推移，来自固定传感器的数据会大量可用，尽管由于空间分布传感器数量的限制，这些数据提供的空间细节相对较少。
• 最后，卫星和遥感技术的最新进展导致空间和时空数据的爆炸式增长。从这些海量数据中提取有用且有趣的信息或模式也是这方面的一项额外挑战。

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MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。