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# 统计代写|时间序列和预测代写Time Series & Prediction代考|Computational Complexity

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## 统计代写|时间序列和预测代写Time Series & Prediction代考|Computational Complexity

The first step of the SSNS segmentation algorithm, i.e., calculation of partition width requires an iteration over each pair of consecutive data points in the time-series. The partitions are stored in order, in an array. Hence, the time complexity for partitioning is $O(n)$ and the space complexity is $O(n+z)$ where $n$ is the number of data points in the time-series and $z$ is the number of partitions made.
Construction of the T-SOT involves iterating over each pair of consecutive data points in the time-series and identifying the partition to which each data point belongs. Since the array of partitions is ordered, a binary search algorithm is employed here. Hence, computing the partition to which a data point belongs takes $O(\log z)$ amount of time. This computation is done for $n$ data points and hence, the total time complexity is $O(n \log z)$. Furthermore, the windowing, window labelling and segmentation steps together, iterate over each character in the T-SOT and place a marker at each segment boundary. The window being of fixed length, the decision to place a segment boundary can be taken in constant time. As the T-SOT contains $n-1$ linguistic characters, this step has a time complexity of $O(n)$. Thus, the overall time complexity is $O(n)+O(n \log z)+O(n) \approx O(n \log z)$ and the space complexity is $O(n+z)$.

Let there be $l$ temporal segments obtained from the time-series. Each segment is approximated as a 10-dimensional point in order for it to be clustered based on the pattern that it represents. In the initialization step of the proposed multi-layered DBSCAN clustering algorithm, a distance matrix is computed from the given points. Given $l$ points, such a distance matrix can be computed in $O\left(l^2\right)$ time. In each density based stratum, the first step involves tuning the value of $\varepsilon$ corresponding to the maximum density clusters. This is done by taking the mean of the least $k$ distances from the distance matrix, where $k$ is an empirically chosen constant. Hence, the time complexity involved is $O\left(l^2\right)$.

## 统计代写|时间序列和预测代写Time Series & Prediction代考|Prediction Experiments and Results

This section proposes an experiment to examine the success of the proposed model in forecasting, using a dynamic stochastic automaton constructed from the TAIEX economic close price time-series. The time-series is first divided into two parts, the first part to be used for knowledge and forecasting and the second part for validation. Here, the first part refers to the TAIEX close price time-series from 1st January, 1990 to 31 st December 2000 (10 years) and the second part includes the period from 1st January, 2001 to 31 st December 2001 (1 year). The experiment involves the following steps.

Step 1. Automaton construction from the time-series: The steps used for segmentation, clustering and construction of the dynamic stochastic automaton introduced earlier are undertaken.
Step 2. Prediction of the most probable sequence of partitions: Steps introduced in Sect. 4.6.2 are invoked for the prediction of the most probable sequence of transitions (MPST) to reach a target state from a given starting state.
Step 3. Validation: In the validation phase, we construct a dynamic stochastic automaton again for the entire span of 11 years from 1st January, 1990 to 31st December 2001. We forecast the most probable sequence of transitions using the automaton constructed previously in step 1 and validate our prediction using the automaton from the data of 11 years (1990-2001).
Here, we partition a time-series into seven equi-spaced partitions, namely, Extremely Low (1), Very Low (2), Low (3), Medium (4), High (5), Very High (6) and Extremely High (7). The dynamic stochastic automaton obtained from the 10 years (1990-2000) data of the TAIEX close price time-series (patterns given in Fig. 4.4) is shown in Fig. 4.9. We carry out the prediction on every day of the first 9 months (assuming a threshold time-limit of 90 days) of the year 2001. In Table 4.3, we show the probabilistic and duration accuracy of our approach by matching the probability and duration of occurrence of the MPST in both training and test phase automata. The results of certain chosen dates where the time-series changes are most prominent have been shown in Table 4.3 Prediction accuracies shown in the table and those obtained for the entire testing period are similar.

## 统计代写|时间序列和预测代写Time Series & Prediction代考|Computational Complexity

SSNS分割算法的第一步，即分区宽度的计算需要对时间序列中的每一对连续数据点进行迭代。分区是按顺序存储在一个数组中。因此，分区的时间复杂度为$O(n)$，空间复杂度为$O(n+z)$，其中$n$是时间序列中的数据点数量，$z$是分区的数量。
T-SOT的构建包括对时间序列中的每一对连续数据点进行迭代，并确定每个数据点所属的分区。由于分区阵列是有序的，这里采用的是二进制搜索算法。因此，计算一个数据点所属的分区需要$O(log z)$的时间。这个计算是针对n$数据点进行的，因此，总的时间复杂度是$O(n\log z)$。此外，窗口化、窗口标记和分割步骤一起迭代了T-SOT中的每个字符，并在每个段的边界处放置了一个标记。窗口的长度是固定的，因此可以在恒定的时间内决定是否放置一个段的边界。由于T-SOT包含n-1$的语言字符，这一步的时间复杂度为$O(n)$。因此，总的时间复杂度为$O(n)+O(n\log z)+O(n)\approx O(n\log z)$，空间复杂度为$O(n+z)$。

## MATLAB代写

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。