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A Novel Time Series Forecasting Approach with

Multi-Level Data Decomposing and Modeling∗

Xuemei Han, Congfu Xu, Huifeng Shen and Yunhe Pan

College of Computer Science,

Zhejiang University,

Hangzhou, 310027, P.R.China

zjuwendy@yahoo.com.cn, xucongfu@cs.zju.edu.cn

useraddcn@yahoo.com.cn, panyh@zju.edu.cn

Abstract—Time series produced in complex systems are

always controlled by multi-level laws, including macroscopic and

microscopic laws. These multi-level laws bring on the combination

of long-memory effects and short-term irregular fluctuations in

the same series. Traditional analysis and forecasting methods do

not distinguish these multi-level influences and always make a

single model for prediction, which has to introduce a lot of

parameters to describe the characteristics of complex systems

and results in the loss of efficiency or accuracy. This paper

goes deep into the structure of series data, decomposes time

series into several simpler ones with different smoothness, and

then samples them with multi-scale sizes. After that, each time

series is modeled and predicated respectively, and their results

are integrated finally. The experimental results on the stock

forecasting show that the method is effective and satisfying, even

for the time series with large fluctuations.

Index Terms—time series forecasting, complex system, data

decomposing, multi-scale sampling

I. INTRODUCTION

Time series is one of the most frequently encountered

forms of data. The forecasting of time series is an important

topic in data mining since it plays great role in the economic

decision making, prevention of natural calamity, and so on.

This paper pays more attention to the time series in com-

plex systems, such as financial data, meteorologic data, etc.

Since these data are controlled by multi-level laws, including

macroscopic and microscopic laws, they are more stochastic

than other types of data, such as engineering data and physical

data. Statisticians, economists and mathematicians have paid

much attention to the analysis of time series structure of

complex systems. Most of economists consider that time series

are the components of trends, cycles, seasonal variations and

irregular fluctuations [1]. In 1951, Hurst [2] investigated the

long-term storage capacity of reservoirs and found the long-

memory trait of the hydrology series firstly. After 1980s,

researchers found this trait was very common in time series of

different areas [3].

On the other hand, Rosenblatt [4] presents the concept, st-

——————————————–

∗This paper is supported by National Natural Science Foundation of China (No. 60402010), Advanced Research Project of China Defense Ministry (No.

413150804), and partially supported by the Aerospace Research Foundation (No. 2003-HT-ZJDX-13).

Congfu Xu is the correspondent author.

rong

process in time series.

Many methods are proposed to cope with time series fore-

casting, such as Box-Jenkins [5][6], Neural Networks [7][8],

Genetic Algorithms [9][10], Kalman filter method [11], etc.

These methods generally construct a single model with compli-

cated parameters on the raw data when describing the complex

system, while ignore preprocessing before modeling. However,

since both long term trends and short term fluctuations coexist

in the same sequence, it is a dilemma to balance the accuracy

and the efficiency: discarding mass historic data which may be

useful for analysis and forecasting will decrease the accuracy,

while giving the same weight to the historic data will increase

the processing time inevitably.

This paper proposes a new forecasting approach with

multi-level decomposing and multi-scale sampling. In our

method, the time series are decomposed into several simpler

ones which are called as separated-series, then we sample

every new separated-series with diverse scales. The new

separated-series can be described with different models or the

same model with different parameters. Then the forecasting of

every new separated-series is conducted. The final forecasting

results of the original series can be obtained by integrating

those of separated-series.

The rest of this paper is organized as follows. In Section

II, we give several definitions and formulations of time series

preprocessing. Section III explains the modeling and forecast-

ing processes. Experimental results of the approach are shown

in Section IV. The last section offers our conclusion.

mixingcondition,which reflectstheshort-term

II. TIME SERIES PREPROCESSING

The data produced in complex systems are often influenced

by multi-level factors, and the influence periods of these

factors are diverse. Some factors may result in a long-term

evolution, for example, the inherent value mechanism of

stock, which is decisive in the long-term trends of stock

price. While some factors’ durations are much shorter, such

as people’s psychological factors in the stock market, which

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may always cause short-term fluctuations. This requests

multi-scale models which can deal with not only long-term

trends forecasting but also short-term fluctuations prediction.

In this paper, original time series are decomposed into

several ones with different cycles. The traits of the new time

series are more visible and the new time series are easier

to be modeled. After decomposing, each new time series is

sampled with different cycle which can bring the gain of

efficiency without decreasing the accuracy. This section gives

the formulation of the approach.

A. Definitions

We start by giving some definitions related to time series

which may be convenient for describing our approach.

Definition 1 (standard sampling cycle): Standard sampl-

ing cycle, which is denoted as δ, is the original interval of

recordings or statistics. This paper assumes that δ is invariable

for a given system, and the other sampling cycles are all

integral multiple of δ.

Definition 2 (operator ⊕ and separated-series): T is an

ordered set {t1, t2, ..., tn, ...} with ti+1− ti = δ,i ∈

N, and 1 ≤ i ≤ n. A time series X = {xti,t ∈ T} can be

decomposed to several time series and this operation is denoted

as ⊕

X = X(1) ⊕ X(2) ⊕ ... ⊕ X(m),

(1)

and the i-th position of the sequence X(j) satisfies

m

?

j=1

x(j)ti= xti,

1 ≤ j ≤ m.

X(j) is named as separated-series which is equinumerous to

X, and the sampling time of the same order as X is the same.

B. Time Series Decomposing

In this subsection we present a method to decompose

time series into several separated-series satisfying equation

(1). This method, which we name as multi-smoothness-factor

decomposing, is an inductive process as below:

a) Set smoothness-factor array: Given an array

L = {l1,l2,...,lm−1}

where li ∈ N, 1 ≤ i ≤ m − 1, L is a monotonically

decreasing sequence, for instance, a binary-base exponent

array like {26,24,22,21}. L can be determined by experiments

or experiences.

b) Evaluation of X(1): By using smoothness-factor l1,

the p-th element in X(1) can be calculated by

⎧

⎪

x(1)tp=

⎪

⎪

⎪

⎩

⎨

1

p

1

l1

p?

q=1xtq,

p?

p ≤ l1+ 1

q=p−l1

xtq,p > l1+ 1

c) Evaluation of X(j), 1 < j < m: Elements of

time series X(j) can be gotten by subtracting elements of

X(1),X(2),...,X(j−1) from X with the same orders. Using

smoothness-factor lj, the p-th element in X(j) can be calculated

by

⎧

⎪

d) Evaluation of X(m): Elements of time series X(m)

can be gotten by subtracting elements of X(1), X(2), ...,

X(m − 1) from X with the same orders.

Algorithm 1 shows the process of evaluating x(j)tp.

x(j)tp=

⎪

⎪

⎪

⎩

⎨

1

p

1

lj(

p?

q=1xtq,

p?

p ≤ lj+ 1

j−1

?

q=p−lj

xtq−

k=1

x(k)tq),p > lj+ 1

Algorithm 1: Evaluate-Elements-of-X(j)

Data: X, L

Result: x(j)tp, 1 ≤ j ≤ m

begin

x(1)tp←− xtp

for j = 1 to m − 1 do

if p ≤ lj+ 1 then

x(j)tp←− (x(j)tp−1∗ (p − 1) + xtp)/p

else

x(j)tp←− (x(j)tp−1∗lj−x(j)tp−lj−1+xtp)/lj

xtp←− xtp− x(j)tp

x(m)tp←− xtp

end

Therefore, the original time series is decomposed into

several ones with different traits. The number of X(j) can

be determined by experiments, in most cases, 3 to 5 may

be appropriate. The separated-series will turn from smooth to

coarse with the increment of j.

C. Multi-Scale Sampling

A typical decomposing result is shown in fig. 1. The orig-

inal time series is decomposed into three separated-series by

two smoothness factors. We can find that the separated series

turn from smooth to coarse and the periods of fluctuations

become shorter with the increment of j. For the smoother

separated-series, since they reflect the extending of long-term

trends, their long-term history must be considered in modeling.

However, each point in the smoother separated-series carries

less information than that in the coarse ones. It is a waste

of time if the smoother separated-series are sampled with

standard sampling cycle δ. Multi-scale sampling has been

discussed in recent years [12]. The essence is that the new

data are sampled with high frequency and the older data with

lower frequency in the same time series. This seems simple,

but the variable measurements always make an extra trouble

for the later processes.

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50100150200 250300350400450500

−5

0

5

10

15

X(1)

X(2)

X(3)

Fig. 1.A typical figure of separated-series.

This paper proposes a new multi-scale sampling method.

We sample the same separated-series with the constant fre-

quency, while the sampling frequencies are variable when

handling the different separated-series. Let the ordered set

S={s1,s2,...,sm} be the sampling cycle array, where sj(1 ≤

j ≤ m) is the sampling cycle of X(j) and the elements of S

are monotonically decreasing. To be convenient for calculation,

we choose sj−1/sj∈ N. The new time series sampling from

X(j) is denoted as˙X(j).

III. MODELING AND FORECASTING

In the following subsections, we show the process of

modeling and forecasting. The process of new data entering

and other details related to modeling and forecasting are also

discussed.

A. New Data Entering

The time series analysis and forecasting is always an online

process. New data will enter continuously in the process of

forecasting, and they should be processed at once. Algorithm 2

shows the update procedure of sampled separated-series when

new data entering.

Algorithm 2: New-Data-In

Data: Sampled separated-series{˙X(j)}, xtk, L,S,

{CountS[j]} is set of sampling counters.

Result: {˙X?(j)} with new data, 1 ≤ j ≤ m

begin

Evaluate-Elements-of-X(j)

for j = 1 to m do

CountS[j] ←− CountS[j] + 1

if CountS[j] mod S[j] = 0 then

˙X(j) ←−˙X(j) + {x(j)tk}

end

B. Construct Forecasting Models

Neural Networks, Genetic Algorithms, etc., can be em-

ployed for the separated-series modeling. Different models

or one model with different parameters can be adopted for

different separated-series. Here we exemplify the modeling and

forecasting process by using Box-Jenkins method.

A stationary time series {yk} of mean zero can be taken

for responses of linear time-invariant random system with the

input of white noise. Then yksatisfies the difference equation

yk=

p

?

i=1

φiyk−i+ εk−

q

?

j=1

θjεk−j.

(2)

which is denoted as ARMA(p,q), where

the weighted sum of the most recent p responses, and

?q

can be obtained by the least squares method or maximum

likelihood estimating method, whereafter, ykcan be predicated

by equation 2. Details of constructing ARMA model can be

found in [3].

Assume that there are H sampling points in ˙X(j), then

the processes of modeling are:

1) Select the most recent h sampling points, h ≤ H. If

˙X(j) is not a stationary sequence, it must be transformed

to be a stationary one by first or multistage difference

method .

2) Construct an ARMA model for the sampling points with

appropriate parameters. The discussion of determining

the order of ARMA model can be seen in [13]. The

reasonability of the model should be tested through the

Box Pierce test [3].

3) Predict the value of next moment by using equation 2.

4) Add new data and compare the difference of new data

with the prediction of the step 3. The residuals should

be calculated.

The model should be verified periodically. For the time-

variant complex system, the parameters may not be appropriate

for prediction any longer with the passage of time. When the

residuals can not satisfy presumable assumptions, the most

recent h sampling points are selected and the model is updated

through step 1 and step 2.

?p

i=1φiyk−i is

j=1θjεk−jis the weighted sum of the recent q white noise.

The estimation of φ1, φ2, ... φp, θ1, θ2, ... θq and σ2

ε

C. Result Integration

Now the prediction of each separated-series is gotten. The

sampling cycle of each˙X(j) is different, the prediction cycle

turns shorter as the sampling frequency goes higher. They are

very useful, for example, from the prediction of the longer

sampling cycle, the potential trend of the time series can be

seen. But for the short-term accurate prediction, the results of

all the separated-series must be integrated to obtain the final

forecasting result.

As shown in fig. 2, the final forecasting results should be

integrated from those of all the frequency sampling sequences.

Assuming to predict the result of the moment τ, the prediction

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Fig. 2.Relations between multi-scale frequency series

of τ of each separated-series should be obtained according to

equation 1. Since the sampling cycles are different, not all

the predictions of τ can be obtained directly, but the nearest

observing and forecasting are available. Since these cases most

happen on the smoother separated-series, the absent predic-

tions can be calculated by polynomial interpolation simply.

Therefore, we get the final result.

IV. EXPERIMENTS AND RESULTS

Stock data are chosen to validate our approach in this

paper. Some forecasting results are demonstrated in Figures

3, 4 and 5. We choose 530 days close prices of Minsheng

Bank for modeling and forecasting. The smoothness array

L = {60,15} , which decompose the original time series into

three separated-series. As fig. 3 shows, the top left subplot

is the fitting curve of the original time series, and the other

subplots are the fitting curves of separated-series.

The sampling cycle array S is chosen as {10,5,1} accord-

ing to the smoothness of each separated-series. The final fitting

result is shown in fig. 4 (Close price as the vertical axis and

time as the horizontal axis).

100200300400 500

5

10

15

100 200300 400500

5

10

15

100 200 300400 500

−5

0

5

100 200300 400500

−5

0

5

Fig. 3.The original time series and the separated-series

The curve of relative errors is shown in fig. 5. The mean

of relative errors is 0.0105, and 92.83% relative errors are

smaller than 3%. Compared with the approaches which are

used for modeling with fixed parameters such as in [14][15],

our approach gets better forecasting results.

50100150 200250300350 400450 500

5

6

7

8

9

10

11

12

13

14

15

16

Observed data

Forecasting data

Fig. 4.The fitting and forecasting curve of Minsheng Bank stock

50 100150200250300350 400450500

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Fig. 5. Fitting curve of errors

V. CONCLUSION

In this paper, we propose a new approach for time series

analysis and forecasting. Through multi-level decomposing,

the complex time series turns to be a series of simpler

ones, then the separated-series are sampled according to their

smoothness and modeled respectively. Treating the long-term

trends and short-term fluctuations with different weight bring

on gains of accuracy and efficiency.

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