Why LINEX Our Explanation (cont-d) Our Explanation (cont-d) - - PowerPoint PPT Presentation

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Why LINEX (Linear Exponential) Loss Functions?

Emilio Ramirez and Vladik Kreinovich

Department of Computer Science University of Texas at El Paso El Paso, TX 79968, USA, ejramirez5@miners.utep.edu, vladik@utep.edu

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1. Empirical Fact

• In design and control, we want to select the values of

the parameters that minimize the losses.

• Traditionally, the dependence of loss on the parameter

is described by a quadratic function.

• However, for this function:

– positive and negative deviations from the optimal value – lead to the exact same increase in loss.

• In practice, the effects are often different.
• E.g., for a refrigerator, a decrease in temperature is

rarely harmful, while an increase can spoil the food.

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2. Empirical Fact (cont-d)

• It turns out that in many practical situations,

– in the next approximation, – the best “asymmetric” loss function is a linear com- bination of a linear and exponential functions: L(x) = c · exp(a · x) − b · x + a0.

• This combination is known as LINEX.
• How can we explain this empirical fact?

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3. Our Explanation

• Quadratic functions are linear combinations of smooth

functions 1, x, and x2.

• The class of such functions does not change if we change

the starting point for measuring x.

• Then, we replace x with x + x0 for some x0.
• Let us thus look for similarly “shift-invariant” smooth

families of the type c1 · f1(x) + c2 · f2(x) + c3 · f3(x).

• Shift-invariance means, in particular, that all shifted

functions fi(x + x0) belong to the same family.

• In particular, this means that, for some value cij(x0),

we have fi(x + x0) =

• j

cij(x0) · fj(x).

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4. Our Explanation (cont-d)

• For each i and x0, let us select three different values xk.
• Then, we get a linear system for three unknowns cij(x0),

j = 1, 2, 3: fi(xk + x0) =

• j

cij(x0) · fj(xk).

• By Cramer’s rule, the values cij(x0) are rational func-

tions of values fi(xk + x0) and fj(xk).

• These values smoothly depend on x0.
• Thus, the functions cij(x0) are also differentiable.
• So, we can differentiate the above equality with respect

to x0.

• After the differentiation, we can take x0 = 0 and get

f ′

i(x) = aij · fj(x), where aij def

= c′

ij(0).

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5. Our Explanation (cont-d)

• Solutions to such linear differential equations with con-

stant coefficients are well known.

• These solutions are linear combinations of terms

xk · exp(λ · x), where:

• λ is an eigenvalue of the matrix cij, and
• k = 0, 1, 2, . . .; k > 0 corresponds to the case of

equal eigenvalues.

• Quadratic functions correspond to the case when all

three eigenvalues are equal to 0.

• The simplest modification is when:
• one of the eigenvalues becomes different from 0,

while

• the other two remain 0s.

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6. Our Explanation (cont-d)

• Reminder: The simplest modification is when:
• one of the eigenvalues becomes different from 0,

while

• the other two remain 0s.
• This corresponds to the loss functions

c1 · exp(λ · x) + c2 · x + c2.

• This is exactly LINEX.
• Thus, the practical efficiency of LINEX loss functions

can be naturally explained.

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7. References

• Yen-Chang Chang and Wen-Liang Hung, “LINEX loss

functions with applications to determining the opti- mum process parameters”, Quality & Quantity, 2007,

• Vol. 41, pp. 291–301.
• H. R. Varian, “A Bayesian approach to real estate as-

sessment”, In: S. E. Fienberg and A. Zellner (eds.), Studies in Bayesian Econometrics and Statistics in Honor

• f Leonard L. Savage, Noth Holland, Amsterdam, 1975,

1975, pp. 195–208.

• A. Zellner, “Bayesian estimation and prediction using

asymmetric loss functions”, Journal of the American Statistical Association, 1986, Vol. 81, pp. 446–451.