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Research Article
 

A New Method on Partially Linear Autoregression Forecasting



Ang Wei, F.U. Chunyao and Wei Wei
 
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ABSTRACT

In this study, a new method to acquire the decision rule of information system with continuous attributes was proposed. The modeling theory and the estimation method of partially linear auto-regression model are discussed and the lag variables, best model, the optimal bandwidth are determined. Partially linear auto-regression models of Shanghai stock index and Shenzhen component index based on partial residual estimate are established and the validity of the model are examined. Finally, forecasting with the model and judged the effect of the model based on the real data.

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  How to cite this article:

Ang Wei, F.U. Chunyao and Wei Wei, 2012. A New Method on Partially Linear Autoregression Forecasting. Information Technology Journal, 11: 730-733.

DOI: 10.3923/itj.2012.730.733

URL: https://scialert.net/abstract/?doi=itj.2012.730.733
 
Received: January 18, 2012; Accepted: February 06, 2012; Published: February 28, 2012



INTRODUTION

Partially linear auto-regression model with parametric variable and nonparametric variable has more adaptability and explanatory ability than the parametric model and the nonparametric model. The wavelet is a perfect tool to deal with instability signals. This thesis has established models of Shanghai stock index and Shenzhen component index based on wavelet and time series models.

Traditional methods discretize the attributes before apply linear auto-regression model to the information system. Fayyad and Irani (1993) use a recursive entropy minimization heuristic for attribute discretization. In (Wang, 2001), naïve and semi naïve scaler methods are proposed to find the appropriate cut point of the continuous attribute. In this study, we construct the indiscernibility set of x by defining the toleration relation on the information systems. Using the indiscernibility set of x, we partitioned U and acquired the decision rule. This decision rule is influence by the system permitted error. The relation between the system permitted error and the effectiveness of the acquired decision rule is discussed through a numerical experiment. And we compared our methods with the traditional ones under the optimum system permitted error. It shows that our method is better than the traditional ones in many cases.

INFORMATION SYSTEM WITH CONT-INUOUS CONDITION ATTRIBUTES

First, let us have a look on the definition of decision information system.

Definition 1: A decision information system including, where:

U is a non-empty finite set of objects

Image for - A New Method on Partially Linear Autoregression Forecasting

A is a non-empty finite set of attributes

Image for - A New Method on Partially Linear Autoregression Forecasting

D is a non-empty finite set of decisions

Image for - A New Method on Partially Linear Autoregression Forecasting

F is the relation set between U and A

Image for - A New Method on Partially Linear Autoregression Forecasting

fj: U→vj (j≤q), Vj is the domain of aj
G is the relation set between U and D

Image for - A New Method on Partially Linear Autoregression Forecasting

gj: U→V’j (j≤q), V’j is the domain of dj

In order to solve the problem of continuous condition attributes discretization, we define the relationship Image for - A New Method on Partially Linear Autoregression Forecasting and Image for - A New Method on Partially Linear Autoregression Forecasting in the following way.

Definition 2: A decision information systems Xt = (Xt1,..., Xtp)T we define the toleration relation as the following:

Image for - A New Method on Partially Linear Autoregression Forecasting

Here δ≥0 is called a system permitted error.

Under this relation, the indiscernibility set of x is Wψf (a,b).

Hence, ∀X⊆U, we can get its approximation under the relation:

Image for - A New Method on Partially Linear Autoregression Forecasting

Let:

Image for - A New Method on Partially Linear Autoregression Forecasting

then RD is equivalent relation in X, its equivalent class is denoted as:

Image for - A New Method on Partially Linear Autoregression Forecasting

DECISION RULE ACQUISITION

According to Zhang et al. (2001, 2003a, b), Let:

Image for - A New Method on Partially Linear Autoregression Forecasting

Definition 1:

Image for - A New Method on Partially Linear Autoregression Forecasting

let:

Image for - A New Method on Partially Linear Autoregression Forecasting

Theorem 1: Suppose that (U,A,F,D,G) is an information system with continuous condition attributes, then all the non-empty: 0≤α≤1 (hj (t) = E (Xj|T = t)) form a partition of U.

Proof: (1) first, to prove that with different Yt = βTXt+ g (Zt)+εt, Xti (I = 1,...,p) ∩ Xt = (Xt1,..., Xtp)T = φ (i≠j), reduction to absurdity is used, supposed that Yt = βYt-1+ g (Yt-2t (t≥3), then Ψ (t) εL2 (R) lead to Image for - A New Method on Partially Linear Autoregression Forecasting:

Image for - A New Method on Partially Linear Autoregression Forecasting

so:

Image for - A New Method on Partially Linear Autoregression Forecasting

we can get the result as Di = Dj, which is obviously a confliction.

So:

Image for - A New Method on Partially Linear Autoregression Forecasting

(2) Then, let us prove that to all:

Image for - A New Method on Partially Linear Autoregression Forecasting

According to the definition of Image for - A New Method on Partially Linear Autoregression Forecasting, we know that Image for - A New Method on Partially Linear Autoregression Forecasting and then Image for - A New Method on Partially Linear Autoregression Forecasting, Whereas, ∀ xεU, suppose that:

Image for - A New Method on Partially Linear Autoregression Forecasting

and then:

Image for - A New Method on Partially Linear Autoregression Forecasting

therefore:

Image for - A New Method on Partially Linear Autoregression Forecasting

From (1) and (2) we can draw a conclusion that:

Image for - A New Method on Partially Linear Autoregression Forecasting

According to theorem 1, we can partition U as U = {C1, C2, C3,..., Ct}, in which:

Image for - A New Method on Partially Linear Autoregression Forecasting

And hence gain the decision rule:

Image for - A New Method on Partially Linear Autoregression Forecasting

Now, we call:

Image for - A New Method on Partially Linear Autoregression Forecasting

the reliability of Image for - A New Method on Partially Linear Autoregression Forecasting, Image for - A New Method on Partially Linear Autoregression Forecasting is called the precision of Image for - A New Method on Partially Linear Autoregression Forecasting. Then we obtain the rule with precision:

Image for - A New Method on Partially Linear Autoregression Forecasting

To be more practical, we normally transform the rule to attribute expression. We denote:

Image for - A New Method on Partially Linear Autoregression Forecasting

where:

Image for - A New Method on Partially Linear Autoregression Forecasting

This formula of rule can be expressed as:

Image for - A New Method on Partially Linear Autoregression Forecasting

To illustrate this decision rule acquisition process clearly, let us see some examples.

Example 1:

Image for - A New Method on Partially Linear Autoregression Forecasting

Let δ = 0.3.

We can accomplish the decision rule acquisition:

Image for - A New Method on Partially Linear Autoregression Forecasting
Image for - A New Method on Partially Linear Autoregression Forecasting

Hence, we have:

Image for - A New Method on Partially Linear Autoregression Forecasting

NUMBERICAL EXPERIMENT

We apply the method of rule acquisition to the famous Iris data.

We conduct the experiment in the following way:

Step 1: Given δ = 0
Step 2: Randomly chose these samples from the whole data set as training samples 15, 30, 45, 60, 75, 90, 105, 120, 135, 150
Step 3: Then acquire the rules by applying our methods on the training samples
Step 4: We gain the rules of f (t) = fm (t)+gm-1 (t)+...+g1 (t)
Step 5: Judge the whole data set by the rules we acquired and compare our judgments with the original decision of the data sets. We can gain the percentage of the data that are being judged correctly
Step 6: Repeat step (2-4) for 500 times. Then, compute the average percentage of correctness
Step 7: δ = δ+2.0 if δ<51 go to step 2

or else end

We convert the result into chart. With the value of δ as abscissa and the average percentage of correctness as y-axis, we plot ten curves according to different capability of training data.

Image for - A New Method on Partially Linear Autoregression Forecasting
Fig. 1: The initial data of (a) stock index and (b) component indexz

Table 1: Comparison of all the methods
Image for - A New Method on Partially Linear Autoregression Forecasting

From Fig. 1, we can see that as the capability of the training data becomes larger, the optimum δ slides away from the point δ = 0. When the capability of the training data exceed 75, most the optimum δ stays at the point δ = 8.0.

After the optimum system permitted error was gained, we compare our methods to the traditional ones. 90 samples from the whole data set were chosen randomly and the decision rules under the optimum system permitted error acquire were acquired. Then, we apply the acquired decision rule to the whole data set and gain the correctness of the rule. The comparison between our methods with traditional methods of discretizing the data is showed as follows.

Comparison: Rules derived from 90 samples applying to the whole data set.

In fact, our methods greatly improved the percentage of correctness in many cases. The Table 1 is only an example.

CONCLUSION

The main work of this study can be summarized as: First, a method of decision rule acquisition on decision information system with continuous condition attributes is given. This method avoids the step of data discretization and hence decreases the information which is lost in pretreatment. Second, the relation between system permitted error and the effectiveness of the decision acquisition is discussed through a numerical experiment. We also demonstrate that our method gets better results than traditional discretizing methods by comparison.

ACKNOWLEDGMENTS

This work is supported by the financial support from the Natural Science Research Project of Jiangsu Ordinary University (09KJB430008), the Opening Project of State Key Laboratory of High Performance Ceramics and Superfine Microstructure (SKL201111SIC) and Education Reform Project of NJUPT (JG00711JX39).

REFERENCES

1:  Zhang, W., J. Mi and W. Wu, 2003. Information System and Knowledge Discovery. Science Press, Beijing, China

2:  Zhang, W., W. Wu and J. Liang, 2001. Theory and Methods in Rough Sets. Science Press, Beijing, China

3:  Zhang, W., J. Mi and W. Wu, 2003. Knowledge reductions in inconsistent information systems. Chin. J. Comput., 26: 12-18.

4:  Fayyad, U. and K. Irani, 1993. Multi-interval discretization of continuous-valued attributes for classification learning. Int. Joint Conf. Artificial Intel., 13: 1022-1029.
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5:  Wang, G., 2001. Theory and Knowledge Acquisition of Rough Sets. 1st Edn., Xi'an Jiaotong University Press, Xi'an, China, Pages: 146

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