# Cardinal Basis Piecewise Hermite Interpolation on Fuzzy Data

Authors: H. Vosoughi; S. Abbasbandy
Journal: Advances in Fuzzy Systems
 (2016)

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## Abstract

A numerical method along with explicit construction to interpolation of fuzzy data through the extension principle results by widely used fuzzy-valued piecewise Hermite polynomial in general case based on the cardinal basis functions, which satisfy a vanishing property on the successive intervals, has been introduced here. We have provided a numerical method in full detail using the linear space notions for calculating the presented method. In order to illustrate the method in computational examples, we take recourse to three prime cases: linear, cubic, and quintic.


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## Body

## 1. Introduction

Fuzzy interpolation problem was posed by Zadeh [1]. Lowen presented a solution to this problem, based on the fundamental polynomial interpolation theorem of Lagrange (see, e.g., [2]). Computational and numerical methods for calculating the fuzzy Lagrange interpolate were proposed by Kaleva [3]. He introduced an interpolating fuzzy spline of order l. Important special cases were l=2, the piecewise linear interpolant, and l=4, a fuzzy cubic spline. Moreover, Kaleva obtained an interpolating fuzzy cubic spline with the not-a-knot condition. Interpolating of fuzzy data was developed to simple Hermite or osculatory interpolation, E(3) cubic splines, fuzzy splines, complete splines, and natural splines, respectively, in [4–8] by Abbasbandy et al. Later, Lodwick and Santos presented the Lagrange fuzzy interpolating function that loses smoothness at the knots at every α-cut; also every α-cut (α≠1) of fuzzy spline with the not-α-knot boundary conditions of order k has discontinuous first derivatives on the knots and based on these interpolants some fuzzy surfaces were constructed [9]. Zeinali et al. [10] presented a method of interpolation of fuzzy data by Hermite and piecewise cubic Hermite that was simpler and consistent and also inherited smoothness properties of the generator interpolation. However, probably due to the switching points difficulties, the method was expressed in a very special case and none of three remaining important cases was not investigated and this is a fundamental reason for the method weakness.In total, low order versions of piecewise Hermite interpolation are widely used and when we take more knots, the error breaks down uniformly to zero. Using piecewise-polynomial interpolants instead of high order polynomial interpolants on the same material and spaced knots is a useful way to diminish the wiggling and to improve the interpolation. These facts, as well as cardinal basis functions perspective, motivated us in [11] to patch cubic Hermite polynomials together to construct piecewise cubic fuzzy Hermite polynomial and provide an explicit formula in a succinct algorithm to calculate the fuzzy interpolant in cubic case as a new replacement method for [4, 10].Now, in this paper, in light of our previous work, we want to introduce a wide general class of fuzzy-valued interpolation polynomials by extending the same approach in [11] applying a very special case of which general class of fuzzy polynomials could be an alternative to fuzzy osculatory interpolation in [4] and so its lowest order case (m=1), namely, the piecewise linear polynomial, is an analogy of fuzzy linear spline in [3]. Meanwhile, when m=2 with exactly the same data, we will simply produce the second lower order form of mentioned general class that was introduced in [11] and the interpolation of fuzzy data in [10].The paper is organized in five sections. In Section2, we have reviewed definitions and preliminary results of several basic concepts and findings; next, we construct piecewise fuzzy Hermite polynomial in detail based on cardinal basis functions and prove some new properties of the introduced general interpolant (Section 3). In Section 4, we have produced three initial, linear, cubic [11], and quintic cases and shown the relationship between some of the mentioned cases and the newly presented interpolants in [3, 4, 10]. Furthermore, to illustrate the method, some computational examples are provided. Finally, the conclusions of this interpolation are in Section 5.


## 2. Preliminaries

To begin, let us introduce some brief account of notions used throughout the paper. We shall denote the set of fuzzy numbers byRF the family of all nonempty convex, normal, upper semicontinuous, and compactly supported fuzzy subsets defined on the real axis R. Obviously, R⊂RF. If u∈RF is a fuzzy number, then uα={x∈R∣u(x)≥α}, 0<α≤1, shows the α-cut of u. For α=0 by the closure of the support, u0=cl{x∣x∈R,u(x)>0}. It is well known the α-cuts of u∈RF are closed bounded intervals in R and we will denote them by uα=[u_α,u¯α]; functions u_(·),u¯(·) are the lower and upper branches of u. The core of u is u1={x∣x∈R,u(x)=1}. In terms of α-cuts, we have the addition and the scalar multiplication: (1)u+vα=uα+vα=x+y∣x∈uα,y∈vαλuα=λuα=λx∣x∈uα0α=0for all 0≤α≤1, u,v∈RF, and λ∈R.u = a , b , c , d specifies a trapezoidal fuzzy number, where a≤b≤c≤d and if b=c we obtain a triangular fuzzy number. For α∈[0,1], we have uα=[a+α(b-a),d-α(d-c)]. In the rest of this paper, we will assume that u is a triangular fuzzy number.Definition 1 (see, e.g., [5]).
An L-R fuzzy numberu=(m,l,r)LR is a function from the real numbers into the interval [0,1] satisfying (2)
u
x
=
R
x
-
m
r
for
m
≤
x
≤
m
+
r
,
L
m
-
x
l
for
m
-
l
≤
x
≤
m
,
0
o
t
h
e
r
w
i
s
e
,where R and L are continuous and decreasing functions from [0,1] to [0,1] fulfilling the conditions R(0)=L(0)=1 and R(1)=L(1)=0. When R(x)=L(x)=1-x, we will have L-L fuzzy numbers that involve the triangular fuzzy numbers. For an L-L fuzzy number u=(m,l,r), the support is the closed interval [m-l,m+r] (see, e.g., [6]).
The linear space of all polynomials of degree at mostN will be designated by PN. Full Hermite interpolation problem defines a unique polynomial, called pN(x), which solves the following problem.Theorem 2 (see [12] (existence and uniqueness)).
Letx0,x1,…,xn be n+1 distinct points, α0,α1,…,αn be positive integers, k=0,1,…,αi, and N=α0+α1+⋯+αn+n. Set w(x)=∏i=0n‍(x-xi)αi+1 and (3)likx=wxx-xik-αik!x-xiαi+1-kdαi-kdxαi-kx-xiαi+1wxx=xipNx=∑i=0nrili0x+∑i=0nri′li1x+⋯+∑i=0nriαiliαixis a unique member of PN for which (4)pNx0=r0,pN′x=r0′,…,pNα0-1x0=r0α0⋮pNxn=rn,pN′xn=rn′,…,pNαn-1xn=rnαn.When α0=α1=⋯=αn=1, the full Hermite interpolation simplifies into simple Hermite or osculatory interpolation.Definition 3.
Given distinct knotsx0,x1,…,xn, associated function values f0,f1,…,fn, and a linear space Φ of specific real functions generated by continuous cardinal basis functions ϕj:R→R, (j=0,1,…,n), ϕj(xi)=δij, (i=0,1,…,n), we say that the function F organized in the shape F(x)=∑j=0n‍fjϕj(x) is an interpolant based on cardinal basis and such a procedure is the cardinal basis functions method.


## 3. Piecewise Fuzzy Hermite Interpolation Polynomial

A special case of full Hermite interpolation is piecewise Hermite interpolation (see, e.g., [13, 14]). Let us assume throughout the paper that Δ:a=x0<x1<⋯<xn=b is a grid of I=[a,b] with knots xi and m is a positive integer. All piecewise Hermite polynomials form a certain finite dimensional smooth linear space which we name H2m-1(Δ;I).Definition 4.
H 2 m - 1 ( Δ ; I ) is a collection of all real-valued piecewise-polynomial functions s(x) of degree at most 2m-1, defined on I, such that s(x)∈Cm-1(I). The associated function to s(x) on successive intervals [xi-1,xi], 1≤i≤n, with knots from Δ, is defined by si(x), that is, a (m-1)-times continuously differentiable piecewise Hermite polynomial of degree 2m-1, on I.Definition 5 (see [15]).
Given any real-valued function,f(x)∈Cm-1(I). Let its unique H2m-1(Δ;I)-interpolate, for m and grid Δ of I, be the element s(x) of degree 2m-1 on each interval [xi-1,xi], 1≤i≤n, such that (5)Dksxi=Dkfxi∀0≤k≤m-1,0≤i≤n,Dk=dkdxk.Existence and uniqueness of full Hermite interpolation is provided in [12]. Because of this, presentation (5) is actually a special case of such interpolation on a gridded interval and it follows that each function belonging to Cm-1(I) has a unique interpolate in H2m-1(Δ;I).
A particular cardinal basis for linear spaceH2m-1(Δ;I) of dimension m(n+1) is B={ϕik(x)}i=0,k=0n,m-1, (see, e.g., [16]), where the basis function ϕik(x) is defined by (6)Dlϕikxj=δklδij,0≤k,l≤m-1,0≤i,j≤n,Dl=dldxl.Some important results based on (6) are simple to see in the sequel, as ϕi0(xi)=1 and ϕi0(xj)=0 at all knots xj and since s(x) outside [xi-1,xi], 1≤i≤n, satisfies zero data, then ϕi0≡0 for all x0≤x≤xi-1 and xi+1≤x≤xn. ϕi1(x) is of degree 2m-1, and ϕi1′(xi)=1 but it is zero at all other knots. Moreover, because outside the interval [xi-1,xi+1]ϕi1(x) interpolates zero data, then ϕi1(x) must be vanished identically for all x≥xi+1 and x≤xi-1 (see, e.g., [13, 14, 17]). Analogous reasoning applies to(7)
ϕ
i
1
x
=
x
-
x
i
-
1
m
x
-
x
i
∑
j
=
0
m
-
2
a
j
x
j
,
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i
-
1
≤
x
≤
x
i
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i
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m
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=
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+
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,
0
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o
t
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r
w
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e
.In the following theorem, we will use the recent features.Theorem 6.
Assume thatϕik∈B and satisfies the piecewise Hermite polynomial cardinal basis function constraints (6). Then, (i)
ϕi0(x)+ϕi+10(x)≥1, for all x∈(xi,xi+1), i=0,1,…,n-1.(ii)
For alli=1,2,…,n-1, ϕi1 changes the sign at xi. The sign of ϕi1 is not positive on any subinterval [xi-1,xi], and that is not negative on [xi,xi+1].(iii)
The sign of all other elements ofB is not negative on I.Proof.
With the assumption of (6), let ϕi0(x)+ϕi+10(x) be polynomial of degree 2m-1 on the interval [xi-1,xi+2] and interpolate the data (xj,fj), where fj=1 for j=i,i+1 and zero on the other knots of partition Δ. Suppose that 0<i<n-1 and ϕi0(x)+ϕi+10(x)<1 for some x∈(xi,xi+1). By the mean value theorem, its derivative has a zero on (xi,xi+1). The derivative has two (m-2)th order zeros at xi and xi+1 and its two other zeros are xi-1, xi+2. Then, it has at least 2m-1 zeros on the interval [xi-1,xi+2], which is a contradiction. The cases i=0 and i=n-1 are treated similarly.
In light of representation (7) and condition (6), the polynomial ϕi1(x) is of degree 2m-1. It has only one minimum point on [xi-1,xi] and a single maximum on the subinterval [xi,xi+1]. Suppose that each of the above points are one more. Then, by the mean value theorem, first derivative of ϕi1(x) has at least three zeros on (xi-1,xi) and three zeros on (xi,xi+1). Also, the derivative has two (m-2)th order zeros at xi-1, xi+1. Then, it has at least 2m+2 zeros, which is a contradiction. Hence, ϕi1′(x) has only one zero on each of the intervals (xi-1,xi) and (xi,xi+1). Now, recall ϕi1′(xi)=1; it follows that ϕi1(x)≤0, on [xi-1,xi] and ϕi1(x)≥0, on [xi,xi+1]. This gives (ii).
A similar proof via definition of basis functions and (6) follows the claim (iii).For a givenf(x)∈Cm-1(I) and its piecewise Hermite interpolate s(x)∈H2m-1(Δ;I), an equivalent explicit representation of s(x) in (5) can be uniquely expressed (see, e.g., [13, 17]); namely, (8)sx=∑k=0m-1∑i=0nfkxiϕikx.Now, we want to construct a fuzzy-valued function ass:I→RF such that s(k)(xi)=f(k)(xi)=uki∈RF, 0≤k≤m-1, 0≤i≤n. Also, if for all 0≤k≤m-1, 0≤i≤n, uki=yi(k) are crisp numbers in R and f(k)(xi)=χyi(k) (see, e.g., [2]), then there is a polynomial of degree 2m-1 on successive intervals [xi-1,xi], 0≤i≤n, with s(k)(xi)=yi(k), 0≤k≤m-1, 0≤i≤n such that s(x)=χf(x) for all x∈R, where {(xi,fi,fi′,…,fi(m-1))∣fi(k)∈RF,0≤k≤m-1,0≤i≤n} is given.We suppose that such a fuzzy function exists and we attempt to find and compute it with respect to interpolation polynomial presented by Lowen [2]. Let, for each x∈[x0,xn], s(x) be a fuzzy piecewise Hermite polynomial and Λ={yi(k)}i=0,k=0n,m-1; then, from Kaleva [3] and Nguyen [18], we obtain the α-cuts of s(x) in a succinctly algorithm as follows:(9)sαx=t∈R∣μsxt≥α=t∈R∣∃yik:μukiyik≥α,0≤k≤m-1,0≤i≤n,sΛx=t=t∈R∣t=sΛx,yik∈ukiα,0≤k≤m-1,0≤i≤n=∑k=0m-1∑i=0nukiαϕikx,where (10)sΛx=∑k=0m-1∑i=0nyikϕikxis a piecewise Hermite polynomial in crisp case and by definition (11)sαx=∑k=0m-1∑i=0nukiαϕikxwe obtain a formula that comprises a simple practical way for calculating s(x): (12)sx=∑k=0m-1∑i=0nukiϕikx.Since, for each0≤k≤m-1, 0≤i≤n, ukiα=[u_kiα,u¯kiα], then we will have sα(x) by solving the following optimization problems: (13)max&min∑k=0m-1∑i=0nyikϕikxsubjecttou_kiα≤yik≤u¯kiα,0≤k≤m-1,0≤i≤n.From the ϕij’s sign that we represented in Theorem 6, these problems have the following optimal solutions:Maximization is as follows:(14)yik=u¯kiαifϕikx≥0u_kiαifϕikx<0,0≤k≤m-1,0≤i≤n.Minimization is as follows:(15)yik=u_kiαifϕikx≥0u¯kiαifϕikx<0,0≤k≤m-1,0≤i≤n.Theorem 7.
Ifs(x)=∑k=0m-1‍∑i=0n‍ukiϕik(x) is an interpolating piecewise fuzzy Hermite polynomial, then for all α∈[0,1], i∈{0,1,…,n-1}, x∈[xi,xi+1], (16)lensαx≥min⁡lensαxi,lensαxi+1,where (17)sαx=s_αx,s¯αx,lensαx=s¯αx-s_αx.Proof.
By using Theorem6 and (11), we have sα(xi)=u0iα, sα(xi+1)=u0i+1α and lensα(xi)=lenu0iα, lensα(xi+1)=lenu0i+1α. Since the addition does not decrease the length of an interval from (11), we can write sα(x)=∑k=0m-1‍∑j=0n‍ukjαϕjk(x); then, (18)lensαx≥∑k=0m-1∑j=0nϕjkxlenukjα≥∑j=0nϕj0xlenu0j≥ϕi0xlenu0i+ϕi+10xlenu0i+1≥min⁡lenu0i,lenu0i+1ϕi0x+ϕi+10x≥min⁡lenu0i,lenu0i+1=min⁡lensαxi,lensαxi+1.Theorem 8.
Letuki=(mki,lki,rki), 0≤k≤m-1, 0≤i≤n, be a triangular L-L fuzzy number; then, also s(x), the piecewise fuzzy Hermite polynomial interpolation, is such a fuzzy number for each x.Proof.
The closed interval[m-l,m+r] is the support of u=(m,l,r), a triangular L-L fuzzy number; then for each x and uki, we have (19)sx=∑k=0m-1∑i=0nukiϕikx=∑∑ϕik≥0mki-lkiϕikx+∑∑ϕik<0mki+rkiϕikx,∑∑ϕik≥0mki+rkiϕikx+∑∑ϕik<0mki-lkiϕikx=∑k=0m-1∑i=0nmkiϕikx-∑∑ϕik≥0lkiϕikx-∑∑ϕik<0rkiϕikx+∑k=0m-1∑i=0nmkiϕikx+∑∑ϕik≥0rkixϕikx-∑∑ϕik<0likϕikx=mx-lx,mx+rx.It follows that ifs(x)=(m(x),l(x),r(x)), is a triangular L-L fuzzy number for each x, then (20)mx=∑k=0m-1∑i=0nmkiϕki,lx=∑∑ϕik≥0lkiϕikx-∑∑ϕik<0rkiϕikx,rx=∑∑ϕik≥0rkiϕik-∑∑ϕik<0likϕikx.


## 4. Piecewise-Polynomial Linear, Cubic, and Quintic Fuzzy Hermite Interpolation

We considerm=1 and compute the piecewise fuzzy linear interpolant as the initial case of the presented method based on (12) and for a given set of fuzzy data {xi,fi∣fi∈RF,0≤i≤n}, as follows: (21)sx=∑i=0nu0iϕi0x,where u0i=fi, 0≤i≤n, and subject to conditions (6), (22)
ϕ
00
x
=
0
,
x
≥
x
1
x
1
-
x
x
1
-
x
0
,
x
0
≤
x
≤
x
1
ϕ
i
0
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=
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≤
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i
-
1
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x
≥
x
i
+
1
x
-
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≤
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+
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+
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≤
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≤
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+
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≤
i
≤
n
-
1
ϕ
n
0
x
=
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,
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≤
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n
-
1
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-
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n
-
1
x
n
-
x
n
-
1
,
x
n
-
1
≤
x
≤
x
n
.The obtained s(x) is the same as fuzzy spline of order l=2 that had been introduced in [3] because the basic splines and the cardinal basis functions in two interpolants are equal.Example 9 (see [4]).
Suppose the data (1, (0, 2, 1), (1, 0, 3)), (1.3, (5, 1, 2), (0, 2, 1)), (2.2, (1, 0, 3), (4, 4, 3)), (3, (4, 4, 3), (5, 1, 2)), (3.5, (0, 3, 2), (1, 1, 1)), (4, (1, 1, 1), and (0, 3, 2)). In Figure1, the dashed line is the 0.5-cut set of piecewise cubic fuzzy interpolation s(x), x∈[1,4] and the solid lines represent the support and the core of s(x).Figure 1
Graph of Example9.Whenm=2, we get the piecewise cubic fuzzy Hermite polynomial interpolant in [11] for a given set of data {(xi,fi,fi′)∣fi,fi′∈RF,0≤i≤n}, (23)sx=∑i=0nu0iϕi0x+∑i=0nu1iϕi1x,where uki=fi(k), k=0,1, 0≤i≤n.An outstanding feature of this study is that, by simply applying the second case of the presented general method and exactly the same data, we have produced an alternative to simple fuzzy Hermite polynomial interpolation in [4]. Heretofore, the mentioned cubic case (23) was independently introduced in [10] but only in very weak conditions and without using the extension principle.The cardinal basis functionsϕik(x), k=0,1, 0≤i≤n, were computed in [17].Example 10.
Suppose the data (1, (0, 2, 1), (1, 0, 3)), (1.5, (5, 1, 2), (0, 2, 1)), (2.7, (1, 0, 3), (4, 4, 3)),(3,(4,4,3),(5,1,2)), (3.7, (0, 3, 2), (1, 1, 1)), and (4, (1, 1, 1), (0, 3, 2)). In Figure 2, the dashed line is the 0.5-cut set of piecewise cubic fuzzy interpolations s(x), x∈[1,4] and the solid lines represent the support and the core of s(x).Figure 2
Graph of Example10.Letm=3; from (6), we shall construct the cardinal basis for H5(Δ;I). The quintic Hermite polynomials ϕi0(x), ϕi1(x), and ϕi2(x) are solving the interpolation problem (24)Dlϕikxj=δklδij,0≤k,l≤2,0≤i,j≤n.To this end, we determine uniquely all the pervious ϕij’s by the (24).For1≤i≤n-1, let (25)
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.The six next functions are similarly defined. In particular, (26)
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.Thus, we can immediately write down piecewise quintic fuzzy Hermite interpolation polynomial s(x) using (27)sx=∑i=0nu0iϕi0x+∑i=0nu1iϕi1x+∑i=0nu2iϕi2x,where {xi,fi,fi′,fi′′∣fi(k)∈BF,0≤k≤2,0≤i≤n}, is given and uki=fi(k).Example 11.
Suppose that (0, (0, 1, 3), (0, 2, 2), (1, 4, 4)), (1.3, (0.05, 1.9, 3.5), (0.3, 3.2, 0.8), (1, 3.1, 3)), (2, (2, 6.7, 5.3), (2, 0.5, 3.5), (1, 2.6, 2.4)), (4, (8, 10.1, 9.9), (4, 4, 0), (1, 0.6, 0.5)), (5.3, (14, 13, 12), (5.3, 0.2, 3.8), (1, 1.5, 1.7)), (6, (18, 13.2, 14.8), (6, 0.9, 3), and (1, 3.4, 3.2)) are the interpolation data. In Figure3, the solid lines denote the support and the core of piecewise quintic fuzzy Hermite interpolation s(x), x∈[0,6], and the dashed line is the 0.5-cut set of s(x).Figure 3
Graph of Example11.


## 5. Conclusions and Further Work

Based on the cardinal basis functions form(n+1) dimension H2m-1(Δ,I) linear space, interpolation of fuzzy data by the fuzzy-valued piecewise Hermite polynomial as the extension of same approach in [11] has been successfully introduced in general case and provided a succinct formula for calculating the new fuzzy interpolant. Moreover, two first cases of the presented method have been applied as an analogy to fuzzy spline of order two in [3] and an alternative to fuzzy osculatory interpolation in [4], respectively. In the guise of a remarkable achievement, the piecewise fuzzy cubic Hermite polynomial interpolation that was constructed with poor conditions and without using the extension principle in [10] has been produced in the role of a very special subdivision for the presented general method in this study. Finally, the third initial case, piecewise fuzzy quintic Hermite polynomial, has been described in detail. The next step to improve this method is interpolation of fuzzy data including switching points by a fuzzy differentiable piecewise interpolant.



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*Source: 8127215-2016-12-29.xml*