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# 统计代写|时间序列和预测代写Time Series & Prediction代考|STAT435 Time-Series Prediction Using Fuzzy Reasoning

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## 统计代写|时间序列和预测代写Time Series & Prediction代考|Time-Series Prediction Using Fuzzy Reasoning

One primary motivation of time-series prediction by fuzzy logic based reasoning is to determine the implication functions that exist between the membership function of a time point and its next time point in a time series. For instance, consider a close price time-series $\mathrm{c}(\mathrm{t})$ where $\mu_{\mathrm{A}}(\mathrm{c}(\mathrm{t}-1))$ and $\mu_{\mathrm{B}}(\mathrm{c}(\mathrm{t}))$ denote the respective membership values of the close price in fuzzy sets A and B. We like to express the implication relation from “c $(\mathrm{t}-1)$ is $\mathrm{A}$ ” to “c( $\mathrm{t})$ is $\mathrm{B}$ ” by a fuzzy relation $\mathrm{R}(\mathrm{c}(\mathrm{t}-1), \mathrm{c}(\mathrm{t}))$. Now, suppose we have $\mathrm{m}$ partitions, and suppose the data points in a partition is represented by a suitable fuzzy set. Thus for $\mathrm{m}$ partitions $\mathrm{P} 1, \mathrm{P} 2, \ldots, \mathrm{Pm}$, we have $\mathrm{m}$ fuzzy sets A1, A2, .., Am. To extract the list of all possible implications, we thus require to mark the transitions of the time-series at all time points $t-1$ and $t$ and then identify the appropriate fuzzy sets to which $c(t-1)$ and $\mathrm{c}(\mathrm{t})$ belong with the highest memberships.

Let $c(t-1)$ and $c(t)$ have highest memberships in $A_p$ and $A_q$ for certain value of $t=t_i$. Then we would construct a fuzzy relation for the rule: If $c(t-1)$ is $A_p$ Then $c$ (t) is $\mathrm{A}_{\mathrm{q}}$. The question then naturally arises as to how to construct the implication relation. In fact there exists quite a few implication relations, a few of these that are worth mentioning includes Mamdani (Min), Lukasiewicz and Diens-Rescher implications. Most of the existing literature on fuzzy time-series, however, utilizes Mamdani (Min-type) implication for its simplicity, low computational complexity and publicity.

Once a fuzzy implication relation $\mathrm{R}(\mathrm{c}(\mathrm{t}-1), \mathrm{c}(\mathrm{t}))$ is developed, we would be able to predict the fuzzy membership value $\mu_{\mathrm{B}}{ }^{\prime}(\mathrm{c}(\mathrm{t}))$ for the next time-series data point $\mathrm{c}(\mathrm{t})$ from the measured membership value $\mu_{\mathrm{A}}{ }^{\prime}(\mathrm{c}(\mathrm{t}-1))$ of the close price at time $\mathrm{c}(\mathrm{t}-1)$. The dash over $\mathrm{A}^{\prime}$ and $\mathrm{B}^{\prime}$ here represents that the fuzzy set $\mathrm{A}^{\prime}$ and $\mathrm{B}^{\prime}$ are linguistically close enough to the respective fuzzy sets A and B respectively. The fuzzy compositional rule of inference has been used to determine $\mu_{\mathrm{B}}{ }^{\prime}(\mathrm{c}(\mathrm{t}))$ by the following step:
$$\mu_{\mathrm{B}}^{\prime}(\mathrm{c}(\mathrm{t}))=\mu_{\mathrm{A}}^{\prime}(\mathrm{c}(\mathrm{t}-1)) \text { o } \mathrm{R}$$
where o is a max-min compositional operation, which is similar to matrix multiplication operation with summation replaced by maximum and product replaced by minimum operators.

## 统计代写|时间序列和预测代写Time Series & Prediction代考|Single and Multi-Factored Time-Series Prediction

Traditional techniques of time-series prediction solely rely on the time-series itself for the prediction. However, recent studies [96] reveal that the predicted value of the time-series cannot be accurately determined by the previous values the series only. Rather, a reference and more influential time-series, such as NASDEQ and DOWJONES can be used along with the original time-series for the latter’s prediction.

Several models of forecasting have been developed in the past to improve forecasting accuracy and also to reduce computational overhead. There exist issues of handling uncertainty in business forecasting by partitioning the intervals of non-uniform length. The work undertaken by $\mathrm{Li}$ and Cheng [104] proposes a novel deterministic fuzzy time-series to determine suitable interval lengths. Experiments have been performed to test the feasibility of the forecasting on enrolment data in Alabama Universities. Experimental results further indicate that the first order time series used here is highly reliable for prediction and thus is appropriate for the present application.

Existing models of fuzzy time-series rely on a first order partitioning of the dynamic range of the series. Such partitioning helps in assigning fuzzy sets to individual partitions. First order partitions being uniform cannot ensure a large cardinality of data points. In fact, occasionally, a fewer partitions have fewer data points. In the worst case, the partition may be empty. One approach to overcome the above limitation is to re-partitioning a partition into two or more partitions. It is important to note that when data density in a partition is non-uniform, we re-partition if for having sub-partitions of near uniform data cardinality. The work by Gangwar et al. [105] is an attempt to achieve re-partitioning of a time-series for efficient prediction.

Most of the works on fuzzy time series presumed time invariant model of the time-series. The work by Aladag et al. [102] proposes a novel time-variant model fuzzy time-series, where they use particle swarm optimization techniques to determine fuzzy relational matrices connecting the membership functions of the of the two consecutive time point values. Besides providing a new approach to construct fuzzy relational matrices, the work also considers employing fuzzy c-means clustering techniques for fuzzification of the time-series.

## 统计代写|时间序列和预测代写Time Series \& Prediction代考|Time-Series Prediction Using Fuzzy Reasoning

$$\mu_{\mathrm{B}}^{\prime}(\mathrm{c}(\mathrm{t}))=\mu_{\mathrm{A}}^{\prime}(\mathrm{c}(\mathrm{t}-1)) \text { o } \mathrm{R}$$

## MATLAB代写

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