New interpolating functions for non-decreasing time series data

Nonlinear Analysis, Theory, Merbods & Applicarions. Vol. 30, No. 2, pp. 995-1006, 1997 Proc. 2nd World Congress of Nonlinear Analysts 0 1997 Ekevier Science Ltd Printed in Great Britain. All tights reswved

Pergamon

0362-546X/97

PII: SO362-546X(97)00065-5

NEW INTERPOLATING NON-DECREASING KUNIO OSHIMA,*

$17.00

+ 0.00

FUNCTIONS FOR TIME SERIES DATA

ICHIRO HOFUKU,**

and KATSUHISA

HORIMOTO*

*Faculty of Science and Engineering,

Science University of Tokyo in Yamaguchi, Daigaku-dori l-l-l, Onoda 756, Japan; and **Department of Mathematics, Tokyo Metropolitan College of Technology, Higashi-Ooi l-10-40, Shinagawa-ku 140, Japan Key words and phrases: Gompertz curve, locally modified Gompertz curve, new interpolating functions, modeling of new interpolating function, generating P-model or Q-model 1. INTRODUCTION

The purpose of this paper is to present an interpolating function which may give us more fitted results than that of an existing one for non-decreasing time series data. There are many exponential like functions to express accumulated data. We will chooseGompertz function among the existing functions and generate a modified function based upon Gompertz function [ 11.The generated function will be more accurate approximated function than that of the original one. In sectiona, we will present the definition and general properties of Gompertz function to build up common back ground. In section3, we will define tools which we will apply later to build up a modified function. These are called P-model and Q-model. It will also be shown the general properties of these models. In section4, the time series data will be presented ( see Table1 in section4). We will first construct a specific Gompertz function by using the data, say, gl(x). Then in the following four subsections, we will find an actual form of P-mode2 or Q-modeE according to the location of the points which the previous function should have intersected. We next will set differential equations, solve them, and find out a modified continuous function at each step, say WI(T), wz(z), w3(x) and W&X), respectively. Finally, we figure out the fitness between the original function and the modified one by means of the discrete a-norm. We also observe the limit of the modified function We as x goesto 00,and compare the limit of original Gompertz function with that of the function w4bh

In section

5, we will

give some comments 2. DEFINITION

and mention

a possible

application

of this method.

AND PROPERTIES

We will introduce the definition and the general properties of Gompertz function. Gompertz function has originally been developed and applied to a prediction for a population problem. Recently, it is also applied to predict cumulative problems such as cumulative soft-ware’s bugs or cumulative number of patients for a certain disease[2]. We will first consider a differential equation whose solution will be a Gompertz function.

995

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DEFINITION

2.1

A non-negative

Congress

of Nonlinear

real valued function,

ix (t) = y czbL

(y>O,

Analysts

(2.1)

O
(2.1)

defined on the interval 0 < t < 00,is said to be a Gompertz function [l]. 2.2 Suppose that r (t) = y ab’ is a Gompertz function. Let a = -1n a and /3 = -In b, then the solution of the differential equation of the form LEMMA

g = apexp ( -p t) d)

(2.2)

is a Gompertz function. Proof; Let x (t) = y ub’

=yexp(-ae-pt). Hence z(t)=yewp(-ae-pt).

1:2. 3 )

Next we differentiate the above function with respect to t, then s={Yexp(-ae-fit)}’ =Yap(-ae-pt)a/3exp(-pt) =ap exp ( -p t) x(t) .

Q.E.D.

Note (1) that the Gompertz function is monotone increasing ftmction, since the conditions (I, b, y imply dx/dt>O (2) and that /@z x(t) =y, since b+ e 0. Secondly, we will find a point of inflection for the function. LEMMA

2.3

A point of inflection for the function is the ordered pair ((Znayp, y / e). 3.GENEFtATING

P-MODEL

AN-D

Q-MODEL

The purpose of this paper is to find out a better modified continuous function with respect to time series data than an existing one which is a Gompertz function in this paper. The outline of this method is that we differentiate Gompertz function gl(x) which is found by applying given time series data and then will modify the differential equation by applying P-model or Q-model We next solve the modified differential equation and repeat this procedure at each step till the end of the data. This procedure will produce a better modified function. We will adopt the discrete a-norm for the distance between the data and a modified function on each point so that the formula of the distance will be

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where gi(r) and t(x) take each value at given time series x = 1,2,3,

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DEFINITION 3.1 The above value found by applying the formula is called the worst value of the function gi(r) with the data.

More precisely, we want to modify the function gi(r) to find a differentiable function whose norm with the data is better than the worst one defined in DEFINITION 3.1. It is noted here that gi(r) is not necessary to be Gompertz function, in general, it can be any function which we want to modify. The main procedures are the followings; [4-l]

Whenever a function passesthrough above the point, we want to make the function pull down locally in order to make it intersect the point. [4-21 Whenever a function passesthrough under the point, we want to make the function pull up locally in order to make it intersect the point. DEFINITION 3.2 The class of the function [4-l] is called P-model [4-21 is called Q-model.

and the class of the function

Note that there is the restriction on P-model that the derivative of a modified function by Pmodel is at most 0, since our data are accumulated one, i.e., the function can never be decreased. We first discuss the P-model which the function gi(z) will be made to pull down locally. Then we will next discuss the Q-model which the function g&c) will be made to pull up locally. DEFINITION

3.3

For any real number m, n, r, the function defined on the interval 0 <:x
is called the p ( m,n, r )(x)-model. The sketch of the models is in Fig.1. We fix r = 5, and n = 5, and take m = 0.5 which is the shallow one, m = 0.7 which is the middle one, and m = 0.9 which is the deepest one, respectively. NOTE 3.4

Since the data are non decreasing, a function which we want to modify forces to be constant from one point ( ~1, a ) to another ( x2, a ), i.e., the derivative of such a function is to 0 on the interval [ xi, ~2 1. Therefore we may have a point x: such that P(m,n, P )(x)=0

Since the function nm,n, r j(x) is symmetric with respect to line x = r - 0.5, we fur m = 1 in order to have pm,n, r )(x)=0. Henceforth we chooseRl,n, p)(x), whenever we want to obtain a constant part of a modifying function on a certain interval. If we solve a differential equation multiplied by pcl,A,r j(z), then its derivative will be 0 on the interval,

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The sketch of such functions is in Fig.2. We fix m = 1 and F = 5 and choose n = respectively. We next define Q( =,t, U)(x)-model and give an example of these functions. DEFINITION

3.5

1,6,11,

For any real number s, t, U, the function defined on the interval [ 0 S:x -ZOO )

is called the Q( S,t, UAx&model. We fix u = 5, and t = 5, and take s = 0.8, 1.0 and 1.2. The sketch of the functions is in Fig.3 where the lowest one, the middle one, and the highest one, respectively. i 0.8 0.6 0.4 0.2

0

2 Fig.

1. These

curves

6 show

the function

8 nm. 5,5 Ax)

m = 0.5, 0.7,0.9.

incaseof

0

4

2 Fig. 2. These incaseofn=1,6,11.

4 curves

show

6 the functiun

8 41, n.

5

$x(x>

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4

2

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6

8

Fig. 3. These curves show the function in case of s = 0.8, 1.0, 1.2

q(s. 5,5 Ax)

NOTE 3.6 The classes of P-model and Q-model contain infinitely the construction of the definitions. The set,

many functions

[ P(,,,, r j(x) : m, n, r are real numbers

1

1 Q(S,t, I )(x1 : s, t, u are real numbers

1

according to

or

is only a subset of the class P-model or Q-model. It is emphasized that there are many functions which make another function pull up or pull down by certain operators other than these defined models. 4. CONSTRUCTION

OF BETTER

FUNCTIONS

THAN

PREVIOUS

ONE

The time series data that we will work with is presented in Tablel: We begin with using the data which is in table1 and construct a Gompertz function, according to Section& we compute y, a, and P from the data then these will be y = 10.2, Let the function

a = exp(1.29802)

, /I = 0.545848.

be gi(x) then the limit y is

Table

x-axis Y-axis

1. Time

series

data.

1

2

3

4

1

4

4

7

that is,

loo0

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y = 10.2 and the correlation

coefficient,

p, between

gl(x) and the data is p = 0.9486863

therefore we may consider that the correlation between gl(x) and the data is reasonable. shows the sketch of Gompertz function gl(x) and the time series data which we apply. In DEFINITION 3.1, we defined the worst value. In this case the value is that

t a(x)

- t(x) I= kl

Our goal is to find out a function

which

lgl(x) -t(x)

improves

Fig. 4. This curve

4.1

FINDING

A FIRST

12= 2.15184.

the worst

shows

MODIFIED

Fig.4

value.

the function

FUNCTION

&)

wl(x)

/

We first refer the point ( x, y )=( 1,1 ) and the Gompertz function gl(x) in the Fig.4 ,then the point ( x, y ) = ( 1, 1) locates a little under the function so that we use P-model to modify the function gl
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l-0.2eq

Pl(~)=p(o,2,5,1)(x)=

Applying pi(x) = p( 0.2,5,1 )(x1 in P-mode& which intersects the point ( 1,l):

I dw ~ dx

of Nonlinear

= a&w

Analysts

-5

(

{

1001

x-

(

l-1

2112]

we now show the actual form of the function WI(X)

(-p3c)~1(3c)P(o.2,5,1)(x)

(4.1)

(-a)

w1(O)=yexp P(O.2,5,1)(4

= l-0.2acp

- 5 (x-o.5)2

It follows from (4.1) that

Wl(X) = 9.20214 exp --&(j~~~ i where

erf means that let t = 6(0.44515-x)

+ 0.154431

fierf

(6(0.445415

and then the error function

f(t) = 4

The sketch of this graph is in Fig.5.

I 1 Fig.

2 5.

3

4

5

This curve shows the function WI(X).

6

-Lx,]

I

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The norm 1wl( 1) - t (1) I= 0.000217306, where Hence

we have chosen m = 0.2 on pi m,n,r j(x). the WI(X)almost intersects the point ( 1,l ) . 4.2. FINDING

A SECOND

MODIFIED

FUNCTION

u&z)

Similarly, we wiII find wz(n) that intersects the point ( 2,4 ). Referring Fig.5, we can see that the graph pass through under the point ( 2,4 ) so that we make use of Q-model to pull the graph up to the point ( 2,4 ): For the case ( x, y ) = ( 2,4 ), we have found the function qdx) as follows;

___ I w(x)

'2

qc o.s,5,2Ax) = 1 + 0.8 exp - 5 (x - 1.5)2

=

-atiP

{

1

(-~dW2(~)P(O.2,5,l)(X)qC

08,5,2)(x)

wz(O)= yexp (-a)

Q(O.8,5,2)(X) = 1 + 0.8ap

(4.2)

-5(X - 1.5)2

It follows from ( 4.2 ) that

w2b)

= 12.1681ap

- 0.35789 fi

- 3.66204 + 0.00766169 @f

(m(O.972708 -LX))

erf (G(1.44542 - x,) - 0.154431@$

(G( - 0.445415 -x))}

The sketch of this graph is in Fig.6. The norm (We-WI=

~~l~w2(+t(x)/2

= 0.00582949

is a possibleminimum value where we have chosen the points ( 1,l) on q( 8, cu $z). 4.3 FINDING

A THIRD

MODIFIED

FUNCTION

and ( 2,4 ) and also vz= 0.2

203(x)

As we found W(X) in the last subsection, we similarly find w3(x) that intersects the point I’ 3,4 ). Referring F’ig.6, we can see that the graph pass through above the point ( 3,4 ) so that we make use of p( 1,n, r j(x) which we noted in Note3.4 :

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~=asQP(-P) (1 x

w3(0)

= yexp

7JJ3 x

PC 0 2,5,

1 ,(x1

qc 1,5, z,(x)

1003

p( 1, 3.4,3kr)

( -a)

P( 1,3.4,3)(x)

(4.3)

= 1 - exp - 3.4 (x - 2.5)2

It follows from ( 4.3 ) that

w3(2)

= 9.25271 exp ,,“;~f~~

- 0.0000103838

lO-‘j merfi4.97926

- 3.6606x))

dZerf (3.70119 - 2.89828x) + 0.130046 Jlterf(5.42636

+ 0.141544 merf(4.46176 - 0.00766169

- 5.62807x

fierf

- 1.84391x)+ 0.357879 (m&O.972708

@f{fi

(-

+ x ,)- 0.154431 fi{G

- 2.89828x)

1.44542 +z )} ( - 0.44515 +3c )}

The sketch of this graph is in Fig.7. The norm 1 w3(x)

is a possible minimum m= 1 ~np(~,~,~)(x).

- t(x)

I=

$,

- t(x) )2 = 0.132146

) w3(2)

value where we have chosen the points ( 1, 1 ), ( 2,4 ) and ( 3,4 ) and also

4.4 FINDING

A FORTH

MODIFIED

FUNCTION

w4(x)

We will similarly find w4(x) that intersects the point ( 4,7 ) . Referring F’ig.7, we can see that the graph passes through under the point ( 4,7 ) so that we make use of q( s, t, u j(x) which pulls a function up the point :

-1 dw4 cl5

- apexp

w4(0)

= yap

( -px,

w4(x)

p( 0.2, 5, 1 j(2)

Q( 1, 5,2)(X)

( -a)

q(1.3,5,4)(x) = 1 + 1.3 exp - 5 (x - 3.5)2 { 1

It follows from ( 4,4 > that

p( 1, 3.4,3)(X)

m1.3

,5,4)(X)

(4.4)

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1004

w4(x)

= 10.5919 exp ,&fgz

- 3.5857~

lo-l3 merfI8.32829 - 4.48952 x)

- 5.6280710m6Jiferf(4.97926-3.6606x) - 2.005804mzr~(7.71105-3.6606~) + 3.31183x 10e6 merf(9.076953.6606x) - 1.03838x 10m5dZe$(3.70119-2.89828x) + 0.141544 merf( 4.46176-1.84341x) +1.48672x lO~“~~{[email protected])} - 1.21462@f{m(2.47271-

LX+--0.195197Jt

x)}

+ LX)>- 0.00766169 af{6(-0.972708

+ 0.357879~erf{~(-1.44542

- 0.154431 flerf{Js(-

erf(G(3.44542-

+ x)}

0.445415 + x,}

The sketch of this graph is in Fig.8.

6

1

2

4

3

Fig. 6. This curve shows the fbnction

0

1

2

3

Fig. 7. This curve shows the fbnction

5

wz(x>.

5

4 w(x).

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6

Fig. 8. This curve shows the function

8 w4(x).

The norm jw4(x)

- t(x)

I=

x$1

] w4(2)

- t(x) 12= 0.14944

is a possible minimum value where we have chosen all four points. The difference

between (gl(x)

the norm - t(x)

I-

1gl(x)

- t(x) 1 and [204(x)

- t(r) I= 2.15184

(204(x)

- t(x) I is that

- 0.14944

= 2.00240

therefore we can find a modified function w4(x) better than the function gl(r) to express the behavior of the data as a continuous function. We also observe that the limit of the w.&) is; fiy..

w4(x)

= 10.7868.

On the other hand, the limit of the e(x) was; f*-

gi(x) = 10.2,

which tells us that the limit of w&r) is more appropriate than that of gl(r) to express the behavior of the data as x goes to m. 5. COMMENTS Modified functions that we have developed have the following four properties; 1. A modified function will be more appropriate function than an original one (Gompertz function, in this case) to interpolate the data as a smooth continuous function. 2. A modified function can be expressed as only one function, have as many as inflection points and is differentiable on the interval.

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3. A modified function can intersect a small neighborhood that we want the function to intersect. 4. A modified function can interpolate all discrete data and is smooth so that the limit, as a virtual extension of the data, can have more meaning than the original limit. There are very few existing interpolations that satisfy the above four conditions. For ex.ample, a polynomial interpolation satisfy the item 2 and 3 but not the item 1 or 4. Most of ,Spline interpolations satisfy item 1 and 3 but not 2 or 4. Actually, the interpolation is a set of piecewise continuous functions. We apply this method to predict the number of male adults who are HIV positive every year from 1996 to 2020 [21. We will mention the choice of the parameters in P-model and Q-model. It has not been found the best theoretical way to find those parameters, We have used a heuristic way, i.e., we first fur r and u, next choosen and t. We finally move m and s by referring the norm at each point (see also Fig.1, Fig.2 and Fig.3). There is room for investigating a rather theoretical way to find the values of those parameters.

REFERENCES 1. K. OSHIMA

Transactions 2. K. OSHIMA Predict

& I. HOFUKU

., Two stq cumulativecurve and

modified

Gompertz

curve,

of JSZAM, 4,259-2’74 (1994). & I. HOFUKU

the Number

of HIV

& K. HORIMOTO Positive,

., A Locally

Transactions

Modilied

Gampertz

of JSZAM, (in process).

Curve

and Its Application

to