Why quadratic log-log dependence is ubiquitous and what next

Abstract

Purpose

In many real-life situations ranging from financial to volcanic data, growth is described either by a power law – which is linear in log-log scale or by a quadratic dependence in the log-log scale. The purpose of this paper is to explain this empirical fact.

Design/methodology/approach

The authors use natural scale invariance requirements.

Findings

In this paper, the authors used natural scale invariance requirement to explain the ubiquity of quadratic log-log dependencies. The authors also explain what to do if quadratic log-log models turn out to be insufficiently accurate. In this case, scale-invariance requirements lead to dependencies which in the log-log scale take cubic, 4th order, etc. form.

Originality/value

To the best of authors’ knowledge, this is the first theoretical explanation of the empirical quadratic log-log dependence.

Citation

Aguilar, S.R., Kreinovich, V. and Pham, U. (2021), "Why quadratic log-log dependence is ubiquitous and what next", Asian Journal of Economics and Banking, Vol. 5 No. 1, pp. 25-31. https://doi.org/10.1108/AJEB-07-2020-0034

1. Formulation of the problem

Predictions: a typical situation. One of the main objective of science and engineering is to predict the future state of the world – i.e. the future values of the quantities that describe this state – and to come up with measures that lead to the most favorable future state. For example, we want to predict tomorrow’s weather – and if it will be catastrophic in a given area, we need to plan corresponding closings and, if needed, evacuations. We want to predict next year’s gross domestic product (GDP) – and if the current trends predict a crisis, we want to come up with measures that would prevent this crisis – or at least decrease its severity.

In some situations, we can predict the future values of some quantities with high accuracy. For example, we can predict solar eclipses centuries ahead.

However, such situations are rare. In most real-life situations, we cannot make exact predictions: e.g. when we predict the weather, there are many factors that affect tomorrow’s weather, and that we, at present, do not know. In such situations, we can predict, at best, the probabilities of future values of the corresponding quantity. These probabilities can be described, e.g. by the probability density function (pdf) ρ(x). Such situations are typical in economics and finance; they are also typical in geosciences – e.g. in predicting volcanic activity, they are typical in many other application areas.

Stationary vs non-stationary situations. In some cases – e.g. in celestial mechanics or, for a reasonably short period, in weather prediction – the corresponding probabilities remain the same day after day and year after year. So, the probability density function ρ_t(x) remains the same for all moments of time t: ρ_t(x) = ρ(x).

However, in many other cases, the situation changes with time: on average, the values x grow with time. This happens with economic characteristics such as GDP or stock prices; this happens with volcanic activity when the volcano becomes more and more active. In many such cases, the shape of the probability distribution remains the same, but the scale changes. In other words, at each moment of time t, the distribution of x is similar to the initial (t = 1) distribution of the quantity xC(t) for some increasing function C(t), i.e. ρ_t(x) should be proportional to ρ1(xC(t)). The coefficient of proportionality can be easily found from the condition that the overall probability should be equal to 1, i.e. that ∫ρt(x) dx=1. Thus, we get:

(1) ρt(x)=1C(t)·ρ(xC(t)).

How C(t) depends on time: empirical fact. In many practical situations, the growth C(t) is described by the power law:

(2) C(t)=A·tb,

for some constants A and b. If we take the logarithm of both sides of this formula, we can conclude that in the log-log scale, the power law becomes a linear dependence:

(3) ln⁡(C(t))=b·ln⁡(t)+ln⁡(A).

In other cases, we have a more complex dependence:

(4) C(t)=A·tb(t)

where:

(5) b(t)=b0+b1·ln⁡(t).

In this case, in log-log-scale, we have:

(6) ln⁡(C(t))=b(t)·ln⁡(t)+ln⁡(A)=b0·ln⁡(t)+b1·(ln⁡(t))2+ln⁡(A).

In other words, we have a quadratic log-log dependence, of which the linear log-log dependence (3) is a particular case corresponding to b₁ = 0.

Such dependencies are really ubiquitous: e.g. an empirical analysis provided in Mariani et al. (2020) has shown that many real-life dependencies, ranging from economic to volcanic data, follow these formulas. This naturally leads to the following questions:

Natural questions:

Q1.

How can we explain the ubiquity of the dependence (4)–(5)?

Q2.

When the match with this formula is not perfect, what more accurate formula should we try?

In this paper, we provide answers to both questions.

2. Analysis of the problem: why power law?

Why power law. In many cases, the dependence of C(t) on t is described by the power law. So, before we analyze the more general question of why the general dependence (4)–(5) is ubiquitous, let us analyze why power law is frequently observed.

Scale invariance: idea. We are interested in learning how the growth function C(t) depends on time t. In our analysis, we use numerical values of C(t) and numerical values of time t. Numerical values of time depend on the measuring unit: we get different values if we use years, quarters, months, days, etc. If we replace the original unit with the one that is λ times smaller, then all numerical values of time t are replaced with new values λ· t. For example, if we replace years with months, then in the new units, the period of two years becomes 12 * 2 = 24 months.

In many cases, we do not have a preferred unit for measuring time. In such situations, it is reasonable to require that the dependence between C and t remains the same if we change the unit for measuring time. This requirement is known as scale invariance.

The requirement of scale invariance is typical in physics: e.g. the formula d = v · t describing the relation between distance d, velocity v and time t remains true, no matter what units we select for measuring time, e.g. hours or seconds.

Of course, we cannot simply assume that the value C(t) remains the same: the growth in two years is clearly different from the growth in two months. This can be easily explained on the example of the above physics formula: if we change the unit for time, then, for the formula to remain valid, we need to also appropriately change the units for other quantities: e.g. replace the velocity unit from km/hour to km/sec. In our cases, this means that if we change a unit for time, then the formula C = C(t) should remain valid if we also appropriately change the unit for C, into a new unit which is μ times smaller, for some μ depending on λ.

In other words, if we have:

(7) C=C(t),

then, for every λ, we should also have:

(8) C′=C(t′),

where:

(9) C′=μ(λ)·C and t′=λ·t.

Scale invariance explains power law. Substituting expression (9) for C′ and t′ into equation (8), we conclude that μ(λ) · C = C(λ· t). Substituting equation (7) for C into this formula, we conclude that:

(10) C(λ·t)=μ(λ)·C(t).

In real life, the dependencies are usually continuous, and for continuous functions, it is known that all solutions of the functional equation (10) have the form C(t) = A·t^b; (Aczél and Dhombres, 2008).

Thus, we indeed have a natural explanation for the power law.

3. How to explain quadratic log-log dependence

Main idea. As we have mentioned earlier, the main reason why the real-life processes are probabilistic is that the actual value x(t) depends also on some characteristics that we do not know. In particular, this means that the value C(t) also depends on such characteristics y₁,…,y_n: C = C(t,y₁,…,y_n).

Let us first consider the simplest such case; when we consider only the dependence of one such characteristics y₁, then C = C(t,y₁).

Let us apply scale invariance to this situation. Now, in addition to time t, we have another quantity y₁ for which we can also select different measuring units. It is, therefore, reasonable to require that no matter how we change both unit for t and unit for y₁, we will get the same dependence.

In precise terms, for every λ > 0 and λ₁ > 0, there exists a value μ(λ, λ₁) such that if:

(11) C=C(t,y1),

then:

(12) C′=C(t′,y′1),

where:

(13) t′=λ·t,  y′1=λ·y1,  C′=μ(λ,λ1)·C.

Let us analyze this situation. Substituting equation (13) into equation (12), we conclude that:

(14) C(λ·t,λ1·y1)=μ(λ,λ1)·C.

Substituting equation (11) for C into equation (14), we get:

(15) C(λ·t,λ1·y1)=μ(λ,λ1)·C(t,y1).

For each y₁, by taking λ₁ = 1, we conclude that:

(16) C(λ·t,y1)=μ(λ,1)·C(t,y1).

Thus, for each y₁, the function Cy1(t)=defC(t,y1) satisfies equation (10). Thus, based on the result cited in the previous section, we have:

(17) Cy1(t)=C(t,y1)=A(y1)·tb(y1),

for some A and b depending on y₁. In particular, in log-log scale, we get:

(18) ln⁡(C(t,y1))=b(y1)·ln⁡(t)+ln⁡(A(y1)).

Similarly, for every t, we can take λ = 1 and get:

(19) C(t,λ1·y)=μ(1,λ1)·C(t,y1).

Thus, for each t, the function Ct(y1)=defC(t,y1) satisfies equation (10). Thus, based on the result cited in the previous section, we have:

(20) Ct(y1)=C(t,y1)=A′(t)·y1b′(t),

for some A′ and b′ depending on t. In particular, in log-log scale, we get:

(21) ln⁡(C(t,y1))=b′(t)·ln⁡(y1)+ln⁡(A′(t)).

Equation (18) and (21) describe the same expression ln(C(t,y₁)). By equating these expressions for two different values t₁ < t₂, we conclude that:

(22) b(y1)·ln⁡(t1)+ln⁡(A(y1))=b′(t1)·ln⁡(y1)+ln⁡(A′(t1));

(23) b(y1)·ln⁡(t2)+ln⁡(A(y1))=b′(t2)·ln⁡(y1)+ln⁡(A′(t2)).

Subtracting equation (22) from equation (23), we get:

(24) b(y1)·(ln⁡(t2)−ln⁡(t1))=(b′(t2)−b′(t1))·ln⁡(y1)+(ln⁡(A′(t2))−ln⁡(A′(t1)).

So, by dividing both sides by the difference ln(t₂) – ln(t₁), we get:

(25) b(y1)=c1·ln⁡(y1)+c2,

where we denoted:

(26) c1=b′(t2)−b′(t1)ln⁡(t2)−ln⁡(t1) and c2=ln⁡(A′(t2))−ln⁡(A′(t1))ln⁡(t2)−ln⁡(t1).

Similarly, if we multiply equation (23) by ln(t₁), equation (22) by ln(t₂) and subtract the results, we get:

(27) ln⁡(A(y1))·(ln⁡(t2)−ln⁡(t1))=(b′(t2)·ln⁡(t1)−b′(t1)·ln⁡(t2))·ln⁡(y1)+(ln⁡(A′(t2))·ln⁡(t1)−ln⁡(A′(t1))·ln⁡(t2)),

Hence:

(28) ln⁡(A(y1))=c3·ln⁡(y1)+c4,

where we denoted:

(29a) c3=b′(t2)·ln⁡(t1)−b′(t1)·ln⁡(t2)ln⁡(t2)−ln⁡(t1)

and:

(29b) c4=ln⁡(A′(t2))·ln⁡(t1)−ln⁡(A′(t1))·ln⁡(t2)ln⁡(t2)−ln⁡(t1).

Substituting equation (25) and (28) into equation (18), we conclude that:

(30) ln⁡(C(t,y1))=(c1·ln⁡(y1)+c2)·ln⁡(t)+(c3·ln⁡(y1)+c4)=c4+c2·ln⁡(t)+c3·ln⁡(y1)+c1·ln⁡(t)·ln⁡(y1).

If we now assume that the dependence of y₁ on t is also scale invariant, then the result of the previous section shows that:

y1=A″·tb″

for some A″ and b″, i.e. in log-log form:

(31) ln⁡(y1)=b″·ln⁡(t)+ln⁡(A″).

Substituting equation (31) into equation (30), we get:

(32) ln⁡(C(t))=ln⁡(C(t,y1(t)))=C0+C1·ln⁡(t)+C2·(ln⁡(t))2,

where:

(33) C0=c4+c3·ln⁡(A″),  C1=c2+c3·b″+c1·ln⁡(A″),  C2=c1·b″.

Thus, we indeed explained the quadratic log-log dependence.

4. What next?

Let us use scale invariance. In general, we have a dependence:

(34) C=C(t,y1,…,yn),

on n ≥ 1 auxiliary quantities. In this general case, scale invariance means for every λ > 0 and for all possible values λ₁ > 0, …, λ_n, there exists a value μ(λ, λ₁,…, λ_n) such that if equation (34) is satisfied, then:

(35) C′=C(t′,y′1,…,y′n),

where:

(36) t′=λ·t,  y′1=λ·y1,…,y′n=λ·yn,  C′=μ(λ,λ1,…,λn)·C.

What we can derive from scale invariance. Similar to the previous section, we can, thus, conclude that the expression ln(C(t,y₁,…,y_n)) is linear in ln(t), linear in ln(y₁), …, and linear in ln(y_n). Thus, it is a multi-linear function:

(37) ln⁡(C(t,y1,…,yn))=c0+ct·ln⁡(t)+∑i=1nci·ln⁡(yi)+∑i=1nct,i·ln⁡(t)·ln⁡(yi)+∑i<jci,j·ln⁡(yi)·ln⁡(yj)+…+ct,1,…,n·ln⁡(t)·ln⁡(y1)·…·ln⁡(yn).

If we assume that the dependence of each auxiliary quantity y_i on t is also scale invariant, then we get:

(38) ln⁡(yi)=b′′i·ln⁡(t)+ln⁡(A′′i)

for some values b″_i and A″_i. Substituting equation (38) into equation (37), we conclude that:

(39) ln⁡(C(t))=C0+C1·ln⁡(t)+C2·(ln⁡(t))2+…+Cn+1·(ln⁡(t))n+1.

Resulting recommendation. So, if quadratic log-log dependence (corresponding to n = 1) is too inaccurate, we need to try cubic log-log dependence (corresponding to n = 2), then, if needed, fourth-order log-log dependence corresponding to n = 3, etc.