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统计代写|时间序列和预测代写Time Series & Prediction代考|Fuzzy Rule Based NN Model

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统计代写|时间序列和预测代写Time Series & Prediction代考|Fuzzy Rule Based NN Model

In this section, we propose a variation in the training scheme of the neural nets from the previous model. It should be noted that a time-series data point cannot be wholly represented by a partition. A partition is a certain continuous portion of the dynamic range of the time-series. The best way to represent a partition is to consider its mid-point. Hence, by approximating a data point lying in a partition with its corresponding mid-point value, we intentionally delve into some approximation error which grows with the width of the partition itself. One way to avoid this error is to consider each partition as a fuzzy set, with the dynamic range of the time-series representing the universe of discourse. Clearly, each partition is then associated with a judiciously chosen membership function and every time-series data point will have some membership value lying in the set $[0,1]$, for a partition.

We want to design a membership function with the peak corresponding to the center of the partition and a gradual fall-off on either side of the peak with a value almost close to 0 towards the bounds of the partition. The membership value need not however necessarily be 0 at the boundaries of the partition. One typical function which satisfies this requirement is the Gaussian membership function as shown in Fig. 5.4. We use these membership values to train the neural networks in this model. The advantage of this approach is the exploitation of the inherent fuzziness involved in the task of assigning a partition to a time-series data point and using it for better prediction results.

Let the ith partition be $p_i=\left[l_i, u_i\right]$,and let the mid-point of the partition be $m_i$. We define the standard deviation for the Gaussian membership function as $\sigma_i=\left(u_i-m_i\right)=\left(m_i-l_i\right)$. With the above terms defined, the membership function corresponding to the ith partition is given in Eq. (5.16) and illustrated in Fig. 5.4.
$$\mu_i(x)=e^{-\frac{\left(x-m_i\right)^2}{2 \sigma_i^2}}$$

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

The sunspot time-series data is used to record solar activity and is a classic example of a real-life chaotic time-series. Due to the effect of solar activity on various aspects of human life like the climate and weather, prediction of the sunspot time-series has become an important and critical challenge for many researchers. In this chapter, we use the smoothed sunspot time-series data from the World Data Center for sunspot index. The time-series from the period of November 1834 to June 2001 has 2000 data-points which is divided into two equal parts of 1000 points each. The first half is used for training the models and the second half is used for testing. The time-series is scaled to the range $[0,1]$. The following steps are carried out for the experiment:

Step 1. Partitioning, First-Order Rule Extraction And Neural Network Training: In this step, we first partition the training phase time-series into 20 partitions. We experimentally choose the value 20 based on best prediction results. The first-order transition rules extracted from the partitioned time-series along with the probability of occurrence of each rule is given in the transition matrix shown in Table 5.1. It should be noted that the entry corresponding to the cell $\left(P_i, P_j\right)$ contains the probability of occurrence of the first-order transition $P_i \rightarrow P_j$.

Following the extraction of first-order transition rules, they are grouped into training sets, each representing a mapping of distinct antecedents to consequents. The groups of transition rules used to train each neural network is shown in Table 5.2. It should be noted that groups with less than 6 transition rules are ignored for training purposes.

The bunched first-order transition rules are further processed to yield training sets for the two proposed neural network ensembles according to Sects. 5.3 and 5.4. The networks are trained and we use the trained ensembles of both the proposed models to make predictions on the test phase time-series.

Step 2. Prediction on Test-Phase Time-Series: In this step, we apply the trained models to make predictions on the test phase Sunspot time-series. Figs. 5.7 and 5.8 illustrate the predictions made by the first-order rule based NN model and the fuzzy rule based NN model respectively on the test phase sunspot series. In order to quantify the prediction accuracy, we use three well-known error metrics, i.e., the mean square error (MSE), the root mean square error (RMSE) and the normalized mean square error (NMSE). Let $c_{\text {test }}(t)$ denote the value of the test-period time-series at the time-instant $t$ and let $c^{\prime}(t)$ be the predicted time-series value for the same time-instant. The above mentioned error metrics can be defined by the following equations:
\begin{aligned} M S E & =\frac{\sum_{t=1}^N\left(c_{\text {test }}(t)-c^{\prime}(t)\right)^2}{N} \ R M S E & =\sqrt{\frac{\sum_{t=1}^N\left(c_{\text {test }}(t)-c^{\prime}(t)\right)^2}{N}} \end{aligned}

统计代写|时间序列和预测代写Time Series & Prediction代考|Fuzzy Rule Based NN Model

$$\mu_i(x)=e^{-\frac{\left(x-m_i\right)^2}{2 \sigma_i^2}}$$

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

\begin{aligned} M S E & =\frac{sum_{t=1}^N\left(c_{text {test }(t)-c^{prime}(t)\right)^2}{N}。\ R M S E & =\sqrt{frac{sum_{t=1}^N\left(c_{text {test }(t)-c^{prime}(t)right)^2}{N}。 \end{aligned}

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。