### 统计代写|贝叶斯统计代写Bayesian statistics代考|Examples of spatio-temporal data

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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|贝叶斯统计代写beyesian statistics代考|Spatio-temporal data types

Data are called spatio-temporal as long as each one of them carries a location and a time stamp. This leaves open the possibility of a huge number of spatio-temporal data types even if we exclude the two extreme degenerate possibilities where data are observed either at a single point of time or at a single location in space. Analysts often opt for one of the two degenerate possibilities when they are not interested in the variation due to either space or time. This may simply be achieved by suitably aggregating the variation in the ignored category. For example, one may report the annual average air pollution levels at different monitoring sites in a network when daily data are available. Aggregating over time or space reduces variability in the data to be analyzed and will also limit the extent of inferences that can be made. For example, it is not possible to detect or apportion monthly trends just by analyzing annual aggregates. This text book will assume that there is spatiotemporal variation in the data, although it will discuss important concepts, as required, for studying spatial or temporal only data.

One of the very first tasks in analyzing spatio-temporal data is to choose the spatial and temporal resolutions at which to model the data. The main issues to consider are the inferential objectives of the study. Here one needs to decide the highest possible spatial and temporal resolutions at which inferences must be made. Such considerations will largely determine whether we are required to work with daily, monthly or annual data, for example. There are other issues at stake here as well. For example, relationships between variables may be understood better at a finer resolution than a coarser resolution, e.g. hourly air pollution level may have a stronger relationship with hourly wind speed than what the annual average air pollution level will have with the annual average wind speed. Thus, aggregation may lead to correlation degradation. However, too fine a resolution, either temporal and/or spatial, may pose a huge challenge in data processing, modeling and analysis without adding much extra information. Thus, a balance needs to be stuck when deciding on the spatio-temporal resolution of the data to be analyzed and modeled. These decisions must be taken at the data pre-processing stage before formal modeling and analysis can begin.

Suppose that the temporal resolution of the data has been decided, and we use the symbol $t$ to denote each time point and we suppose that there are $T$ regularly spaced time points in total. This is a restrictive assumption as often there are cases of irregularly spaced temporal data. For example, a patient may be followed up at different time intervals; there may be missed appointments during a schedule of regular check-ups. In an air pollution monitoring example, there may be more monitoring performed during episodes of high air pollution. For example, it may be necessary to decide whether to model air pollution at an hourly or daily time scales. In such situations, depending on the main purposes of the study, there are two main strategies for modeling. The first one is to model at the highest possible regular temporal resolution, in which case there would be many missing observations for all the unobserved time points. Modeling will help estimate those missing observations and their uncertainties. For example, modeling air pollution at the hourly resolution will lead to missing observations for all the unmonitored hours. The second strategy avoiding the many missing observations of the first strategy is to model at an aggregated temporal resolution where all the available observations within a regular time window are suitably averaged to arrive at the observation that will correspond to that aggregated time. For example, all the observed hourly air pollution recordings are averaged to arrive at the daily average air pollution value. Although this strategy is successful in avoiding many missing observations, there may still be challenges in modeling since the aggregated response values may show un-equal variances since those have been obtained from different numbers of observations during the aggregated time windows. In summary, careful considerations are required to select the temporal resolution for modeling.

## 统计代写|贝叶斯统计代写beyesian statistics代考|New York air pollution data set

We use a real-life data set, previously analyzed by Sahu and Bakar (2012a), on daily maximum 8-hour average ground-level ozone concentration for the months of July and August in 2006 , observed at 28 monitoring sites in the state of New York. We consider three important covariates: maximum temperature (xmaxtemp in degree Celsius), wind speed (xwdsp in nautical miles), and percentage average relative humidity (xrh) for building a spatiotemporal model for ozone concentration. Further details regarding the covariate values and their spatial interpolations for prediction purposes are provided in Bakar (2012). This data set is available as the data frame nysptime in the bmstdr package. The data set has 12 columns and 1736 rows. The $R$ help command ?nysptime provides detailed description for the columns.

In this book we will also use a temporally aggregated spatial version of this data set. Available as the data set nyspatial in the bmstdr package, it has data for 28 sites in 28 rows and 9 columns. The $R$ help command ?nyspatial provides detailed descriptions for the columns. Values of the response and the covariates in this data set are simple averages (after removing the missing observations) of the corresponding columns in nysptime. Figure $1.1$ represents a map of the state of New York together with the 28 monitoring locations.The two data sets nyspatial and nysptime will be used as running examples for all the point referenced data models in Chapters 6 and 7. The spatiotemporal data set nysptime is also used to illustrate forecasting in Chapter 9 . Chapter 3 provides some preliminary exploratory analysis of these two data sets.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Air pollution data from England and Wales

This example illustrates the modeling of daily mean concentrations of nitrogen dioxide $\left(\mathrm{NO}{2}\right)$, ozone $\left(\mathrm{O}{3}\right)$ and particles less than $10 \mu m\left(\mathrm{PM}{10}\right)$ and $2.5 \mu m$ $\left(\mathrm{PM}{2.5}\right)$ which were obtained from $n=144$ locations from the Automatic Urban and Rural Network (AURN, http://uk-air.defra.gov.uk/networks) in England and Wales. The 144 locations are categorized into three site types: rural (16), urban (81) and RKS (Road amd Kerb Side) (47). The locations of these 144 sites are shown in Figure 1.2.

Mukhopadhyay and Sahu (2018) analyze this data set comprehensively and obtain aggregated annual predictions at each of the local and unitary authorities in England and Wales. It takes an enormous amount of computing effort and data processing to reproduce the work done by Mukhopadhyay and Sahu (2018) and as a result, we do not attempt this here. Instead, we illustrate spatio-temporal modeling for $\mathrm{NO}_{2}$ for 365 days in the year 2011 . We also obtain an annual average prediction map based on modeling of the daily data in Section 8.2.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Air pollution data from England and Wales

Mukhopadhyay 和 Sahu（2018 年）全面分析了该数据集，并获得了英格兰和威尔士每个地方和单一当局的汇总年度预测。重现 Mukhopadhyay 和 Sahu (2018) 所做的工作需要大量的计算工作和数据处理，因此，我们不在这里尝试。相反，我们说明时空建模ñ这22011 年 365 天。我们还根据第 8.2 节中的每日数据建模获得了年平均预测图。

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## MATLAB代写

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