INTRODUTION
Partially linear autoregression 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 autoregression
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 CONTINUOUS 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 nonempty finite set of objects 
• 
A is a nonempty finite set of attributes 
• 
D is a nonempty finite set of decisions 
• 
F is the relation set between U and A 
• 
f_{j}: U→v_{j} (j≤q), V_{j}
is the domain of a_{j} 
• 
G is the relation set between U and D 
• 
g_{j}: U→V’_{j} (j≤q), V’_{j}
is the domain of d_{j} 
In order to solve the problem of continuous condition attributes discretization,
we define the relationship
and
in the following way.
Definition 2: A decision information systems X_{t} = (X_{t1},..., X_{tp})^{T} we define the toleration relation as the following:
Here δ≥0 is called a system permitted error.
Under this relation, the indiscernibility set of x is W_{ψ}f (a,b).
Hence, ∀X⊆U, we can get its approximation under the relation:
Let:
then R_{D} is equivalent relation in X, its equivalent class is denoted as:
DECISION RULE ACQUISITION
According to Zhang et al. (2001, 2003a,
b), Let:
Definition 1:
let:
Theorem 1: Suppose that (U,A,F,D,G) is an information system with continuous condition attributes, then all the nonempty: 0≤α≤1 (h_{j} (t) = E (X_{j}T = t)) form a partition of U.
Proof: (1) first, to prove that with different Y_{t} = β^{T}X_{t}+
g (Z_{t})+ε_{t}, X_{ti} (I = 1,...,p) ∩ X_{t}
= (X_{t1},..., X_{tp})^{T} = φ (i≠j), reduction
to absurdity is used, supposed that Y_{t} = βY_{t1}+ g
(Y_{t2}+ε_{t} (t≥3), then Ψ (t) εL^{2}
(R) lead to :
so:
we can get the result as D^{i} = D^{j}, which is obviously a confliction.
So:
(2) Then, let us prove that to all:
According to the definition of ,
we know that
and then ,
Whereas, ∀ xεU, suppose that:
and then:
therefore:
From (1) and (2) we can draw a conclusion that:
According to theorem 1, we can partition U as U = {C_{1}, C_{2}, C_{3},..., C_{t}}, in which:
And hence gain the decision rule:
Now, we call:
the reliability of ,
is called the precision of .
Then we obtain the rule with precision:
To be more practical, we normally transform the rule to attribute expression. We denote:
where:
This formula of rule can be expressed as:
To illustrate this decision rule acquisition process clearly, let us see some examples.
Example 1:
Let δ = 0.3.
We can accomplish the decision rule acquisition:
Hence, we have:
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) = f_{m} (t)+g_{m1} (t)+...+g_{1}
(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 (24) 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 yaxis, we plot ten curves according to different capability of training data.

Fig. 1: 
The initial data of (a) stock index and (b) component indexz 
Table 1: 
Comparison of all the methods 

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).