[PPT] - Why Fuzzy Interpretation of . . . Bernstein Polynomials Fuzzy PowerPoint Presentation

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Why Bernstein Polynomials Are Better: Fuzzy-Inspired Justification

Jaime Nava, Olga Kosheleva, and Vladik Kreinovich

University of Texas at El Paso El Paso, TX 79968, USA nava.jaime@gmail.com, olgak@utep.edu, vladik@utep.edu

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1. Functional Dependencies are Ubiquitous

In practice, we are often interested in a quantity y

which is difficult to measure directly.

Examples: distance to a faraway star, amount of oil in

a well.

To estimate such quantities, we:

– measure related quantities x1, . . . , xn, and – use the known dependence y = f(x1, . . . , xn) to es- timate y.

To predict a future value y of a quantity, we use the

known relation between y and current values xi.

In all these cases, we need, given xi, to compute

y = f(x1, . . . , xn).

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2. Need for Polynomial (and Piece-Wise Polyno- mial) Approximations

In a computer, only addition, subtraction, and multi-

plication are hardware supported.

All other operations with real numbers, including divi-

sion, are implemented as a sequence of +, −, ·.

A composition of +, −, · is a polynomial.
Also hardware supported are logical operations; so, we

can have different expressions in different sub-domains.

Thus, whatever we want to compute, we are computing

a (piece-wise) polynomial function (spline).

In other words, every computation means that we (lo-

cally) approximate functions by polynomials.

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3. Possibility of a Polynomial Approximation

The possibility to approximate functions by polynomi-

als was first proven by Weierstrass in 19 cent.

Specifically, Weierstrass showed that:

– for every continuous function f(x1, . . . , xn), – for every box [a1, b1] × . . . × [an, bn], – for every real number ε > 0, – there exists a polynomial P(x1, . . . , xn) which is, on this box, ε-close to the original function f(x1, . . . , xn): |P(x1, . . . , xn) − f(x1, . . . , xn)| ≤ ε for all xi ∈ [ai, bi].

Polynomial approximations are ubiquitous in physics:

– many dependencies are analytical, i.e., representable as Taylor series, – so, to get a good approximation, we keep a few first terms in the Taylor series.

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4. How to Represent Polynomials in a Computer: Traditional Approach

In the physics case, it is natural to write

f(x) = c0 + c1 · x + c2 · x2 + . . . + cd · xd.

So, we can represent this polynomial by its coefficients

c0, c1, c2, . . . , cd.

A general polynomial has the form

f(x1, . . . , xn) =

d1,...,dn

cd1...dn · xd1

1 · . . . · xdn n .

It is natural to represent it as a corresponding multi-D

array of coefficients cd1...dn.

This is how polynomials are traditionally represented.

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5. Bernstein Polynomials: an Alternative Com- puter Representation

In practice, alternative computer representations often

work better.

For example, Bernstein proposed to represent a poly-

nomial as f(x1, . . . , xn) =

k1...kn

ck1...kn · pk11(x1) · . . . · pknn(xn), where pkii(xi)

def

= (xi − ai)ki · (bi − xi)d−ki.

In this representation, in the computer, we store the

coefficients ck1...kn.

Bernstein polynomials are often successful, but they

are the least studied.

Our objective is to study them.

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6. Bernstein Polynomials Are Useful in Interval and Fuzzy Computations

Bernstein polynomials are useful in interval computa-

tions, when we compute the range f([a1, b1], . . . , [an, bn])

def

= {f(x1, . . . , xn) : x1 ∈ [a1, b1], . . . , xn ∈ [an, bn]}.

Interval computations are used in fuzzy computations:

– we know fuzzy values µi(xi) of the inputs x1, . . . , xn, – we want to find the fuzzy value of y = f(x1, . . . , xn).

It is known that for every α, the α-cut for y is the range
f f(x1, . . . , xn) over α-cuts xi(α):

y(α) = f(x1(α), . . . , xn(α)).

This is how fuzzy computations are usually performed:

we perform interval computations over the α-cuts.

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7. Bernstein Polynomials: Explaining Success

Fact: Bernstein polynomials are successful.
Problem: there is no clear explanation for this success.
Idea: use fuzzy logic.
Preliminary step: to make fuzzy interpretation possi-

ble, let us reduce each interval [a, b] to [0, 1]: – when x ∈ [a, b], – we take t

def

= x − a b − a ∈ [0, 1].

For each f(x1, . . . , xn), we take

F(t1, . . . , tn) = f(a1+t1·(b1−a1), . . . , an+tn·(bn−an)).

Then, once we get an approximation

F for F, we can approximate f(x1, . . . , xn) by

f(x1, . . . , xn) =

F x1 − a1 b1 − a1 , . . . , xn − an bn − an

.

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8. Fuzzy Approximations: Reminder

Fuzzy logic has a natural approach for approximation

– rules: “if x1 is P1, x2 is P2, . . . , and xn is Pn, then y = c.”

For each rule k, the degree to which this rule is acti-

vated is: dk = f&(µ1(x1), . . . , µn(xn)).

In the simplest case f&(a, b) = a · b, we have:

dk = µ1(x1) · . . . · µn(xn).

The conclusions ck of several rules can be combined as

a weighted average c1 · d1 + . . . + cr · dr.

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9. Fuzzy Interpretation of the Usual Polynomial Representation c0 + c1 · x + c2 · x2 + . . . + cm · xm

This representation corresponds to fuzzy rules:
c0 (with no condition);
if x, then c1;
if x2, then c2; . . .
If we take x as the degree to which x ∈ [0, 1] is large,

then x2 is usually interpreted as “very large”, etc.

Thus, the above rules have the form:
c0;
if x is large, then c1;
if x is very large, then c2, etc.
Similarly, c012 · x2 · x2

3 means the following rule:

“if x2 is large and x3 is very large, then c012.”

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10. Fuzzy Interpretation Reveals Two Limitations

f the Traditional Computer Representation
First, an accurate representation requires polynomials
f higher degrees.
So, we need several distinct coefficients corresponding

to different hedges such as “very”, “very very”, etc.

In practice, we can only meaningfully distinguish be-

tween a small number of hedges.

Second, for computational efficiency, it is desirable to

have as few terms as possible to represent each f-n.

This can be achieved if in some important cases, some

coefficients are close to 0 and can, therefore, be ignored.

In the above fuzzy representation, all the terms are

meaningful.

There is no reason why some terms can be ignored.

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11. How Can We Overcome Limitations of the Traditional Computer Representation

Traditional computer representation of polynomials cor-

responds to opinions of a single expert.

We can achieve high accuracy if an expert can distin-

guish between “very large”, “very very large”, etc.

However, most experts are not very good in such a

distinction.

Since we cannot get a good approximation by using a

single expert, why not use d > 1 experts?

Then, there is no need for an expert to make difficult

distinctions, e.g., “very large” vs. “very very large”.

So, we can as well use each expert where he/she is the

strongest: separating “large” and “not large”.

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12. Fuzzy Analysis (cont-d)

In general, ki out of k experts believe that xi is large

and d − ki that x is not large.

Since we use product for “and”, the degree to which

this condition is satisfied is xki

i ·(1−xi)d−ki, i.e., pkii(xi).

The degree to which all n variables satisfy the corre-

sponding condition is also equal to the product pk11(x1) · . . . · pknn(xn).

Thus, the corresponding fuzzy rules lead to polynomi-

als of the type

k1,...,kn

ck1...kn · pk11(x1) · . . . · pknn(xn).

So, we indeed get a fuzzy explanations for the emer-

gence of Bernstein polynomials.

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13. Why Bernstein Polynomials Are More Com- putationally Efficient: A Fuzzy Explanation

The traditional polynomials correspond to rules in which

conditions are “x is large”, “x is very large”, etc.

There is no reason to conclude that some of the corre-

sponding terms become small.

In contrast, each Bernstein term xki

i ·(1−xi)d−ki (except

for ki = 0 and ki = d), corresponds to the condition

“xi is large (very large, etc.) and
xi is not large (very not large, etc.)”.
In practice, we are often confident that xi is large or

that xi is not large.

In this case, the above degree is close to 0, so the cor-

responding terms can be safely ignored.

Thus, computations become indeed more efficient.

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14. From Fuzzy Explanation to a More Precise Explanation

Let us pick ε > 0 and ignore all terms t for which

max t < ε.

In the traditional representation, the maximum of each

term xki

i is 1, so none of the terms can be ignored.

Each ki takes d + 1 possible values 0, 1, . . . , d.
Thus, we need (d + 1)n terms.
For a Bernstein term ti = xki

i · (1 − xi)d−ki, maximum

is attained for xi = ki d , and is equal to max ti = ki d ki ·

1 − ki

d d−ki .

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15. From Fuzzy to a More Precise Explanation (cont-d)

For ki ≈ 0, we have
1 − ki

d d−ki ≈

1 − ki

d d ≈ exp(−ki).

Thus, ln(max ti) ≈ −ki · (ln(d) − ln(ki) + 1).
Since ki ≪ d, we get ln(max ti) ≈ −ki · ln(d).
For ki ≈ d, we get ln(max ti) ≈ −(d − ki) · ln(d).
The condition max t ≥ ε is equivalent to

ln(max t) =

n

i=1

(max ti) ≥ ln(ε).

Analysis shows that we need to keep 2n · nC terms.
This is much less than (d + 1)n terms needed for the

traditional representation.

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16. Conclusions

In practice, we often (piece-wise) approximate a de-

pendence y = f(x1, . . . , xn) by a polynomial f(x1, . . . , xn) = c0 +

n

i=1

ci · xi +

n

i=1

n

j=1

cij · xi · xj + . . .

Traditionally, polynomials are represented by coeffi-

cients c0, c1, . . . , cn, c11, . . . , c1n, . . . , cn1, . . . , cnn, . . .

It is often more computationally efficient to represent

polynomials as linear combinations of Bernstein terms (x1 −a1)k1 ·(b1 −x1)d−k1 ·. . .·(xn −an)kn ·(bn −xn)d−kn.

There has been no intuitively convincing theoretical

explanations for this efficiency.

In this paper, we use fuzzy logic to provide an intuitive

theoretical explanation for this efficiency.

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17. Acknowledgment

This work was supported in part:

– by the National Science Foundation grant HRD- 0734825 (Cyber-ShARE Center of Excellence), – by the National Science Foundation grant DUE- 0926721, and – by Grant 1 T36 GM078000-01 from the National Institutes of Health.

The authors are thankful to anonymous referees for