### 统计代写|贝叶斯统计代写Bayesian statistics代考|Jargon of spatial and spatio-temporal modeling

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
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• 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|Stochastic processes

We are familiar with the term random variable to denote measurements of outcomes of random experiments. For example, a random variable, say $Y$, could denote the height of a randomly chosen individual in a population of interest, such as a class of seven years old pupils in a primary school. If we

measure the height of $n$ randomly chosen children, then we denote these heights by notations $y_{1}, \ldots, y_{n}$ which are the recorded numerical values. We use the corresponding upper case letters $Y_{1}, \ldots, Y_{n}$ to denote the physical random variable: heights of $n$ randomly selected children.

The concept of random variables is sound and adequate enough when we intend to model and analyze data which are not associated with a continuous domain. For example, the population of seven years old is countable and finite – so the associated domain is not continuous. Further examples include hourly air pollution values recorded at the top of the hour at a particular monitoring site within a city. However, this concept of random variables is not adequate enough when we allow the possibility of having an uncountably infinite collection of random variables associated with a continuous domain such as space or time, or both. For example, suppose that we are interested in modeling an air pollution surface over a city. Because the spatial domain is continuous here, having an uncountably infinite number of locations, we shall require a richer concept of uncountably infinite number of random variables. Hence, we welcome the arrival of the concept of stochastic processes.

A stochastic process is an uncountably infinite collection of random variables defined on a continuous domain such as space, time, or both. Hence, the discrete notation, $Y_{i}, i=1,2, \ldots$ for random variables is changed to either $Y(\mathbf{s}), Y(t)$ or $Y(\mathbf{s}, t)$ where $s$ denotes a spatial location, described by a finite number of dimensions such as latitude, longitude, altitude, etc. and $t$ denotes a continuously measured time point depending on the data collection situation: only spatial, only temporal, and spatio-temporal, respectively. In the spatial only case, we shall use $Y($ s ) to denote a spatial stochastic process or simply a spatial process defined over a domain $\mathbb{D}$, say. In the temporal only case, $Y(t)$ is used to denote the temporally varying stochastic process over a continuous time period, $0 \leq t \leq T$. When time is considered to be discrete, e.g. hourly, then it is notationally convenient to use the notation $Y_{t}$ instead of the more general $Y(t)$. In this case, $Y_{t}$ is better referred to as a time series.

In this book we shall not consider time in the continuous domain at all since such analyses are much more theoretically rich, requiring deeper theoretical understanding but practically not so common in the subject area of the examples described previously in Chapter 1 . Henceforth, we will use the notation $t$ to denote time in a discrete sense and domain. With $t$ being discrete, which of the two notations: $Y(\mathbf{s}, t)$ and $Y_{t}(\mathbf{s})$, should be adopted to denote our spatio-temporal data? Both the notations make sense, and it will be perfectly fine to use either. In this book we adopt the first, slightly more elaborate, notation $Y(s, t)$ throughout, although the subscript $t$ will be used to denote vector-valued random variables as necessary. Hence, the notation $y(\mathbf{s}, t)$ will denote a realization of a spatial stochastic process at location s and at a discrete-time point $t$.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Stationarity

An object is stationary if it does not move from a fixed position. To be stationary, a stochastic process must possess certain stable behavior. A stochastic process, constituting of an infinite collection of random variables, cannot be a constant everywhere since otherwise, it will not be stochastic at all. Hence it makes sense to define stationarity of particular properties, e.g. mean and variance. The type of stationarity depends on the stationarity of the particular property of the stochastic process. In the discussion below, and throughout, we shall assume that the stochastic process under consideration has finite mean and variance, respectively denoted by $\mu(\mathbf{s})$ and $V(\mathbf{s})$ for all values of $s$ in $\mathbb{D}$.
A stochastic process, $Y(\mathbf{s})$, is said to be mean stationary if its mean is constant over the whole domain $\mathbb{D}$. For a mean stationary process $Y(\mathbf{s})$, $\mu(\mathbf{s})=E(Y(\mathbf{s}))$ is a constant function of s. Thus, the mean surface of a mean stationary stochastic process will imply a one-color map depicting the mean over the domain $\mathbb{D}$. Such a map will not exhibit any spatial trend in any direction. Note that this does not mean that a particular realization of the stochastic process, $Y(s)$ at $n$ locations $\mathbf{s}{1}, \mathbf{s}{2}, \ldots, \mathbf{s}{n}$ will yield a constant surface, $y\left(\mathbf{s}{1}\right)=y\left(\mathbf{s}{2}\right)=\ldots=y\left(\mathbf{s}{n}\right)$. Rather, mean stationarity of a process $Y(\mathbf{s})$ means that $\mu\left(\mathbf{s}{1}\right)=\mu\left(\mathbf{s}{2}\right)=\cdots=\mu\left(\mathbf{s}{n}\right)$ at an arbitrary set of $n$ locations, $\mathbf{s}{1}, \mathbf{s}{2}, \ldots, \mathbf{s}{n}$, where $n$ itself is an arbitrary positive integer. Similarly, we say that a time series, $Y_{t}$ is mean stationary if $E\left(Y_{t}\right),\left(=\mu_{t}\right.$, say), does not depend on the value of $t$. A mean stationary process is rarely of interest since, often, the main interest of the study is to investigate spatial and/or temporal variation. However, we often assume a zero-mean stationary process for the underlying error distribution or a prior process in modeling.

In spatial and temporal investigations often it is of interest to study the relationships, described by covariance or correlation, between the random variables at different locations. For example, one may ask, “will the covariance between two random variables at two different locations depend on the two locations as well as the distance between the two?” A lot of simplification in analysis is afforded when it is assumed that the covariance only depends on the simple difference (given by the separation vector $\mathbf{s}-\mathbf{s}^{\prime},=\mathbf{h}$, say) between two locations $s$ and $\mathbf{s}^{\prime}$ and not on the actual locations $\mathbf{s}$ and $\mathbf{s}^{\prime}$. A stochastic process $Y(\mathbf{s})$ is said to be covariance stationary if $\operatorname{Cov}\left(Y(\mathbf{s}), Y\left(\mathbf{s}^{\prime}\right)\right)=C(\mathbf{h})$ where $C$ is a suitable function of the difference $\mathbf{h}$. The function $C(\mathbf{h})$ is called the covariance function of the stochastic process and plays a crucial role in many aspects of spatial analysis. The global nature of the covariance function $C(\mathbf{h})$, as it is free of any particular location in the domain $\mathbb{D}$, helps

tremendously to simplify modeling and analysis and to specify joint distributions for the underlying random variables.

A stochastic process, $Y(\mathrm{~s})$, is said to be variance stationary if its variance, $V(\mathbf{s})$, is a constant, say $\sigma^{2}$, over the whole domain $\mathbb{D}$. For a variance stationary process, no heterogeneity arises due to variation either in space or time. This is again a very strong assumption that may not hold in practice. However, while modeling we often assume that the underlying error distribution has a constant spatial variance. Other methods and tricks, such as data transformation and amalgamation of several processes are employed to model non-constant spatial variance.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Variogram and covariogram

The quantity $\operatorname{Var}(Y(\mathbf{s}+\mathbf{h})-Y(\mathbf{s}))$ is called the variogram of the stochastic process, $Y$ (s) as it measures the variance of the first difference in the process at two different locations $\mathbf{s}+\mathbf{h}$ and $\mathbf{s}$. Our desire for a simplified analysis,

using intrinsic stationarity, would dictate us to suppose that the variogram depends only on the separation vector $h$ and not on the actual location $\mathbf{s}$.
There is a one-to-one relationship between the variogram under the assumption of mean and variance stationarity for a process $Y(\mathbf{s})$. Assuming mean and variance stationary we have $E(Y(\mathbf{s}+\mathbf{h}))=E(Y(\mathbf{s}))$ and $\operatorname{Var}(Y(\mathbf{s}+\mathbf{h}))=\operatorname{Var}(Y(\mathbf{s}))=C(\mathbf{0})$, the spatial variance. For an intrinsically stationary process, we have:
\begin{aligned} \operatorname{Var}(Y(\mathbf{s}+\mathbf{h})-Y(\mathbf{s})) &=E{Y(\mathbf{s}+\mathbf{h})-Y(\mathbf{s})-E(Y(\mathbf{s}+\mathbf{h})+Y(\mathbf{s}))}^{2} \ &=E{(Y(\mathbf{s}+\mathbf{h})-E(Y(\mathbf{s}+\mathbf{h})))-(Y(\mathbf{s})-E(Y(\mathbf{s})))}^{2} \ &=E\left{(Y(\mathbf{s}+\mathbf{h})-E(Y(\mathbf{s}+\mathbf{h})))^{2}\right}+E\left{(Y(\mathbf{s})-E(Y(\mathbf{s})))^{2}\right} \ &=-2 E{(Y(\mathbf{s}+\mathbf{h})-E(Y(\mathbf{s}+\mathbf{h})))(Y(\mathbf{s})-E(Y(\mathbf{s})))} \ &=C(\mathbf{0}(Y(\mathbf{s}+\mathbf{h}))+\operatorname{Var}(Y(\mathbf{s}))-2 \operatorname{Cov}(Y(\mathbf{s}+\mathbf{h}), Y(\mathbf{s}))-2 C(\mathbf{h})\ &=2(C(\mathbf{0})-C(\mathbf{h})) . \end{aligned}
This result states that:
Variogram at separation $\mathbf{h}=2 \times{$ Spatial variance $-$ Spatial covariance function at separation $\mathbf{h}}$.

Clearly, we can easily find the variogram if we already have a specification for the spatial covariance function for all values of its argument. However, it is not easy to retrieve the covariance function from a specification of the variogram function, $\operatorname{Var}(Y(\mathbf{s}+\mathbf{h})-Y(\mathbf{s}))$. This needs further assumptions and limiting arguments, see e.g. Chapter 2 of Banerjee et al. (2015).

In order to study the behavior of a variogram, $2(C(\mathbf{0})-C(\mathbf{h}))$, as a function of the covariance function $C(\mathbf{h})$, we see that the multiplicative factor 2 is only a distraction. This is why, the semi-variogram, which is half of the variogram is conventionally studied in the literature. We use the notation $\gamma(\mathbf{h})$ to denote the semi-variogram, and thus $\gamma(\mathbf{h})=C(\mathbf{0})-C(\mathbf{h})$.

In practical modeling work we assume a specific valid covariance function $C(\mathbf{h})$ for the stochastic process and then the semi-variogram, $\gamma(\mathbf{h})$ is automatically determined. The word “valid” has been included in the previous sentence since a positive definiteness condition is required to ensure nonnegativeness of variances of all possible linear combinations of the random variables $Y\left(\mathbf{s}{1}\right), \ldots, Y\left(\mathbf{s}{n}\right)$. The simplification provided by the assumption of intrinsic stationarity is still not enough for practical modeling work since it is still very hard to specify a valid multi-dimensional function $C(\mathbf{h})$ as a function of the separation vector $\mathbf{h}$. The crucial concept of isotropy, defined and discussed below, accomplishes this task of specifying the covariance function as a one-dimensional function.

## 统计代写|贝叶斯统计代写beyesian statistics代考|Variogram and covariogram

\begin{aligned} \operatorname{Var}(Y(\mathbf{s}+\mathbf{h})-Y(\mathbf{s})) &=E{Y(\mathbf{s}+\mathbf{ h})-Y(\mathbf{s})-E(Y(\mathbf{s}+\mathbf{h})+Y(\mathbf{s}))}^{2} \ &=E{( Y(\mathbf{s}+\mathbf{h})-E(Y(\mathbf{s}+\mathbf{h})))-(Y(\mathbf{s})-E(Y(\mathbf {s})))}^{2} \ &=E\left{(Y(\mathbf{s}+\mathbf{h})-E(Y(\mathbf{s}+\mathbf{h}) ))^{2}\right}+E\left{(Y(\mathbf{s})-E(Y(\mathbf{s})))^{2}\right} \ &=-2 E{ (Y(\mathbf{s}+\mathbf{h})-E(Y(\mathbf{s}+\mathbf{h})))(Y(\mathbf{s})-E(Y(\mathbf {s})))} \ &=C(\mathbf{0}(Y(\mathbf{s}+\mathbf{h}))+\operatorname{Var}(Y(\mathbf{s}))- 2 \operatorname{Cov}(Y(\mathbf{s}+\mathbf{h}), Y(\mathbf{s}))-2 C(\mathbf{h})\ &=2(C(\mathbf {0})-C(\mathbf{h})) .\end{aligned}\begin{aligned} \operatorname{Var}(Y(\mathbf{s}+\mathbf{h})-Y(\mathbf{s})) &=E{Y(\mathbf{s}+\mathbf{ h})-Y(\mathbf{s})-E(Y(\mathbf{s}+\mathbf{h})+Y(\mathbf{s}))}^{2} \ &=E{( Y(\mathbf{s}+\mathbf{h})-E(Y(\mathbf{s}+\mathbf{h})))-(Y(\mathbf{s})-E(Y(\mathbf {s})))}^{2} \ &=E\left{(Y(\mathbf{s}+\mathbf{h})-E(Y(\mathbf{s}+\mathbf{h}) ))^{2}\right}+E\left{(Y(\mathbf{s})-E(Y(\mathbf{s})))^{2}\right} \ &=-2 E{ (Y(\mathbf{s}+\mathbf{h})-E(Y(\mathbf{s}+\mathbf{h})))(Y(\mathbf{s})-E(Y(\mathbf {s})))} \ &=C(\mathbf{0}(Y(\mathbf{s}+\mathbf{h}))+\operatorname{Var}(Y(\mathbf{s}))- 2 \operatorname{Cov}(Y(\mathbf{s}+\mathbf{h}), Y(\mathbf{s}))-2 C(\mathbf{h})\ &=2(C(\mathbf {0})-C(\mathbf{h})) .\end{aligned}

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## MATLAB代写

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