Scale-Invariance: A . . . Heavy-Tailed . . . What Is Usually Done Multi-D Case Why Student Distributions? A Combination . . . Why Matern’s Covariance Main Result Derivation of Student . . . Model? A Symmetry-Based What Next? Alternative Symmetry- . . . Explanation Home Page Title Page on 1 , Gael Kermarrec 1 , Boris Kargoll 1 Stephen Sch¨ Ingo Neumann 1 , Olga Kosheleva 2 , and Vladik Kreinovich 2 ◭◭ ◮◮ ◭ ◮ 1 Leibniz University Hannover, 30167 Hannover, Germany schoen@ife.uni-hannover.de, gael.kermarrec@web.de Page 1 of 40 kargoll@gih.uni-hannover.de, neumann@gih.uni-hannover.de 5 University of Texas at El Paso, USA, olgak@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit
Scale-Invariance: A . . . Heavy-Tailed . . . 1. Scale-Invariance: A Natural Property of the What Is Usually Done Physical World Multi-D Case • Scientific laws are described in terms of numerical val- A Combination . . . ues of the corresponding quantities, be it Main Result Derivation of Student . . . – physical quantities such as distance, mass, or ve- What Next? locity, Alternative Symmetry- . . . – or economic quantities such as price or cost. Home Page • These numerical values, however, depend on the choice Title Page of a measuring unit: ◭◭ ◮◮ – if we replace the original unit by a new unit which ◭ ◮ is λ times smaller, Page 2 of 40 – then all the numerical values of the corresponding quantity get multiplied by λ . Go Back Full Screen Close Quit
Scale-Invariance: A . . . Heavy-Tailed . . . 2. Scale-Invariance (cont-d) What Is Usually Done • For example: Multi-D Case A Combination . . . – if instead of meters, we start using centimeters – a Main Result 100 smaller unit – to describe distance, Derivation of Student . . . – then all the distances get multiplied by 100, so that, What Next? e.g., 2 m becomes 2 · 100 = 200 cm. Alternative Symmetry- . . . • It is reasonable to require that: Home Page – the fundamental laws describing objects from the Title Page physical world ◭◭ ◮◮ – do not change if we simply change the measuring ◭ ◮ unit. Page 3 of 40 • In other words, it is reasonable to require that the laws Go Back be invariant with respect to scaling x → λ · x . Full Screen Close Quit
Scale-Invariance: A . . . Heavy-Tailed . . . 3. Scale-Invariance (cont-d) What Is Usually Done • Of course: Multi-D Case A Combination . . . – if we change a measuring unit for one quantity, Main Result – then we may need to also correspondingly change Derivation of Student . . . the measuring unit for related quantities as well. What Next? • For example, in a simple motion, the distance d is equal Alternative Symmetry- . . . to the product v · t of velocity v and time t . Home Page • If we simply change the unit of t without changing the Title Page units of d or v , the formula stops working. ◭◭ ◮◮ • However, the formula remains true if we accordingly ◭ ◮ change the unit for velocity. Page 4 of 40 Go Back Full Screen Close Quit
Scale-Invariance: A . . . Heavy-Tailed . . . 4. Scale-Invariance (cont-d) What Is Usually Done • For example: Multi-D Case A Combination . . . – if we started with seconds and m/sec, and we Main Result change seconds to hours, Derivation of Student . . . – then we should also change the measuring unit for What Next? velocity from m/sec to m/hr. Alternative Symmetry- . . . • Thus, scale-invariance means that: Home Page – if we arbitrarily change the units of one or more Title Page fundamental quantities, ◭◭ ◮◮ – then after an appropriate re-scaling of related units, ◭ ◮ – we should get, in the new units, the exact same Page 5 of 40 formula as in the old units. Go Back Full Screen Close Quit
Scale-Invariance: A . . . Heavy-Tailed . . . 5. Heavy-Tailed Distributions: A Situation in What Is Usually Done Which We Expect Scale-Invariance Multi-D Case • Measurements are rarely absolutely accurate. A Combination . . . Main Result • Usually, the measurement result � x is somewhat differ- Derivation of Student . . . ent from the actual (unknown) value x . What Next? • In many cases, we know the upper bound of the mea- Alternative Symmetry- . . . surement error. Home Page • Then, the probability of exceeding this bound is either Title Page equal to 0 or very small (practically equal to 0). ◭◭ ◮◮ • Often, however, the probability of large measurement ◭ ◮ def errors ∆ x = � x − x is not negligible. Page 6 of 40 • In such cases, we talk about heavy-tailed distributions. Go Back • Such distributions are ubiquitous in physics, in eco- Full Screen nomics, etc. Close Quit
Scale-Invariance: A . . . Heavy-Tailed . . . 6. Heavy-Tailed Distributions (cont-d) What Is Usually Done • Interestingly, they have the same shape in different ap- Multi-D Case plication areas. A Combination . . . Main Result • This ubiquity seems to indicate that there is a funda- Derivation of Student . . . mental reason for such distributions. What Next? • It therefore seems reasonable to expect that for this Alternative Symmetry- . . . fundamental law, we have scale-invariance. Home Page • So, for the corresponding pdf ρ ( x ), for every λ > 0, Title Page there exists µ ( λ ) for which ◭◭ ◮◮ ρ ( λ · x ) = µ ( λ ) · ρ ( x ) . ◭ ◮ Page 7 of 40 Go Back Full Screen Close Quit
Scale-Invariance: A . . . Heavy-Tailed . . . 7. Alas, No Scale-Invariant pdf Is Possible What Is Usually Done • At first glance, the above scale-invariance criterion Multi-D Case sounds reasonable, but, alas, it is never satisfied. A Combination . . . Main Result • Indeed, the pdf should be measurable and have � Derivation of Student . . . ρ ( x ) dx = 1. What Next? • It is known that every measurable solution of the above Alternative Symmetry- . . . equation has the form ρ ( x ) = c · x α for some c and α . Home Page • For this function, the integral over the real line is al- Title Page ways infinite: ◭◭ ◮◮ – for α ≥ − 1, it is infinite in the vicinity if 0, while ◭ ◮ – for α ≤ − 1, it is infinite for x → ∞ . Page 8 of 40 Go Back Full Screen Close Quit
Scale-Invariance: A . . . Heavy-Tailed . . . 8. A Simple Explanation of Why Power Laws Are What Is Usually Done the Only Scale-Invariant Ones Multi-D Case • If we assume that ρ ( x ) is differentiable, then the power A Combination . . . laws c · x α can be easily derived. Main Result • Indeed, µ ( λ ) = ρ ( λ · x ) Derivation of Student . . . is differentiable, as a ratio of ρ ( x ) What Next? two differentiable functions ρ ( λ · x ) and ρ ( x ). Alternative Symmetry- . . . Home Page • Since both functions ρ ( x ) and µ ( λ ) are differentiable, we can differentiate both sides of the equation by λ . Title Page • For λ = 1, we get x · dρ = dµ def ◭◭ ◮◮ dx = α · ρ , where α dλ | λ =1 . ◭ ◮ • By moving all the terms containing ρ to the left-hand Page 9 of 40 side and all others to the right, we get dρ ρ = α · dx x . Go Back • Integrating both sides, we get ln( ρ ) = α · ln( x ) + C . Full Screen • Hence for ρ = exp(ln( ρ )), we get ρ ( x ) = c · x α . Close Quit
Scale-Invariance: A . . . Heavy-Tailed . . . 9. What Is Usually Done What Is Usually Done • A usual idea is to abandon scale-invariance completely. Multi-D Case A Combination . . . • For example: Main Result – one of the most empirically successful ways to de- Derivation of Student . . . scribe heavy-tailed distributions What Next? – is to use non-scale-invariant Student distributions , Alternative Symmetry- . . . with the probability density Home Page ρ ( x ) = c · (1 + a · x 2 ) − ν for some c, a, and ν. Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 40 Go Back Full Screen Close Quit
Scale-Invariance: A . . . Heavy-Tailed . . . 10. What We Show in This Talk What Is Usually Done • In this paper, we “rehabilitate” scale-invariance: we Multi-D Case show that: A Combination . . . Main Result – while the distribution cannot be “directly” scale- Derivation of Student . . . invariant, What Next? – it can be “indirectly” scale-invariant. Alternative Symmetry- . . . • Namely. it can be described as a scale-invariant com- Home Page bination of two scale-invariant functions. Title Page • Interestingly, under a few reasonable additional condi- ◭◭ ◮◮ tions, we get exactly Student distributions. ◭ ◮ • Thus, indirect scale-invariance explains their empirical Page 11 of 40 success. Go Back Full Screen Close Quit
Scale-Invariance: A . . . Heavy-Tailed . . . 11. What We Show in This Talk (cont-d) What Is Usually Done • This line of reasoning also provides us with a reason- Multi-D Case able next approximation. A Combination . . . Main Result • Namely, we should try a scale-invariant combination of Derivation of Student . . . three or more scale-invariant functions. What Next? • This approximation is worth trying if we want a more Alternative Symmetry- . . . accurate description. Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 40 Go Back Full Screen Close Quit
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