### 统计代写|贝叶斯网络代写Bayesian network代考|CSCl252

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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|Why SpaBN for Spatial Time Series Prediction

Spatio-temporal variables are not independent. In most of the cases, these are dependent on various other co-located variables. For example, consider a scenario

of predicting water level at a reservoir in the flow of any river. The water level at the reservoir depends on many factors, like the volume of inflow and outflow of water, seepage into ground, evaporation, meteorological condition, and so on. One of the significant meteorological factors in this regard is the environmental precipitation. Now, the level of water in the reservoir depends not only on the precipitation at the reservoir location, but also that in the various other locations in the whole watershed of the corresponding river. Moreover, based on the various topographical factors, like soil type, land cover category, land slope etc., the precipitation at the different locations may have different influence on the water level of the reservoir. Thus, modeling the spatial effect/influence of precipitation or any such meteorological factor on the reservoir water level becomes a challenging issue, inasmuch as the watershed of any river is in general large and consists of locations with varying topographic characteristics.

Although the graphical models, like Bayesian networks, are highly suitable for representing such inter-variable influences, yet, for each such influencing variable, introducing representative node corresponding to each spatial location will lead to a very complicated causal dependency graph structure consisting of a large number of nodes and edges. One such example scenario has been illustrated through the Fig. 4.3, which shows a graphical model representing influence from three variables $V_{i}$, $(i=1,2,3)$, distributed at $K=8$ number of spatial locations. This eventually leads to extremely high time and space complexities during parameter learning and inference process. The spatial Bayesian network (SpaBN) can handle this situation efficiently, by using the composite node representation for each spatially distributed variable.

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

Let us assume a directed acyclic graph $G\left(V_{s}, V_{c}, E\right)$, as shown in the Fig. 4.1, where $V_{s}=\left{V_{2}, V_{6}\right}$ denotes the set of standard nodes; $V_{c}=\left{V_{1}, V_{3}, V_{4}, V_{5}\right}$ denotes the set of composite nodes; and $E$ is the set of edges $\left{V_{1} \rightarrow V_{2}, V_{1} \rightarrow\right.$ $\left.V_{3}, \quad V_{1} \rightarrow V_{4}, V_{2} \rightarrow V_{4}, V_{2} \rightarrow V_{5}, V_{3} \rightarrow V_{4}, V_{3} \rightarrow V_{6}, V_{4} \rightarrow V_{5}, V_{4} \rightarrow V_{6}\right}$. An edge from $V_{i}$ to $V_{j}$ can be interpreted as $V_{i}$ has influence on $V_{j}$. Let us also consider that the variables corresponding to the composite nodes are spatially distributed over $K$ ( $=8$ as per the figure) number of locations.

According to the learning principle of SpaBN $[6]$, the marginal probabilities of the composite nodes $\in V_{c}$ in this scenario are calculated with consideration to the spatial importance of each neighboring location:
\begin{aligned} &\boldsymbol{P}\left(V_{1}\right)=\gamma \cdot\left[\sum_{i=1}^{K} P\left(V_{1}^{i}\right) \cdot S W_{i}\right] \ &P\left(V_{3}\right)=\gamma \cdot\left[\sum_{i=1}^{K} P\left(V_{3}^{i}\right) \cdot S W_{i}\right] \ &P\left(V_{4}\right)=\gamma \cdot\left[\sum_{i=1}^{K} P\left(V_{4}^{i}\right) \cdot S W_{i}\right] \ &P\left(V_{5}\right)=\gamma \cdot\left[\sum_{i=1}^{K} P\left(V_{5}^{i}\right) \cdot S W_{i}\right] \end{aligned}
where, $\gamma$ is a normalization constant such that the sum of marginal probabilities corresponding to all possible values of the variable becomes $1 . P\left(V_{j}^{i}\right)$ is the marginal probability of singular component $V_{j}^{i}$ in $V_{j}$, for $\mathrm{j}=1,3,4,5$, and $S W_{i}$ is the spatial weight/importance of the $i$ th neighboring location. For example, considering

the example scenario in the Chap. 1 (Fig. 1.2) and assuming the prediction location is Location-3, the probability distribution for the variable $T$ for the year 2011 can be estimated as: $P(T 1)=\gamma \cdot\left[\left(T_{1}^{\text {Locl }} \times S W_{L o c 1}\right)+\left(T_{1}^{\text {Loc } 2} \times S W_{L o c 2}\right)+\left(T_{1}^{\text {Loc } 3} \times\right.\right.$ $\left.\left.S W_{\text {Lac3 }}\right)\right]=\gamma \cdot[(0.8 \times 0.03)+(0.0 \times 0.97)+(0.0 \times 1.0)]=0.024 \gamma$. In similar way, we can estimate $P(T 2)=0.006 \gamma, P(T 3)=0.2 \gamma, P(T 4)=1.38 \gamma$, and $P(T 5)=0.39 \gamma$, where $\gamma$ is the normalization constant and can be determined as $0.5$. Thus, the normalized probability distribution for the spatially distributed variable $T$ becomes $P(T 1)=0.012, P(T 2)=0.003, P(T 3)=0.1, P(T 4)=0.69$, $P(T 5)=0.195$

The conditional probabilities, involving composite nodes $\in V_{c}$, are calculated similarly, considering spatial importance of the nearby locations, as follows:
\begin{aligned} P\left(V_{2} \mid V_{1}\right)=\gamma \cdot & {\left[\sum_{i=1}^{K} \frac{n\left(V_{2}, V_{1}^{i}\right)}{n\left(V_{1}^{i}\right)} \cdot S W_{i}\right] } \ P\left(V_{3} \mid V_{1}\right)=\gamma \cdot & {\left[\sum_{i=1}^{K} \frac{n\left(V_{3}^{i}, V_{1}^{i}\right)}{n\left(V_{1}^{i}\right)} \cdot S W_{i}\right] } \ P\left(V_{4} \mid V_{1}, V_{2}, V_{3}\right)=\gamma \cdot & {\left[\sum_{i=1}^{K} \frac{n\left(V_{1}^{i}, V_{2}, V_{3}^{i}, V_{4}^{i}\right)}{n\left(V_{1}^{i}, V_{2}, V_{3}^{i}\right)} \cdot S W_{i}\right] } \ P\left(V_{5} \mid V_{2}, V_{4}\right)=\gamma \cdot & {\left[\sum_{i=1}^{K} \frac{n\left(V_{5}^{i}, V_{2}, V_{4}^{i}\right)}{n\left(V_{2}, V_{4}^{i}\right)} \cdot S W_{i}\right] } \ P\left(V_{6} \mid V_{3}, V_{4}\right)=\gamma \cdot & {\left[\sum_{i=1}^{K} \frac{n\left(V_{6}, V_{3}^{i}, V_{4}^{i}\right)}{n\left(V_{3}^{i}, V_{4}^{i}\right)}-S W_{i}\right] } \end{aligned}
where, $n(<,>)$ represents the total number of observation for the variable combination $<\cdot>$. Considering the example scenario presented in Chap. 1 (Fig. 1.2) and assuming the prediction location to be Location-3, the calculation of conditional probability distribution of the variable humidity $(H)$ for the year 2011 are explained through Fig. 4.4, in comparison with standard BN based probability calculation. Here, the structure of SpaBN is considered to be as depicted in Fig. 4.2.

It is to be noted that the causal dependency graph of the SpaBN does not contain any of the spatial attributes (SAs) as described while discussing ST relationship learning in the Chap. 3. Rather, for any variable under study, the network considers relevant node corresponding to each of the associated spatial locations explicitly, and the appropriate spatial attributes are utilized in spatial weight/importance $(S W)$ calculation. The overall process of ST relationship learning using SpaBN is presented through the Algorithm $3 .$

## 统计代写|贝叶斯网络代写Bayesian network代考|SpaBN-Based Prediction

Once the inferred value is produced, it is further processed to finally generate the predicted value of the variable. Among all the inferred values of the prediction variable, the predicted value becomes the one which is associated with the highest probability estimates $P(\cdot)$. Therefore, if pred $V_{j}$ is the predicted value of the variable $V_{j}$, then $P\left(\operatorname{pred}{V{j}} \mid e\right)=\max \left{P\left(V_{j} \mid e\right)\right}$, where $e$ indicates the given combination of values for the set of evidence variables, and $P\left(V_{j} \mid e\right)$ represents the inferred probability distribution of the variable $V_{j}$ given $e$. With respect to the above example, $P\left(\right.$ pred $\left.V_{6} \mid V_{1}, V_{2}, \ldots, V_{4}\right)=\max \left{P\left(V_{6} \mid V_{1}, V_{2}, \ldots, V_{4}\right)\right}$. Now, since the overall SpaBN analysis is performed considering discretized value of the variables, the predicted value pred ${ }{V j}$ is also obtained in the form of range of values $\left[L B{j}, U B_{j}\right]$. Hence, in order to obtained a single value for the prediction variable, the mid value of the range may be considered. Therefore, finally, $\operatorname{pred}{V{j}}=\left(L B_{j}+U B_{j}\right) / 2$

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