Abstract— Exponential curve prediction model is one of

the most common prediction models[1]. This paper is mainly introduce improved model which add one-time items based on modified exponential curve prediction model. In this article, we not only proved the characteristics of the historical data which consistent with this model in theory but also describes how to solve the unknown parameters according to the given time series. Improved model overcomes the shortcomings of modified exponential curve prediction model which is only a monotonic function model. So the improved model has more extensive application value. Then, we promoted the conclusions at last.

I. INTRODUCTION TATISTICS show that the development of socio-economic is progressive which have a certain

regularity with the respect of time. Therefore, when the prediction object showing a trend of rise or a downward, and no significant seasonal fluctuations according to the time, we can use a right curve of function to reflect this trend, that means we can use trend extrapolation to predict the future value. Four trend prediction models have been proposed by predecessors, they are as follows: polynomial curve prediction model[1][2],exponential curve prediction model[3]-[5], logarithmic curve prediction model[6]and growth curve prediction model[7].The modified exponential curve prediction model have a wide range of application in engineering and economic fields[8]-[12]. Each model has their own advantages. In this paper, we make further improvements based on the modified exponential curve prediction model, we add polynomial to the modified exponential curve prediction model. Here, we take one polynomial for

instance: tty a bt kc

∧= + + , where t is a time

variable,^

ty is a timing values. In fact the improved model is a combination of polynomial curve prediction model and exponential curve prediction model. Next, we will introduce the improved model as follows: the model proposed, the model identification, determination of model parameters as well as the advantages and disadvantages of the model, the promotion of the model also will be mentioned at last. Our biggest regret is that we haven’t found the right data to verify the improved prediction model of the modified exponential curve. However, we believe that there must be exist this model in our life and the model must have values.

Yuzhen Lu and Qing Li are with the Department of Mathematics, The Dalian Maritime University, Dalian, Liaoning, China (email: {dlyuzhen,liqing912809839}@sina.com).

II． MODEL PROPOSED Actually, modified exponential curve prediction model can be seen as an improved exponential curve prediction model. The following is a brief presentation of modified exponential curve prediction.

tty a bc

∧= + (1)

Where t is a time variable,^

ty is a timing values. Although model (1) has a widely used in our life, there are still many limitations for itself. Modified exponential curve prediction model only fit for a monotonically increasing or monotonically decreasing characteristic time series because it is a monotonic function. In order to overcome this shortcoming, we make this model for further improvement. Therefore, here we propose the improved prediction model of the modified exponential curve:

tty a bt kc

∧= + + (2)

Where t is a time variable,^

ty is a timing values. In order to analysis model (2), we get the derivation of function (2):

'

lnc tty b k c

∧= + (3)

From the expressions of the derivation of the function (2) we can find that (1) changed greatly after add polynomial to modified exponential curve prediction model. (2) is no longer simply describes a monotonically increasing or monotonically decreasing time sequence because (3) exist null point when parameters have appropriate values. That means model (2) exists increasing and decreasing time interval when parameters have appropriate values.

III. IDENTIFICATION OF THE MODEL For a given time series, we should select the appropriate forecasting model first, but how to select it? This required us to identify the characteristics of respective model.

1.Image Recognition Method This method is to draw a scatter plot with given data,

then compare the scatter plot with the roughly graphics of model (2), If their images are similar to each other, you can try to use this model as our prediction model. Roughly graphics of the function are as follows when parameter values in different ranges.

Improved Prediction Model of Modified Exponential Curve

Yuzhen Lu, Qing Li and Yuman Guo

S

2012 Third International Conference on Intelligent Control and Information Processing July 15-17, 2012 - Dalian, China

978-1-4577-2143-4/12/$26.00 ©2012 IEEE 513

In order to highlight the trend of the curve, we let the parameters of model (2) very small. Although the image recognition method is intuitive, it is not always reliable. It can only help us quickly and intuitively exclude inappropriate prediction model. Therefore, we need a rigorous mathematical derivation to illustrate the characteristics of the data which may satisfy the model (2). (i) as 0 1c< < .

① 0, 0b k> > ② 0, 0b k> <

③ 0,0 >< kb ④ 0,0 << kb (ii) as 1c > .

① 0,0 <> kb ② 0, 0b k> >

③ 0,0 >< kb ④ 0,0 << kb

For the improved prediction model of modified exponential curve, we still use difference method to Identify. The specific method is as follows.

2.Difference Method In order to select the correct model based on historical

data. We often use the difference method to change the original time series into stationary series, we use the first order ratio of second-order difference of to identify the improved prediction model of modified exponential curve. Listed below are the difference calculate table.

Ti-ming ）t

model(2)

ty a bt= + tkc+

first order difference

1'

−−= ttt yyy

second-order differential

''''−−= ttt yyy

diff-eren-tial ratio

''

''1

t

t

yy −

1

a bkc+

+

-

-

-

2 2

2a bkc+

+

)1( −+ ckcb

-

-

3 3

3a bkc+

+

)1(2 −+ ckcb

2)1( −ckc -

4 4

4a bkc+

+

)1(3 −+ ckcb

22 )1( −ckc c

1t −

( 1)a t b+ −1tkc −+

)1(2 −+ − ckcb t

23 )1( −− ckct

c

t t

a tbkc+

+

)1(1 −+ − ckcb t

22 )1( −− ckct

c

It easily can be seen from the table that the first order ratio of second-order difference of model (2) is equal to a constant. In other words, we can use this model to predict as long as the first-order ratio of the second-order difference of a time sequence is close to a constant. In a word. For a given time series, firstly, we can use scatter plots to preliminary prediction model, then use difference method to determine the final prediction model. If the first-order ratio of second-order difference is close to a constant, then we can choose the improved prediction model of the modified exponential curve as prediction model.

IV. DETERMINATION OF MODEL PARAMETERS

, ,a b c and k are undetermined parameters in model (2).Here we use the grouping method to solve undetermined parameters of model (2).There are four parameters in model（2）. So we divided the entire time series into four groups by dividing group method, and each group have the same number of items, then simultaneous equations to get the result. Let the number of each items is n , let the sum of first group is

1

0

n

ii

Ay−

=∑ ,the second group is

2 1n

ii n

By−

=∑ ,the third group

is3 1

2

n

ii n

Cy−

=∑ and the fourth group is

4 1

3

n

ii n

Dy−

=∑ ,specific

solution procedure is as follows:

514

11

0( 1)

nn

ii

Ay na b n b k kc−

−

== + + + − + + +∑

( 1) ( 1)=

2 1

nn n cna b kc

− −+ +−

(4)

As the same reason we can get:

2 1 (3 1) ( 1)

2 1

n nn

ii n

n n c cBy na b kc

−

=

− −= + +−∑ (5)

23 1

2

(5 1) ( 1)2 1

n nn

ii n

n n c cCy na b kc

−

=

− −= + +−∑ (6)

34 1

3

(7 1) ( 1)2 1

n nn

ii n

n n c cDy na b kc

−

=

− −= + +−∑ (7)

The result of (5) minus (4) is as follows: 22 1 1

2

0

( 1)1

nn n

i ii n i

cBy Ay n b kc

− −

= =

−− = +−∑ ∑ (8)

The result of (6) minus (5) is as follows: 23 1 2 1

2

2

( 1)1

n nn n

i ii n i n

c cCy By n b kc

− −

= =

−− = +−∑ ∑ (9)

The result of (7) minus (6) is 2 24 1 3 1

2

3 2

( 1)1

n nn n

i ii n i n

c cDy Cy n b kc

− −

= =

−− = +−∑ ∑ (10)

The ratio of (10) minus (9) and (9) minus (8) is 4 3 1 2 1 2

3 23 1 2 1 1 2

2 0

( 1)21

( 1)21

n i n n n n

i i ini n i n i n

n n n n

i i ii n i n i

c c kDy Cy Byc c

c kCy By Ayc

− − −

= = =− − −

= = =

−− +−= =

−− +−

∑ ∑ ∑

∑ ∑ ∑ (11)

According to (11) we can get that 1

4 1 3 1 2 1

3 23 1 2 1 1

2 0

2

2

n n n n

i i ii n i n i n

n n n

i i ii n i n i

Dy Cy Byc

Cy By Ay

− − −

= = =− − −

= = =

⎛ ⎞− +⎜ ⎟⎜ ⎟=⎜ ⎟− +⎜ ⎟⎝ ⎠

∑ ∑ ∑

∑ ∑ ∑ (12)

So the result of , , ,a b c k is as follows: 1 2 1 1

20 0

2 1 1

02

3

14 1 3 1 2 1

3 23 1 2 1 1

2 0

( 1) ( )2 1 ( 1)

1

( 1)( 1)

2

2

n n n

i i i n ni i n i

n n

i i ni n i

n

n n n n

i i ii n i n i n

n n n

i i ii n i n i

n p pAy By Ayn c ca

npBy Ay

cbn

p ckc

Dy Cy Byc

Cy By Ay

− − −

= = =

− −

= =

− − −

= = =− − −

= = =

⎧ −− − − −⎪ − −=

− −−=

⎨ −=−

⎛ ⎞− +⎜ ⎟⎜ ⎟=⎜ ⎟− +⎜ ⎟⎝ ⎠

∑ ∑ ∑

∑ ∑

∑ ∑ ∑

∑ ∑ ∑

⎪⎪⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪⎪⎪⎩

Where3 1 2 1 1

2 0

2n n n

i i ii n i n i

p Cy By Ay− − −

= = =

= − +∑ ∑ ∑ . In order to

save the layout, where , ,a b k has not be replaced by the expression of c .

V. ADVANTAGES AND DISADVANTAGES OF THE MODEL Advantages: (i) Due to the improved of exponential

curve predict model added a one-time items, so it overcomes the traditional exponential curve prediction model that can only predict monotonous time series. (ii) Due to the improved of exponential curve predict model added a one-time items, so it can increase or mitigate the speed of the curve within a time frame which can be used to predict the increase or decrease in range of asymmetric time series, overcome the limitation of polynomial prediction. Disadvantages:(i) As the model contains an exponential function, so with the change of time increases the curve will be obvious, which may emerge a phenomenon: the longer time, the forecast results the worse, therefore, it is necessary to update the data when we use this model for a long-term forecasts.(ii) This model requires more historical data because it used the difference method and the grouping method in solving process,.

VI．THE PROMOTION OF THE MODEL The model of add one-time items based on model (1)

was Introduced above. However, if the polynomial we add have n times, then what will happen? Here only give a brief introduction about this problem. We can use the same method which described above to identify and determine the parameters of this promotion of the model, and the result is similar to model (2).If we add n time polynomial to the right of model (1), then the characteristics of time series is that the first order ratio of

+1n order difference is closed to a constant. We can also use the grouping method to determine the parameters. Finally, we get the promotion of the model:

21 2 3 1

n tnty a a t a t a t kc

∧

−= + + + + + .

Where ia 1, 2, , 1 2fori n n= − ≥， ） and k are unknown parameters of this model.

VII. CONCLUSIONS For a given time series, we can use scatter plots to

choose prediction model, then use difference method to determine the right prediction model. If the first-order ratio of +1n order differential is close to a constant, then

we can choose 21 2 3 1

n tnty a a t a t a t kc

∧

−= + + + + + 1, 2, , 1 2fori n n= − ≥， ） as the prediction model.

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