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General Systems: . . . The Notion of a . . . Growth under Interval . . . What Is Known New Algorithm Justification of the . . . Case of Fuzzy Uncertainty Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 1 of 9 Go Back Full Screen Close Quit

Growth Rates under Interval Uncertainty

Janos Hajagos1 and Vladik Kreinovich2

1Applied Biomathematics,

100 North Country Road Setauket, New York 11733, USA logistic@sdf-eu.org

2Department of Computer Science,

University of Texas at El Paso, El Paso, TX 79968, USA vladik@utep.edu http://www.cs.utep.edu/vladik

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1. General Systems: Linear Approximation

• State: is described by the values of the parameters

x1, . . . , xn.

• Dynamic for continuous time: ˙

xi = fi(x1, . . . , xn).

• Dynamic for discrete time: xi(t+1) = fi(x1(t), . . . , xn(t)).
• Linearization: fi(x1, . . . , xn) is smooth, so in a small

region, fi(x1, . . . , xn) ≈ bi +

n

• j=1

aij · xj.

• Further simplification: an appropriate shift xi → xi−si

leads to fi(x1, . . . , xn) =

n

• j=1

aij · xj.

• Resulting dynamic: ˙

xi =

n

• j=1

aij · xj; xi(t + 1) =

n

• j=1

aij · xj(t).

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2. The Notion of a Growth Rate

• New variables: coefficients yi in the expansion of xi(t)

in eigenvectors of A = (aij).

• Generic case (all eigenvalues different):

˙ yi = λi · yi, or yi(t + 1) = λi · yi(t).

• General solution: a linear combination of

yi(t) = yi(0) · exp(λi · t) or yi(t) = yi(0) · λt

i.

• Asymptotically: xi(t) ∼ exp(λ · t) or xi(t) ∼ λt, where

λ – growth rate – is the largest of the eigenvalues.

• Applications:

– population growth, – growth in animals and plants, – spread of an epidemic, – financial growth.

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3. Growth under Interval Uncertainty

• Idealized case – exactly known coefficients: find the

eigenvalues and find the largest of them.

• In practice: aij are only known with uncertainty.
• Case of interval uncertainty: we know

– approximate values aij, and – upper bounds ∆ij on the approx. error | aij − aij|.

• Hence aij ∈ aij = [aij, aij]

def

= [ aij − ∆ij, aij + ∆ij].

• Problem: find the interval [λ, λ] of possible values of λ

when aij ∈ aij.

• Alternative: at least find an interval I ⊇ [λ, λ].
• Comment: upper bound λ is of special interest: it de-

termines, e.g., how fast a disease can spread.

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4. What Is Known

• Case of small uncertainty: ∆aij

def

= aij − aij ≪ aij.

• Solution: linearize the dependence λk(aij), and get a

range for λ.

• General case: the problem is computationally intractable

(NP-hard).

• Important case: non-negative matrix aij ≥ 0.
• Examples:

– population growth, – spread of disease, – financial situations.

• What we propose: a new efficient algorithm that com-

putes λ for non-negative matrices.

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5. New Algorithm

• We use: a known algorithm A that computing λ(A)

for exact matrices A.

• Input: interval-valued matrix A = aij = [aij, aij].
• Algorithm:

– First, we apply A to A = aij; the resulting value is returned as λ. – Then, we apply A to A = aij; the resulting value is returned as λ.

• Comment: this idea only works for non-negative ma-

trices.

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6. Justification of the New Algorithm

• Notation: A ≤ B means aij ≤ bij for all i, j.
• Notation: x ≥ 0 means xi ≥ 0 for all i.
• Known fact: λ(A) = max

x=0

Ax2 x2 , where x2

def

=

• n
• i=1

x2

i

is the length of x.

• Perron-Frobenius Theorem:

for A ≥ 0, one of the eigenvectors x corresponding to the largest λ(A) is non- negative: x ≥ 0.

• Conclusion: λ(A) =

max

x≥0 & x=0

Ax2 x2 .

• If 0 ≤ A ≤ B and x ≥ 0, then 0 ≤ Ax ≤ Bx hence

Ax2 ≤ Bx2, Ax2 x2 ≤ Bx2 x2 , and λ(A) ≤ λ(B).

• Interval matrix: for all A ∈ [A, A], we have

A ≤ A ≤ A, hence λ(A) ≤ λ(A) ≤ λ(A).

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7. Case of Fuzzy Uncertainty

• Situation: often, we have uncertain expert estimates.
• Fuzzy set description: for each x, describe the degree

µ(x) ∈ [0, 1] to which this value is possible.

• Reducing fuzzy uncertainty to interval uncertainty:

– for each degree of certainty α, values possible with uncertainty ≥ α form an interval (α-cut) x(α) = {x | µ(x) ≥ α}; – if we know α-cuts for all α, we can reconstruct µ(x).

• Computation under fuzzy uncertainty: for each level α,

we apply the interval algorithm to the α-cuts.

• Case of growth rate: for α = 0, 0.1, . . . , 1, we apply our

algorithm to α-cuts aij(α).

• Result: intervals [λ, λ](α) form the fuzzy set for λ.

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8. Acknowledgments This work was supported in part by:

• NASA under cooperative agreement NCC5-209,
• NSF grants EAR-0225670 and DMS-0532645,
• Star Award from the University of Texas System, and
• Texas Department of Transportation grant No. 0-5453.

The authors are thankful to Arnold Neumaier for his valu- able advise.