Linear Dependencies . . .
Linear Dependencies . . .
Skedactic Functions
Problems and What . . .
Invariance Explains Multiplicative and
Exponential Skedactic Functions
Natural Invariance: . . .
Shift and Shift-Invariance
Scale Invariance: Main . . .
Vladik Kreinovich1 , Olga Kosheleva1 ,
Hung T. Nguyen2,3 , and Songsak Sriboonchitta3
1
University of Texas at El Paso,
El Paso, TX 79968, USA
vladik@utep.edu, olgak@utep.edu
2
Department of Mathematical Sciences
New Mexico State University
Las Cruces, NM 88003, USA, hunguyen@nmsu.edu
3
Faculty of Economics, Chiang Mai University
Chiang Mai, Thailand,
songsakecon@gmail.com
Shift Invariance: Main . . .
General Case: Some . . .
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Linear Dependencies . . .
1.
Linear Dependencies . . .
Linear Dependencies Are Ubiquitous
Skedactic Functions
• In many practical situations, a quantity y depends on
several other quantities x1 , . . . , xn : y = f (x1 , . . . , xn ).
(0)
• Often, the ranges of xi are narrow: xi ≈ xi for some
def
(0)
(0)
xi , so ∆xi = xi − xi are relatively small.
• Then, we can expand the dependence of y on xi =
(0)
xi + ∆xi in Taylor series and keep only linear terms:
y=
(0)
f (x1
+
∆x1 , . . . , x(0)
n
+ ∆xn ) ≈ a0 +
n
X
Problems and What . . .
Natural Invariance: . . .
Shift and Shift-Invariance
Scale Invariance: Main . . .
Shift Invariance: Main . . .
General Case: Some . . .
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ai · ∆xi ,
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i=1
def
where a0 = f
(0)
x1 , . . . , x(0)
n
∂f
and ai =
.
∂xi
def
(0)
• Substituting ∆xi = xi − xi into this formula, we get
n
n
P
P
def
(0)
y ≈c+
ai · xi , where c = a0 −
ai · x i .
i=1
i=1
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Linear Dependencies . . .
2.
Linear Dependencies Are Approximate
Linear Dependencies . . .
Skedactic Functions
• Usually,
Problems and What . . .
– in addition to the quantities x1 , . . . , xn that provide
the most influence on y,
– there are also many other quantities that (slightly)
influence y,
– so many that it is not possible to take all of them
into account.
Natural Invariance: . . .
Shift and Shift-Invariance
Scale Invariance: Main . . .
Shift Invariance: Main . . .
General Case: Some . . .
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• Since we do not take these auxiliary quantities into
account, the linear dependence is approximate.
n
P
def
• The approximation errors ε = y − c +
ai · xi dei=1
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pend on un-observed quantities.
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• So, we cannot predict ε based only on the observed
quantities x1 , . . . , xn .
• It is therefore reasonable to view ε as random variables.
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Linear Dependencies . . .
3.
Linear Dependencies . . .
Skedactic Functions
Skedactic Functions
• A natural way to describe a random variable is by its
moments.
• If the first moment is not 0, then we can correct this
bias by appropriately updating the constant c.
Problems and What . . .
Natural Invariance: . . .
Shift and Shift-Invariance
Scale Invariance: Main . . .
Shift Invariance: Main . . .
• Since the mean is 0, the second moment coincides with
the variance v.
• The dependence v(x1 , . . . , xn ) is known as the skedactic
function.
• In econometric applications, two major classes of
skedactic functions have been empirically successful:
n
Q
– multiplicative v(x1 , . . . , xn ) = c ·
|xi |γi and
i=1 n
P
– exponential v(x1 , . . . , xn ) = exp α +
γi · xi .
i=1
General Case: Some . . .
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Linear Dependencies . . .
4.
Problems and What We Do
Linear Dependencies . . .
Skedactic Functions
• Problems:
– Neither of the empirically successful skedactic functions has a theoretical justification.
Problems and What . . .
Natural Invariance: . . .
Shift and Shift-Invariance
Scale Invariance: Main . . .
– In most situations, the multiplication function results in more accurate estimates.
– This fact also does not have an explanation.
• What we do: we use invariance ideas to:
– explain the empirical success of multiplicative and
exponential skedactic functions, and
– come up with a more general class of skedactic functions.
Shift Invariance: Main . . .
General Case: Some . . .
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Linear Dependencies . . .
5.
Natural Invariance: Scaling
Linear Dependencies . . .
Skedactic Functions
• Many economics quantities correspond to prices,
wages, etc. and are thus expressed in terms of money.
Problems and What . . .
• The numerical value of such a quantity depends on the
choice of a monetary unit.
Shift and Shift-Invariance
Natural Invariance: . . .
Scale Invariance: Main . . .
Shift Invariance: Main . . .
• For example, when a European country switches to
Euro from its original currency,
– the actual incomes do not change, but
– all the prices and wages get multiplied by the corresponding exchange rate k: xi → x0i = k · xi .
• Similarly, the numerical amount (of oil or sugar),
changes when we change units.
• For example, for oil, we can use barrels or tons.
• When the numerical value of a quantity is multiplied
by k, its variance gets multiplied by k 2 .
General Case: Some . . .
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Linear Dependencies . . .
6.
Linear Dependencies . . .
Scaling (cont-d)
Skedactic Functions
• Changing the measuring units for x1 , . . . , xn does not
change the economic situations.
• So, it makes sense to require that the skedactic function
also does not change under such re-scaling: namely,
Problems and What . . .
Natural Invariance: . . .
Shift and Shift-Invariance
Scale Invariance: Main . . .
Shift Invariance: Main . . .
– for each combination of re-scalings on inputs,
– there should be an appropriate re-scaling of the output after which the dependence remains the same.
• In precise terms, this means that:
– for every combination of numbers k1 , . . . , kn ,
– there should exist a value k = k(k1 , . . . , kn ) with
the following property:
0
v(x01 , . . . , x0n ),
v = v(x1 , . . . , xn ) if and only if v =
where v 0 = k · v and x0i = ki · xi .
General Case: Some . . .
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Linear Dependencies . . .
7.
Shift and Shift-Invariance
Linear Dependencies . . .
Skedactic Functions
• While most economic quantities are scale-invariant,
some are not.
Problems and What . . .
• For example, the unemployment rate is measured in
percents, there is a fixed unit.
Shift and Shift-Invariance
• Many such quantities can have different numerical values depending on how we define a starting point.
• For example, we can measure unemployment:
– either by the usual percentage xi , or
– by the difference xi − ki , where ki > 0 is what
economists mean by full employment.
• It is thus reasonable to consider shift-invariant skedactic functions:
Natural Invariance: . . .
Scale Invariance: Main . . .
Shift Invariance: Main . . .
General Case: Some . . .
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∀k1 , . . . , kn ∃k (v = v(x1 , . . . , xn ) ⇔ v 0 = f (x01 , . . . , x0n )),
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where v 0 = k · v and x0i = xi + ki .
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Linear Dependencies . . .
8.
Linear Dependencies . . .
Scale Invariance: Main Result
Skedactic Functions
• Definition. We say that a non-negative measurable
function v(x1 , . . . , xn ) is scale-invariant if:
– for every n-tuple of real numbers (k1 , . . . , xn ),
Problems and What . . .
Natural Invariance: . . .
Shift and Shift-Invariance
Scale Invariance: Main . . .
– there exists a real number k = k(k1 , . . . , kn ) for
which, for every x1 , . . . , xn and v:
v=
0
v(x1 , . . . , xn ) ⇔ v = v(x01 , . . . , x0n ),
v 0 = k · v and x0i = ki · xi .
where
• Proposition. A skedactic function is scale-invariant
if and only it has the form
v(x1 , . . . , xn ) = c ·
n
Y
|xi |γi for some c and γi .
i=1
• Discussion. Thus, scale-invariance explains the use
of multiplicative skedactic functions.
Shift Invariance: Main . . .
General Case: Some . . .
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Linear Dependencies . . .
9.
Shift Invariance: Main Result
Linear Dependencies . . .
Skedactic Functions
• Definition. We say that a non-negative measurable
function v(x1 , . . . , xn ) is shift-invariant if:
– for every n-tuple of real numbers (k1 , . . . , kn ),
– there exists a real number k = k(k1 , . . . , kn ) for
which, for every x1 , . . . , xn and v:
v = v(x1 , . . . , xn ) ⇔ v 0 = v(x01 , . . . , x0n ), where
v 0 = k · v and x0i = xi + ki .
Problems and What . . .
Natural Invariance: . . .
Shift and Shift-Invariance
Scale Invariance: Main . . .
Shift Invariance: Main . . .
General Case: Some . . .
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• Result. A skedactic function is shift-invariant ⇔
!
n
Y
v(x1 , . . . , xn ) = exp α +
γi · xi for some α and γi .
i=1
• Discussion. Thus, shift-invariance explains the use of
exponential skedactic functions.
• It also explains why multiplicative functions are more
often useful: scaling is ubiquitous, shift is rarer.
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Linear Dependencies . . .
10.
General Case:
Some Inputs Are ScaleInvariant and Some Are Shift-Invariant
Linear Dependencies . . .
Skedactic Functions
Problems and What . . .
• Definition. We say that a non-negative measurable
function v(x1 , . . . , xn ) is m-invariant if:
– for every n-tuple of real numbers (k1 , . . . , kn ),
– there exists a real number k = k(k1 , . . . , kn ) for
which, for every x1 , . . . , xn and v:
Natural Invariance: . . .
Shift and Shift-Invariance
Scale Invariance: Main . . .
Shift Invariance: Main . . .
General Case: Some . . .
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v
v =k
0
0
= v(x1 , . . . , xn ) ⇔ v = v(x01 , . . . , x0n ), where
· v, x0i = ki · xi for i ≤ m, x0i = xi + ki for i >
m.
• Result. A skedactic function is m-invariant ⇔
!
m
n
X
X
γi · ln(|xi |) +
γi · xi .
v(x1 , . . . , xn ) = exp α +
i=1
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i=m+1
• For m = n, we get multiplicative skedactic function,
for m = 0, we get the exponential one.
• For other m, we get new possibly useful expressions.
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Linear Dependencies . . .
11.
Acknowledgments
Linear Dependencies . . .
Skedactic Functions
• We acknowledge the partial support of:
– the Center of Excellence in Econometrics, Faculty
of Economics, Chiang Mai University, Thailand,
Problems and What . . .
Natural Invariance: . . .
Shift and Shift-Invariance
Scale Invariance: Main . . .
– the National Science Foundation grants:
∗ HRD-0734825 and HRD-1242122
(Cyber-ShARE Center of Excellence) and
∗ DUE-0926721.
Shift Invariance: Main . . .
General Case: Some . . .
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Linear Dependencies . . .
12.
Linear Dependencies . . .
Proof of the First Result
Skedactic Functions
• It is easy to check that the multiplicative skedactic
n
Q
function is scale-invariant: take k =
|ki |γi .
i=1
• Vice versa, the equivalence condition means that
k(k1 , . . . , kn ) · v(x1 , . . . , xn ) = v(k1 · x1 , . . . , kn · xn ).
v(k1 · x1 , . . . , kn · xn )
• Thus, k(k1 , . . . , kn ) =
is a ratio
v(x1 , . . . , xn )
of measurable functions hence measurable.
• Let us consider two tuples (k1 , . . . , kn ), (k10 , . . . , kn0 ).
• If we first use the first re-scaling, i.e., go from xi to
x0i = ki · xi , we get
v(x01 , . . . , x0n )
= k(k1 , . . . , kn ) · v(x1 , . . . , xn ).
Problems and What . . .
Natural Invariance: . . .
Shift and Shift-Invariance
Scale Invariance: Main . . .
Shift Invariance: Main . . .
General Case: Some . . .
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Linear Dependencies . . .
13.
Linear Dependencies . . .
Proof (cont-d)
Skedactic Functions
• If we then apply, to the new values x0i , an additional
re-scaling x0i → x00i = ki0 · x0i , we similarly conclude that
v(x001 , . . . , x00n ) = k(k10 , . . . , kn0 ) · v(x01 , . . . , x0n ) =
k(k10 , . . . , kn0 )
· k(k1 , . . . , kn ) · v(x1 , . . . , xn ).
• We could also get the values x00i if we directly multiply
def
each value xi by the product ki00 = ki0 · ki , thus
v(x001 , . . . , x00n ) = k(k10 · k1 , . . . , kn0 · kn ) · v(x1 , . . . , xn ).
k(k10
k1 , . . . , kn0
• Thus,
·
· kn ) · v(x1 , . . . , xn ) =
k(k10 , . . . , kn0 ) · k(k1 , . . . , kn ) · v(x1 , . . . , xn ).
• If the skedactic function is always equal to 0, then it is
multiplicative, with c = 0.
• If it is not everywhere 0, this means that its value is different from 0 for some combination of values x1 , . . . , xn .
Problems and What . . .
Natural Invariance: . . .
Shift and Shift-Invariance
Scale Invariance: Main . . .
Shift Invariance: Main . . .
General Case: Some . . .
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Linear Dependencies . . .
14.
Linear Dependencies . . .
Proof (cont-d)
Skedactic Functions
• For this tuple, we get k(k10 · k1 , . . . , kn0 · kn ) =
k(k10 , . . . , kn0 ) · k(k1 , . . . , kn ).
Problems and What . . .
• When ki = ki0 = −1 for some i and kj0 = kj = 1 for all
j 6= i, i, we get 1 = k(1, . . . , 1) = k 2 (k1 , . . . , kn ).
Shift and Shift-Invariance
Natural Invariance: . . .
Scale Invariance: Main . . .
Shift Invariance: Main . . .
• Since the function ki is non-negative, this means that
k(k1 , . . . , kn ) = 1.
• Thus, the value k(k1 , . . . , kn ) does not change if we
change the signs of ki : k(k1 , . . . , kn ) = k(|k1 |, . . . , |kn |).
• Thus, ln(k(k10 · k1 , . . . , kn0 · kn )) = ln(k(k10 , . . . , kn0 )) +
ln(k(k1 , . . . , kn )).
• For an auxiliary function K(K1 , . . . , Kn )
ln(k(exp(K1 ), . . . , exp(Kn )), we thus get
General Case: Some . . .
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def
=
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K(K10 +K1 , . . . , Kn0 +Kn )
=
K(K10 , . . . , Kn0 )+K(K1 , . . . , Kn ).
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15.
Proof: Conclusion
Linear Dependencies . . .
Linear Dependencies . . .
Skedactic Functions
• We know that
Problems and What . . .
K(K10 +K1 , . . . , Kn0 +Kn ) = K(K10 , . . . , Kn0 )+K(K1 , . . . , Kn ).
• It is known that measurable functions with this propn
P
erty are linear: K(K1 , . . . , Kn ) =
γi · Ki , so
i=1
k(k1 , . . . , kn ) = exp
n
X
Natural Invariance: . . .
Shift and Shift-Invariance
Scale Invariance: Main . . .
Shift Invariance: Main . . .
General Case: Some . . .
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!
γi · ln(|ki |)
i=1
=
n
Y
|ki |γi .
i=1
• We also have v(x1 , . . . , xn ) = k(x1 , . . . , xn ) · v(1, . . . , 1).
• Thus, we get the desired formula for the multiplicative
n
Q
skedastic function v(x1 , . . . , xn ) = c ·
|xi |γi .
i=1
• Similar proofs hold for shift-invariance and for the general case.
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