# The kind of puzzle that Karl Friedrich was after

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Over the last few updates, I have been indulging in the mathematical logic of Gaussian process, eating it with the spoon of mean-reversion. My so-far experience with using the logic of Gaussian process is that of my personal strategy as regards investment in the stock market, and especially as regards those short, periodical episodes of reshuffling in my investment portfolio, when I am exposed to, and I frequently yield to the gambling-like temptation of short trade (see Acceptably dumb proof. The method of mean-reversion , Fast + slower = compound rhythm, the rhythm of life, and We really don’t see small change ). Gambling-like is the key concept here. I engage into quick trade, and I feel that special flow, peculiar to gambling behaviour, and yet I want that flow to weave around a rational strategy, very much in the spirit of Abraham de Moivre’s ‘The doctrine of chances: or, A method of calculating the probabilities of events in play’, published in 1756. A bit of gambling, yes, but informed gambling.

I am trying to understand why a neural network based on mean-reversed prices as input consistently underestimates the real price, and why the whole method of mean-reversion fails with super-stable prices, such as those of cobalt or uranium (see We really don’t see small change).

I like understanding things. I like understanding the deep logic of the things I do and the methods I use. Here comes the object of my deep intellectual dive, the normal distribution. In the two pictures below, you can see the initial outline of the problem.

How does a function, namely that of normal distribution, assist my process of decision making? Of course, the first-order answer is simple: ‘it gives you numbers, bro’, and when you see those numbers you essentially know what to do’. Good, great, but I want to understand HOW EXACTLY those numbers, thus the function I use, match with my thinking and my action.

Good. I have a function, i.e. that of normal distribution, and for some reason that function works. It works geometrically. The whole mathematical expression serves to create a fraction. If you look carefully at the equation, you will understand that with given mean value μ and standard deviation σ, there is no way this function can go above 1. It is always a fraction. A fraction can be seen from different angles. Firstly, it is a portion of something, like a / b, where a < b. There is a bigger something, the denominator of the fraction, σ[(2π)0,5] = σ* 2,506628275. (elevation to power 0,5 replaces the sign of square root, which I cannot reproduce exactly from the keyboard, as a font).  Secondly, as we talk about denominators, a fraction is a change in units of measurement. Instead of measuring reality in units of 1 – the smallest prime number – we measure reality in units of whatever we put in the denominator of the fraction. Thirdly, a fraction is a proportion between two sides of a rectangle, namely the proportion between the shorter side and the longer side.

Good, so what this function of normal distribution represents is a portion cut of a bigger something equal to σ[(2π)0,5], and that something is my unit of measurement, and, in the same time, it is the longer side of a rectangle. The expression σ[(2π)0,5] is something like one dimension of my world, whilst the whole equation of normal distribution, i.e. the value of that function, makes the other dimension. Is the Gaussian world a rectangular world? I need to know. I start talking to dead people. Usually helps. This time, my interlocutor is Karl Friedrich Gauss, in his General Investigations of Curved Surfaces, presented to the Royal Society, October 8th, 1827.

What many people ignore today is that what we call a Gaussian curve is the outcome of a mathematical problem, which, initially, had virtually nothing to do with probability. What Karl Friedrich Gauss (almost) solved was the problem of geodetic measurements, i.e. the distinction between the bird’s flight distance, and the actual length of the same distance on the rugged and uneven surface of the Earth. I know, when we go through mountains, it is sometimes uphill, sometimes downhill, and, on average, it is flat. Still, when you have to build a railroad through the same mountains, the actual length (spell: cost) of rails to put on the ground is much greater than what would be needed for building the same railroad in the plain. That’s the type of puzzle that Karl Friedrich was after.

Someone could say there is no puzzle. You want to know how long a rail do you need to go over a mountain, you send surveyors and they measure it. Splendid. Yet, civil engineering involves some kind of interference with the landscape. I can come up with the idea of putting my railroad alongside like the half-height of the mountain (instead of going right over its top), or maybe we could sort of shave off the top, couldn’t we, civilised people whom we are? Yes, those ideas are all valid, and I can have a lot of them. Sending surveyors each time I come up with a new concept can become terribly time- and money-consuming. What I could do with is a method of approximating each of those alternative distances on a curved surface, a method which finds good compromise between exactitude and simplicity.

Gauss assumed that when we convert the observation of anything curved – rugged land, or the orbit of a planet – into linear equations, we lose information. The challenge is to lose as little an amount thereof as possible. And here the story starts. Below, you will find a short quote from Gauss: the first paragraph of the introduction.

1.

Investigations, in which the directions of various straight lines in space are to be considered, attain a high degree of clearness and simplicity if we employ, as an auxiliary, a sphere of unit radius described about an arbitrary centre, and suppose the different points of the sphere to represent the directions of straight lines parallel to the radii ending at these points. As the position of every point in space is determined by three coordinates, that is to say, the distances of the point from three mutually perpendicular fixed planes, it is necessary to consider, first of all, the directions of the axes perpendicular to these planes. The points on the sphere, which represent these directions, we shall denote by (1), (2), (3). The distance of any one of these points from either of the other two will be a quadrant; and we shall suppose that the directions of the axes are those in which the corresponding coordinates increase.’

Before I go further, a disclaimer is due. What follows is my own development on Karl Friedrich Gauss’s ideas, not an exact summary on his thoughts. If you want to go to the source, go to the source, i.e. to Gauss’s original writings.

In this introductory paragraph, reality is a sphere. Question: what geometrical shape does my perception of reality have? Do I perceive reality as a flat surface, as a sphere (as it is the case with Karl Friedrich Gauss), or maybe is it a cone, or a cube? How can I know what is the geometrical shape of my perception? Good. I feel my synapses firing a bit faster. There is nothing like an apparently absurd, mindf**king question to kick my brain into higher gear. If I want to know what shape of reality I am perceiving, it is essentially about distance.

I approach the thing scientifically, and I start by positing hypotheses. My perceived reality is just a point, i.e. everything could be happening together, without any perceived dimension to it. Sort of a super small and stationary life. I could stretch into a segment, and thus giving my existence at least one dimension to move along, and yet within some limits. If I allow the unknown and the unpredictable into my reality, I can perceive it in the form of a continuous, endless, straight line. Sometimes, my existence can be like a bundle of separate paths, each endowed with its own indefiniteness and its own expanse: this is reality made of a few straight lines in front of me, crossing or parallel to each other. Of course, I can stop messing around with discontinuities and I can generalise those few straight lines into a continuous plane. This could make me ambitious, and I could I come to the conclusion that flat is boring. Then I bend the plane into a sphere, and, finally things get really interesting and I assume that what I initially thought is a sphere is actually a space, i.e. a Russian doll made of a lot of spheres with different radiuses, packed one into the other.

I am pretty sure that anything else can be made out of those seven cases. If, for example, my perceived reality is a tetrahedron (i.e. any of the Egyptian pyramids after having taken flight, as any spaceship should, from time to time; just kidding), it is a reality made of semi-planes delimited by segments, thus the offspring of a really tumultuous relationship between a segment and a plane etc.

Most likely is different from exact. By my own experience of having applied triangulation in real outdoors (back in the day, before Google Maps, and almost right after Gutenberg printed his first Bible), I know that triangulating with two points is sort of tricky. If my map is really precise (low scale, like military grade), and if it is my lucky day, two points yield a reliable positioning. Still, what used to happen more frequently, were doubtful situations. Is the mountain I can see on the horizon the mountain I think it is on the map? Sometimes it is, sometimes not quite. The more points I triangulate my position on, the closer I come to my exact location. If I have like 5 points or more, triangulating on them can even compensate slight inexactitude in the North-positioning of my map.

The partial moral of the fairy tale is that representing my reality as a sphere around me comes with some advantages: I can find my place in that reality (the landscape) by using just an imperfect representation thereof (the map), and some thinking (the pencil, the ruler, and the compass).  I perceive my reality as a sphere, and I assume, following the intuitions of William James, expressed in his ‘Essays in Radical Empiricism’ that “there is only one primal stuff or material in the world, a stuff of which everything is composed, and if we call that stuff ‘pure experience,’ then knowing can easily be explained as a particular sort of relation towards one another into which portions of pure experience may enter. The relation itself is a part of pure experience; one of its ‘terms’ becomes the subject or bearer of the knowledge, the knower,[…] the other becomes the object known.” (Excerpt From: William James. “Essays in Radical Empiricism”. Apple Books).

Good. I’m lost. I can have two alternative shapes of my perceptual world: it can be a flat rectangle, or a sphere, and I keep in mind that both shapes are essentially my representations, i.e. my relations with the primal stuff of what’s really going on. The rectangle serves me to measure the likelihood of something happening, and the unit of likelihood is σ[(2π)0,5]. The sphere, on the other hand, has an interesting property: being in the centre of the sphere is radically different from being anywhere else. When I am in the centre, all points on the sphere are equidistant from me. Whatever happens is always at the same distance from my position: everything is equiprobable. On the other hand, when my current position is somewhere else than the centre of the sphere, points on the sphere are at different distances from me.

Now, things become a bit complicated geometrically, yet they remain logical. Imagine that your world is essentially spherical, and that you have two complementary, perceptual representations thereof, thus two types of maps, and they are both spherical as well. One of those maps locates you in its centre: it is a map of all the phenomena which you perceive as equidistant from you, thus equiprobable as for their possible occurrence. C’mon, you know, we all have that thing: anything can happen, and we don’t even bother which exact thing happens in the first place. This is a state of mind which can be a bit disquieting – it is essentially chaos acknowledged – yet, once you get the hang of it, it becomes interesting. The second spherical map locates you away from its centre, and automatically makes real phenomena different in their distance from you, i.e. in their likelihood of happening. That second map is more structured than the first one. Whilst the first is chaos, the second is order.

The next step is to assume that I can have many imperfectly overlapping chaoses in an otherwise ordered reality. I can squeeze, into an overarching, ordered representation of reality, many local, chaotic representations thereof. Then, I can just slice through the big and ordered representation of reality, following one of its secant planes. I can obtain something that I try to represent graphically in the picture below. Each point under the curve of normal distribution can correspond to the centre of a local sphere, with points on that sphere being equidistant from the centre. This is a local chaos. I can fit indefinitely many local chaoses of different size under the curve of normal distribution. The sphere in the middle, the one that touches the very belly of the Gaussian curve, roughly corresponds to what is called ‘standard normal distribution’, with mean μ = 0, and standard deviation σ =1. This is my central chaos, if you want, and it can have indefinitely many siblings, i.e. other local chaoses, located further towards the tails of the Gaussian curve.

An interesting proportion emerges between the sphere in the middle (my central chaos), and all the other spheres I can squeeze under the curve of normal distribution. That central chaos groups all the phenomena, which are one standard deviation away from me; remember: σ =1. All the points on the curve correspond to indefinitely many intersections between indefinitely many smaller spheres (smaller local chaoses), and the likelihood of each of those intersections happening is always a fraction of σ[(2π)0,5] = σ* 2,506628275. The normal curve, with its inherent proportions, represents the combination of all the possible local chaoses in my complex representation of reality.

Good, so when I use the logic of mean-reversion to study stock prices and elaborating a strategy of investment, thus when I denominate the differences between those prices and their moving averages in units of standard deviation, it is as if I assumed that standard deviation makes σ =1. In other words, I am in the sphere of central chaos, and I discriminate stock prices into three categories, depending on the mean-reversed price. Those in the interval -1 ≤ mean-reversed price ≤ 1 are in my central chaos, which is essentially the ‘hold stock’ chaos. Those, which bear a mean-reversed price < -1, are in the peripheral chaos of the ‘buy’ strategy. Conversely, those with mean-reversed price > 1 are in another peripheral chaos, that of ‘sell’ strategy.

Now, I am trying to understand why a neural network based on mean-reversed prices as input consistently underestimates the real price, and why the whole method of mean-reversion fails with super-stable prices, such as those of cobalt or uranium (see We really don’t see small change). When prices are super-stable, thus when the moving standard deviation is σ = 0, mean-reversion, with its denomination in standard deviations, yields the ‘Division by zero!’ error, which is the mathematical equivalent of ‘WTF?’. When σ = 0, my central chaos (the central sphere under the curve) shrinks a point, devoid of any radius. Interesting. Things that change below the level of my perception deprive me of my central sphere of chaos. I am left just with the possible outliers (peripheral chaoses) without a ruler to measure them.

As regards the estimated output of my neural network (I mean, not the one in my head, the one I programmed) being consistently below real prices, I understand it as a proclivity of said network to overestimate the relative importance of peripheral chaoses in the [x < -1] [buy] zone, and, on the other hand, to underestimate peripheral chaoses existing in the [x > 1] [sell] zone. My neural network is sort of myopic to peripheral chaoses located far above (or to the right of, if you prefer) the center of my central chaos. If, as I deeply believe, the logic of mean-reversion represents an important cognitive structure in my mind, said mind tends to sort of leave one gate unguarded. In the case of price estimation, it is the gate of ‘sell’ opportunities, which, in turn, leads me to buy and hold whatever I invest in, rather than exchanging it back into money (which is the exact economic content of what we call ‘selling’).

Interesting. When I use the normal distribution to study stock prices, one tail of the distribution – the one with abnormally high values – is sort of neglected to the benefit of the other tail, that with low values. It looks like the normal distribution is not really normal, but biased.

# The stubbornly recurrent LCOE

I am thinking about those results I got in my last two research updates, namely in “The expected amount of what can happen”, and in “Contagion étonnement cohérente”. Each time, I found something intriguingly coherent in mathematical terms. In “The expected amount of what can happen”, I have probably nailed down some kind of cycle in business development, some 3 – 4 years, as regards the FinTech industry. In “Contagion étonnement cohérente”, on the other hand, I have seemingly identified a cycle of behavioural change in customers, like around 2 months, which allows to interpolate two distinct, predictive models as for the development of a market: the epidemic model based on a geometric-exponential function, and the classical model of absorption based on the normal distribution. That cycle of behavioural change looks like the time lap to put into an equation, where the number of customers is a function of time elapsed, like n(t) = e0,69*t.  Why ‘0,69’ in n(t) = e0,69*t? Well, the 0,69 fits nicely, when the exponential function n(t) = eß*tneeds to match a geometric process that duplicates the number of customers at every ‘t’ elapsed, like n(t) = 2*n(t-1) + 1.

I have identified those two cycles of change, thus, and they both look like cycles of behavioural change. It takes a FinTech business like 3+ years to pass from launching a product to stabilizing it, and it apparently takes the customers some 2 months to modify significantly their behaviour – or to take a distinctive, noticeable step in such behavioural change – regarding a new technology. I am trying to wrap my mind around each of those cycles separately, as well as around their mutual connection. It seems important for continuing to write that business plan of mine for the EneFinproject, that FinTech concept for the market of energy, where households and small businesses would buy their energy through futures contracts combined with participatory deeds in the balance sheet of the energy provider.

Now, before I go further, a little explanation for those of you, who might not quite grasp the way I run this blog. This is a research log in the most literal sense of the term. I write and publish as I think about things and as I channel my energy into the thinking. This blog is the living account of what I do, not a planned presentation. As for what I do the latter category, you can find it under the heading of “Your takeaways / Vos plats à emporter“. The approach I use, the one from the side of raw science on the make, is the reason why you can see me coming and going about ideas, and this is why I write in two languages: English and French. I found out that my thinking goes just sort of better when I alternate those two.

Anyway, I am trying to understand what I have discovered, I mean those two intriguing cycles of behavioural change, and I want to incorporate that understanding in the writing of my business plan for the EneFinproject. Cycle of change spells process: there is any point of talking about a cycle if it happens like recurrently, with one cycle following a previous cycle.

So, I do what I need to do, namely I am sketching the landscape. I am visualising urban networks composed of wind turbines with vertical axis, such as I started visualising in « Something to exploit subsequently». Each network has a different operator, who maintains a certain number of turbines scattered across the city. Let this city be Lisbon, Portugal, one of my favourite places in Europe, which, on the top of all its beauty, allows experiencing that shortest interval of time in the universe, i.e. the time elapsing between the traffic lights turning greed, for vehicles, and someone from among said vehicles hooting impatiently.

We are in Lisbon, and there are local operators of urban wind turbines, and with the wind speed being 4,47 m/s on average, each turbine, such as described in the patent application no. EP 3 214 303 A1, generates an electric power averaging 47,81 kilowatts. That makes 47,81 kilowatts * 8760 hours in the normal calendar year = 418 815,60 kilowatt hoursof energy a year. At €0,23 for each kWh at the basic price for households, in Portugal, the output of one turbine is worth like € 96 327,59. According to the basic scheme of EneFin, those € 96 327,59 further split themselves in two, and make:

 € 50 257,87in Futures contracts on energy, sold to households at the more advantageous rate of €0,12, normally reserved for the big institutional end users € 46 069,72in Participatory deeds in the balance sheet of the operator who currently owns the turbine

Thus, each local operator of those specific wind turbines has a basic business unit – one turbine – and the growth of business is measured at the pace of developing such consecutive units. Now, the transactional platform « EneFin» implants itself in this market, as a FinTech utility for managing financial flows between the local operators of those turbines, on the one hand, and the households willing to buy energy from those turbines and invest in their balance sheet. I assume, for the moment, that EneFin takes 5% of commissionon the trading of each complex contract. One turbine generates 5%*€ 96 327,59 =  € 4 816,38 of commission to EneFin.

I am progressively make the above converge with those cycles I have identified. In the first place, I take those two cycles I have identified, i.e. the ≈ 2 months of behavioural change in customers, and the ≈ 3+ years of business maturation. On the top of that, I take the simulations of absorption, as you can see in « Safely narrow down the apparent chaos». That means I take into account still another cycle, that of 7 years = 84 months for the absorption of innovation in the market of renewable energies. As I am having a look at the thing, I am going to start the checking with the last one. Thus, I take the percentages of the market, calculated « Safely narrow down the apparent chaos», and I apply them to the population of Lisbon, Portugal, i.e. 2 943 000 peopleas for the end of 2017.

The results of this particular step in my calculations are shown in Table 1 below. Before I go interpreting and transforming those numbers, further below the table, a few words of reminder and explanation for those among the readers, who might now have quite followed my previous updates on this blog. Variability of the population is the coefficient of proportion, calculated as the standard deviation divided by the mean, said mean being the average time an average customer needs in order to switch to a new technology. This average time, in the calculations I have made so far, is assumed to be 7 years = 84 months. The coefficient of variability reflects the relative heterogeneity of the population. The greater its value, the more differentiated are the observable patterns of behaviour. At v = 0,2it is like a beach, in summer, on the Mediterranean coast, or like North Korea, i.e. people behaving in very predictable, and very recurrent ways. At v = 2, it is more like a Halloween party: everybody tries to be original.

Table 1

 Number of customers acquired in Lisbon [a] [b] [c] [d] Variability of the population 12th month 24th month 36th month 0,1 0 0 0 0,2 30 583 6 896 0,3 5 336 25 445 86 087 0,4 29 997 93 632 212 617 0,5 61 627 161 533 310 881 0,6 85 978 206 314 365 497 0,7 100 653 229 546 387 893 0,8 107 866 238 238 390 878 0,9 110 200 238 211 383 217 1 109 574 233 290 370 157 1,1 107 240 225 801 354 689 1,2 103 981 217 113 338 471 1,3 100 272 208 016 322 402 1,4 96 397 198 958 306 948 1,5 92 525 190 184 292 331 1,6 88 753 181 821 278 638 1,7 85 134 173 925 265 878 1,8 81 695 166 513 254 020 1,9 78 446 159 577 243 014 2 75 386 153 098 232 799

Now, I do two things to those numbers. Firstly, I try to make them kind of relative to incidences of epidemic contagion. Mathematically, it means referring to that geometric process, which duplicates the number of customers at every ‘t’ elapsed, like n(t) = 2*n(t-1) + 1, which is nicely (almost) matched by the exponential function n(t) = e0,69*t. So what I do now is to take the natural logarithm out of each number in columns [b] – [d]in Table 1, and I divide it by 0,69. This is how I get the ‘t’, or the number of temporal cycles in the exponential function n(t) = e0,69*tso as to obtain the same number as shown in Table 1. Then, I divide the time frames in the headings of those columns, thus, respectively, 12, 24, and 36, by the that number of temporal cycles. As a result, I get the length of one period of epidemic contagion between customers, expressed in months.

Good, let’s diagnose this epidemic contagion. Herr Doktor Wasniewski (this is me) has pinned down the numbers shown in Table 2 below. Something starts emerging, and I am telling you, I don’t really like it. I have enough emergent things, which I have no clue what they mean, on my hands. One more emergent phenomenon is one more pain in my intellectual ass. Anyway, what is emerging, is a pattern of decreasing velocity. When I take the numbers from Table 1, obtained with a classical model of absorption, and based on the normal distribution, those numbers require various paces of epidemic contagion in the behaviour of customers. In the beginning, the contagion need to be f***ing fast, like 0,7 ÷ 0,8 of a month, so some 21 – 24 days. Only in very homogenous populations, with variability sort of v = 0,2, it is a bit longer.

One thing: do not really pay attention to the row labelled ‘Variability of the population 0,1’. This is very homogenous a population, and I placed it here mostly for the sake of contrast. The values in brackets in this particular row of Table 2 are negative, which essentially suggests that if I want that few customers, I need going back in time.

So, I start with quite vivacious a contagion, something to put in the scenario of an American thriller, like ‘World War Z no. 23’. Subsequently, the velocity of contagion is supposed to curb down, to like 1,3 ÷ 1,4 months in the second year, and almost 2 months in the 3rdyear. It correlates surprisingly with that 3+ years cycle of getting some stance in the business, which I have very intuitively identified, using Euclidean distances, in «The expected amount of what can happen». I understand that as the pace of contagion between clients is to slow down, my marketing needs to be less and less aggressive, ergo my business gains in gravitas and respectability.

Table 2

 The length of one temporal period « t » in the epidemic contagion n(t) = 2*n(t-1) + 1 ≈ e0,69*t, in the local market of Lisbon, Portugal [a] [b] [c] [d] Variability of the population 12th month 24th month 36th month 0,1 (0,34) (1,26) (6,55) 0,2 2,44 2,60 2,81 0,3 0,96 1,63 2,19 0,4 0,80 1,45 2,02 0,5 0,75 1,38 1,96 0,6 0,73 1,35 1,94 0,7 0,72 1,34 1,93 0,8 0,71 1,34 1,93 0,9 0,71 1,34 1,93 1 0,71 1,34 1,94 1,1 0,71 1,34 1,94 1,2 0,72 1,35 1,95 1,3 0,72 1,35 1,96 1,4 0,72 1,36 1,97 1,5 0,72 1,36 1,97 1,6 0,73 1,37 1,98 1,7 0,73 1,37 1,99 1,8 0,73 1,38 2,00 1,9 0,73 1,38 2,00 2 0,74 1,39 2,01

The second thing I do to numbers in Table 1 is to convert them into money, and more specifically into: a) the amount of transaction-based fee of 5%, collected by the EneFin platform, when b) the amount of capital collected by the suppliers of energy via the EneFin platform. I start by assuming that my customers are not really single people, but households. The numbers in Table 1, referring to single persons, are being divided by 2,6, which is the average size of one household in Portugal.

In the next step, I convert households into energy. Easy. One person in Portugal consumes, for the strictly spoken household use, some 4 288,92 kWh a year. That makes 11 151,20 kWh per household per year. Now, I convert energy into money, which, in financial terms, means €1 338,14a year in futures contracts on energy, at €0,12 per kWh, and €1 226,63in terms of capital invested in the supplier of energy via those complex contracts in the EneFin way. The commission taken by EneFin is 5%*(€1 338,14+ €1 226,63) =  €128,24. Those are the basic steps that both the operator of urban wind turbines, and the EneFin platform will be taking, in this scenario, as they will attract new customers.

People converted into money are shown in Tables 3 and 4, below, respectively as the amount of transaction-based fee collected by EneFin, and as the capital collected by the suppliers of energy via those complex contracts traded at EneFin. As I connect the dots, more specifically tables 2 – 4, I can see that time matters. Each year, out of the three, makes a very distinct phase. During the 1styear, I need to work my ass off, in terms of marketing, to acquire customers very quickly. Still, it does not make much difference, in financial terms, which exact variability of population is the context of me working my ass off. On the other hand, in the 3rdyear, I can be much more respectable in my marketing, I can afford to go easy on customers, and, in the same time, the variability of the local population starts mattering in financial terms.

Table 3

 Transaction-based fee collected by EneFin in Lisbon Variability of the population 1st year 2nd year 3rd year 0,1 € 0,00 € 0,00 € 1,11 0,2 € 1 458,22 € 28 752,43 € 340 124,01 0,3 € 263 195,64 € 1 255 033,65 € 4 246 097,13 0,4 € 1 479 526,18 € 4 618 201,31 € 10 486 926,46 0,5 € 3 039 639,48 € 7 967 324,44 € 15 333 595,20 0,6 € 4 240 693,13 € 10 176 019,80 € 18 027 422,81 0,7 € 4 964 515,36 € 11 321 936,93 € 19 132 083,67 0,8 € 5 320 300,96 € 11 750 639,54 € 19 279 326,77 0,9 € 5 435 424,51 € 11 749 281,67 € 18 901 432,22 1 € 5 404 510,95 € 11 506 577,11 € 18 257 283,50 1,1 € 5 289 424,10 € 11 137 214,92 € 17 494 337,16 1,2 € 5 128 672,87 € 10 708 687,77 € 16 694 429,35 1,3 € 4 945 700,41 € 10 259 985,98 € 15 901 851,61 1,4 € 4 754 575,54 € 9 813 197,53 € 15 139 607,38 1,5 € 4 563 606,09 € 9 380 437,89 € 14 418 674,83 1,6 € 4 377 570,97 € 8 967 947,88 € 13 743 280,35 1,7 € 4 199 088,86 € 8 578 519,11 € 13 113 914,13 1,8 € 4 029 458,58 € 8 212 936,36 € 12 529 062,43 1,9 € 3 869 177,26 € 7 870 840,04 € 11 986 204,76 2 € 3 718 261,64 € 7 551 243,62 € 11 482 385,83

Table 4

 Capital collected by the suppliers of energy via EneFin, in Lisbon Variability of the population 1st year 2nd year 3rd year 0,1 € 0,00 € 0,00 € 10,63 0,2 € 13 948,06 € 275 020,26 € 3 253 324,36 0,3 € 2 517 495,89 € 12 004 537,77 € 40 614 395,82 0,4 € 14 151 834,00 € 44 173 614,09 € 100 308 629,20 0,5 € 29 074 492,97 € 76 208 352,96 € 146 667 559,88 0,6 € 40 562 705,95 € 97 334 772,00 € 172 434 323,50 0,7 € 47 486 146,88 € 108 295 598,06 € 183 000 528,68 0,8 € 50 889 276,10 € 112 396 186,64 € 184 408 925,42 0,9 € 51 990 445,74 € 112 383 198,48 € 180 794 321,60 1 € 51 694 754,11 € 110 061 702,11 € 174 632 966,74 1,1 € 50 593 935,49 € 106 528 711,32 € 167 335 299,39 1,2 € 49 056 331,91 € 102 429 800,96 € 159 684 091,36 1,3 € 47 306 179,81 € 98 137 917,98 € 152 102 996,27 1,4 € 45 478 048,96 € 93 864 336,33 € 144 812 044,65 1,5 € 43 651 404,71 € 89 724 941,73 € 137 916 243,78 1,6 € 41 871 957,84 € 85 779 428,52 € 131 456 019,80 1,7 € 40 164 756,47 € 82 054 498,57 € 125 436 061,23 1,8 € 38 542 223,78 € 78 557 658,50 € 119 841 889,05 1,9 € 37 009 114,98 € 75 285 468,80 € 114 649 394,41 2 € 35 565 590,09 € 72 228 493,11 € 109 830 309,84

Now, I do one final check. I take the formula of LCOE, or the levelized cost of energy, as shown in the formula below:

Symbols in the equation have the following meaning: a) Itis the capital invested in period t b) Mtstands for the cost of maintenance in period t c) Ftsymbolizes the cost of fuel in period t and d) Etis the output of energy in period t. I assume that wind is for free, so my Ftis zero. I further assume that It+ Mtmake a lump sum of capital, acquired by the supplier of energy, and equal to the amounts of capital calculated in Table 4. Thus I take those amounts from Table 4, and I divide each of them by the energy consumed in the corresponding headcount of households. Now, it becomes really strange: whatever the phase in time, and whatever the variability of behaviour assumed in the local population, the thus-computed LCOE is always equal to €0,11. Always! Can you understand? Well, if you do, you are smarter than me, because I don’t. How can so differentiated an array of numbers, in Tables 1 – 4, yield one and the same cost of energy, those €0,11? Honestly, I don’t know.

Calm down, Herr Doktor Wasniewski. This is probably how those Greeks hit their π. Maybe I am hitting another one. I am trying to take another path. I take the number(s) of people from Table 1, I take their average consumption of energy, as official for Portugal – 4 288,92 kWh a year per person – and, finally, I take the 47,81 kilowattsof capacity in one single wind turbine, as described in the patent application no. EP 3 214 303 A1, in Lisbon, with the wind speed 4,47 m/s on average. Yes, you guessed right: I want to calculate the number of such wind turbines needed to supply energy to the given number of people, as shown in Table 1. The numerical result of this particular path of thinking is shown in Table 5 below.

The Devil never sleeps, as we say in Poland. Bloody right. He has just tempted me to take the capital amounts from Table 4 (above) and divide them by the number of turbines from Table 5. Guess what. Another constant. Whatever the exact variability in behaviour, and whatever the year, it is always €46 069,64. I can’t help it, I continue. I take that constant €46 069,64 of capital invested per one turbine, and I divide it by the constant LCOE €0,11 per kWh, and it yields  418 815,60 kWh, or 37,56 households (2,6 person per household) per turbine, in order to make it sort of smooth in numbers.

Table 5

 Number of wind turbines needed for the number of customers as in Table 1 Variability of the population 1st year 2nd year 3rd year 0,1 0 0 0 0,2 0 6 71 0,3 55 261 882 0,4 307 959 2 177 0,5 631 1 654 3 184 0,6 880 2 113 3 743 0,7 1 031 2 351 3 972 0,8 1 105 2 440 4 003 0,9 1 129 2 439 3 924 1 1 122 2 389 3 791 1,1 1 098 2 312 3 632 1,2 1 065 2 223 3 466 1,3 1 027 2 130 3 302 1,4 987 2 037 3 143 1,5 948 1 948 2 994 1,6 909 1 862 2 853 1,7 872 1 781 2 723 1,8 837 1 705 2 601 1,9 803 1 634 2 489 2 772 1 568 2 384

Another thing to wrap my mind around. My brain needs some rest. Enough science for today. I am consistently delivering good, almost new science to my readers, and love doing it, and I am working on crowdfunding this activity of mine. As we talk business plans, I remind you that you can download, from the library of my blog, the business plan I prepared for my semi-scientific project Befund  (and you can access the French versionas well). You can also get a free e-copy of my book ‘Capitalism and Political Power’ You can support my research by donating directly, any amount you consider appropriate, to my PayPal account. You can also consider going to my Patreon pageand become my patron. If you decide so, I will be grateful for suggesting me two things that Patreon suggests me to suggest you. Firstly, what kind of reward would you expect in exchange of supporting me? Secondly, what kind of phases would you like to see in the development of my research, and of the corresponding educational tools?

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# Safely narrow down the apparent chaos

There is that thing about me: I like understanding. I represent my internal process of understanding as the interplay of three imaginary entities: the curious ape, the happy bulldog, and the austere monk. The curious ape is the part of me who instinctively reaches for anything new and interesting. The curious ape does basic gauging of that new thing: ‘can kill or hopefully not always?’, ‘edible or unfortunately not without risk?’ etc. When it does not always kill and can be eaten, the happy bulldog is released from its leash. It takes pleasure in rummaging around things, sniffing and digging in the search of adjacent phenomena. Believe me, when my internal happy bulldog starts sniffing around and digging things out, they just pile up. Whenever I study a new topic, the folder I have assigned to it swells like a balloon, with articles, books, reports, websites etc. A moment comes when those piles of adjacent phenomena start needing some order and this is when my internal austere monk steps into the game. His basic tool is the Ockham’s razor, which cuts the obvious from the dubious, and thus, eventually, cuts bullshit off.

In my last update in French, namely in Le modèle d’un marché relativement conformiste, I returned to that business plan for the project EneFin, and the first thing my internal curious ape is gauging right now is the so-called absorption by the market. EneFin is supposed to be an innovative concept, and, as any innovation, it will need to kind of get into the market. It can do so as people in the market will opt for shifting from being just potential users to being the actual ones. In other words, the success of any business depends on a sequence of decisions taken by people who are supposed to be customers.

People are supposed to make decisions regarding my new products or technologies. Decisions have their patterns. I wrote more about this particular issue in an update on this blog, entitled ‘And so I ventured myself into the realm of what people think they can do’, for example. Now, I am interested in the more marketing-oriented, aggregate outcome of those decisions. The commonly used theoretical tool here is the normal distribution(see for example Robertson): we assume that, as customers switch to purchasing that new thing, the population of users grows as a cumulative normal fraction (i.e. fraction based on the normal distribution) of the general population.

As I said, I like understanding. What I want is to really understandthe logic behind simulating aggregate outcomes of customers’ decisions with the help of normal distribution. Right, then let’s do some understanding. Below, I am introducing two graphical presentations of the normal distribution: the first is the ‘official’ one, the second, further below, is my own, uncombed and freshly woken up interpretation.

So, the logic behind the equation starts biblically: in the beginning, there is chaos. Everyone can do anything. Said chaos occurs in a space, based on the constant e = 2,71828, known as the base of the natural logarithm and reputed to be really handy for studying dynamic processes. This space is ex. Any customer can take any decision in a space made by ‘e’ elevated to the power ‘x’, or the power of the moment. Yes, ‘x’ is a moment, i.e. the moment when we observe the distribution of customers’ decisions.

Chaos gets narrowed down by referring to µ, or the arithmetical average of all the moments studied. This is the expression (x – µ)2or the local variance, observable in the moment x. In order to have an arithmetical average, and have it the same in all the moments ‘x’, we need to close the frame, i.e. to define the set of x’s. Essentially, we are saying to that initial chaos: ‘Look, chaos, it is time to pull yourself together a bit, and so we peg down the set of moments you contain, we draw an average of all those moments, and that average is sort of the point where 50% of you, chaos, is being taken and recognized, and we position every moment xregarding its distance from the average moment µ’.

Thus, the initial chaos ‘e power x’ gets dressed a little, into ‘e power (x – µ)2‘. Still, a dressed chaos is still chaos. Now, there is that old intuition, progressively unfolded by Isaac Newton, Gottfried Wilhelm Leibnizand Abraham de Moivreat the verge of the 17thand 18thcenturies, then grounded by Carl Friedrich Gauss, and Thomas Bayes: chaos is a metaphysical concept born out of insufficient understanding, ‘cause your average reality, babe, has patterns and structures in it.

The way that things structure themselves is most frequently sort of a mainstream fashion, that most events stick to, accompanied by fringe phenomena who want to be remembered as the rebels of their time (right, space-time). The mainstream fashion is observable as an expected value. The big thing about maths is being able to discover by yourself that when you add up all the moments in the apparent chaos, and then you divide the so-obtained sum by the number of moments added, you get a value, which we call arithmetical average, and which actually doesn’t exist in that set of moments, but it sets the mainstream fashion for all the moments in that apparent chaos. Moments tend to stick around the average, whose habitual nickname is ‘µ’.

Once you have the expected value, you can slice your apparent chaos in two, sort of respectively on the right, and on the left of the expected value that doesn’t actually exist. In each of the two slices you can repeat the same operation: add up everything, then divide by the number of items in that everything, and get something expected that doesn’t exist. That second average can have two, alternative properties as for structuring. On the one hand, it can set another mainstream, sort of next door to that first mainstream: moments on one side of the first average tend to cluster and pile up around that second average. Then it means that we have another expected value, and we should split our initial, apparent chaos into two separate chaoses, each with its expected value inside, and study each of them separately. On the other hand, that second average can be sort of insignificant in its power of clustering moments: it is just the average (expected) distance from the first average, and we call it standard deviation, habitually represented with the Greek sigma.

We have the expected distance (i.e. standard deviation) from the expected value in our apparent chaos, and it allows us to call our chaos for further tidying up. We go and slice off some parts of that chaos, which seem not to be really relevant regarding our mainstream. Firstly, we do it by dividing our initial logarithm, being the local variance (x – µ)2, by twice the general variance, or two times sigma power two. We can be even meaner and add a minus sign in front of that divided local variance, and it means that instead of expanding our constant e = 2,71828, into a larger space, we are actually folding it into a smaller space. Thus, we get a space much smaller than the initial ‘e power (x – µ)2‘.

Now, we progressively chip some bits out of that smaller, folded space. We divide it by the standard deviation. I know, technically we multiply it by one divided by standard deviation, but if you are like older than twelve, you can easily understand the equivalence here. Next, we multiply the so-obtained quotient by that funny constant: one divided by the square root of two times π. This constant is 0,39894228 and if my memory is correct is was a big discovery from the part of Carl Friedrich Gauss: in any apparent chaos, you can safely narrow down the number of the realistically possible occurrences to like four tenths of that initial chaos.

After all that chipping we did to our initial, charmingly chaotic ‘e power x‘ space, we get the normal space, or that contained under the curve of normal distribution. This is what the whole theory of probability, and its rich pragmatic cousin, statistics, are about: narrowing down the range of uncertain, future occurrences to a space smaller than ‘anything can happen’. You can do it in many ways, i.e. we have many different statistical distributions. The normal one is like the top dog in that yard, but you can easily experiment with the steps described above and see by yourself what happens. You can kick that Gaussian constant 0,39894228 out of the equation, or you can make it stronger by taking away the square root and just keep two times π in its denominator; you can divide the local variance (x – µ)2just by one time its cousin general variance instead of twice etc. I am persuaded that this is what Carl Friedrich Gaussdid: he kept experimenting with equations until he came up with something practical.

And so am I, I mean I keep experimenting with equations so as to come up with something practical. I am applying all that elaborate philosophy of harnessed chaos to my EneFinthing and to predicting the number of my customers. As I am using normal distribution as my basic, quantitative screwdriver, I start with assuming that however many customers I got, that however many is always a fraction (percentage) of a total population. This is what statistical distributions are meant to yield: a probability, thus a fraction of reality, elegantly expressed as a percentage.

I take a planning horizon of three years, just as I do in the Business Planning Calculator, that analytical tool you can download from a subpage of https://discoversocialsciences.com. In order to make my curves smoother, I represent those three years as 36 months. This is my set of moments ‘x’, ranging from 1 to 36. The expected, average value that does not exist in that range of moments is the average time that a typical potential customer, out there, in the total population, needs to try and buy energy via EneFin. I have no clue, although I have an intuition. In the research on innovative activity in the realm of renewable energies, I have discovered something like a cycle. It is the time needed for the annual number of patent applications to double, with respect to a given technology (wind, photovoltaic etc.). See Time to come to the ad rem, for example, for more details. That cycle seems to be 7 years in Europe and in the United States, whilst it drops down to 3 years in China.

I stick to 7 years, as I am mostly interested, for the moment, in the European market. Seven years equals 7*12 = 84 months. I provisionally choose those 84 months as my average µfor using normal distribution in my forecast. Now, the standard deviation. Once again, no clue, and an intuition. The intuition’s name is ‘coefficient of variability’, which I baptise ßfor the moment. Variability is the coefficient that you get when you divide standard deviation by the mean average value. Another proportion. The greater the ß, the more dispersed is my set of customers into different subsets: lifestyles, cities, neighbourhoods etc. Conversely, the smaller the ß, the more conformist is that population, with relatively more people sailing in the mainstream. I casually assume my variability to be found somewhere in 0,1 ≤ ß ≤ 2, with a step of 0,1. With µ = 84, that makes my Ω (another symbol for sigma, or standard deviation) fall into 0,1*84 ≤ Ω ≤ 2*84 <=> 8,4 ≤ Ω ≤ 168. At ß = 0,1 => Ω = 8,4my customers are boringly similar to each other, whilst at ß = 2 => Ω = 168they are like separate tribes.

In order to make my presentation simpler, I take three checkpoints in time, namely the end of each consecutive year out of the three. Denominated in months, it gives: the 12thmonth, the 24thmonth, and the 36thmonth. I Table 1, below, you can find the results: the percentage of the market I expect to absorb into EneFin, with the average time of behavioural change in my customers pegged at µ = 84, and at various degrees of disparity between individual behavioural changes.

Table 1 Simulation of absorption in the market, with the average time of behavioural change equal to µ = 84 months

 Percentage of the market absorbed Variability of the population Standard deviation with µ = 84 12th month 24 month 36 month 0,1 8,4 8,1944E-18 6,82798E-13 7,65322E-09 0,2 16,8 1,00458E-05 0,02% 0,23% 0,3 25,2 0,18% 0,86% 2,93% 0,4 33,6 1,02% 3,18% 7,22% 0,5 42 2,09% 5,49% 10,56% 0,6 50,4 2,92% 7,01% 12,42% 0,7 58,8 3,42% 7,80% 13,18% 0,8 67,2 3,67% 8,10% 13,28% 0,9 75,6 3,74% 8,09% 13,02% 1 84 3,72% 7,93% 12,58% 1,1 92,4 3,64% 7,67% 12,05% 1,2 100,8 3,53% 7,38% 11,50% 1,3 109,2 3,41% 7,07% 10,95% 1,4 117,6 3,28% 6,76% 10,43% 1,5 126 3,14% 6,46% 9,93% 1,6 134,4 3,02% 6,18% 9,47% 1,7 142,8 2,89% 5,91% 9,03% 1,8 151,2 2,78% 5,66% 8,63% 1,9 159,6 2,67% 5,42% 8,26% 2 168 2,56% 5,20% 7,91%

I think it is enough science for today. That sunlight will not enjoy itself. It needs me to enjoy it. I am consistently delivering good, almost new science to my readers, and love doing it, and I am working on crowdfunding this activity of mine. As we talk business plans, I remind you that you can download, from the library of my blog, the business plan I prepared for my semi-scientific project Befund  (and you can access the French versionas well). You can also get a free e-copy of my book ‘Capitalism and Political Power’ You can support my research by donating directly, any amount you consider appropriate, to my PayPal account. You can also consider going to my Patreon pageand become my patron. If you decide so, I will be grateful for suggesting me two things that Patreon suggests me to suggest you. Firstly, what kind of reward would you expect in exchange of supporting me? Secondly, what kind of phases would you like to see in the development of my research, and of the corresponding educational tools?

Support this blog

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# If I want to remain bluntly quantitative

### My editorial

I am still mining my database in order to create some kind of theoretical model for explaining the relative importance of renewable energies in a given society. Now, I am operating with two variables for measuring said importance. Firstly, it is the percentage of renewables in the final consumption of energy (https://data.worldbank.org/indicator/EG.FEC.RNEW.ZS ). This is renewable energy put into the whole dish of energy that we, humans, use in ways other than feeding ourselves: driving around, air-conditioning, texting to girlfriends and boyfriends, launching satellites, waging war on each other and whatnot. The second estimate is the percentage of renewable energy in the primary production of electricity (https://data.worldbank.org/indicator/EG.ELC.RNEW.ZS ). That one is obviously, much nicer and gentler a variable, less entangled sociologically and biologically. These two are correlated with each other. In my database, they are Pearson-correlated at r = 0,676. This is a lot, for a Pearson-correlation of moments, and still it means that these two mutually explain their respective variances at more or less R2 = 0,6762 = 0,456976. Yes, this is the basic meaning of that R2 coefficient of determination, which kind of comes along whenever I or someone else presents the results of quantitative regression. I take each explanatory variable in my equation, so basically each variable I can find on the right side, and I multiply it, for each empirical observation, by the coefficient of regression attached to this variable.

When I am done with multiplication, I do addition and subtraction: I sum up those partial products, and I subtract this sum, for each particular observation, from the actual value of the explained variable, or the one on the left side of the equation. What I get is a residual constant, so basically the part of the actually observed explained variable, which remains unmatched by this sum of products ‘coefficient of regression times the value of the explanatory variable’. I make an arithmetical average out of those residuals, and I have the general constant in those equations I use to present you on this blog whenever I report the results of my quantitative tests. Once I have that general function, I trace it as a line, and I compute the correlation between this line, and the actual distribution of my left-hand variable, the explained one. This correlation tells me how closely my theoretical line follows the really observed variable.

Now, why the heck elevating this coefficient of correlation to power two, so why converting the ‘r’ into the capital ‘R2’? Well, correlations, as you probably know, can be positive or negative in their sign. What I want, though, is kind of a universal indicator of how close did I land to real life in my statistical simulations. As saying something like ‘you are minus 30% away from reality’ sounds a bit awkward, and as you cannot really have a negative distance, the good idea is to get rid of the minuses by elevating them power two. It can be any even power, by the way. There is no mathematical reason for calculating R2 instead of R22, for instance, only the coefficients of correlation are fractions, whose module is always smaller than one. If you elevate a decimal smaller than one to power 22, you get something so tiny you even have problems thinking about it without having smoked something interesting beforehand. Thus, R2 is simply handier than R22, with no prejudice to the latter.

Oh, I started doctoring, kind of just by letting myself being carried away. All right, so this is going to be a didactic one. I don’t mind: when I write as if I were doctoring you, I am doctoring myself, as a matter of fact, and it is always a good thing to learn something valuable from someone interesting, for one. For two, this blog is supposed to have educational value. Now, the good move consists in asking myself what exactly do I want to doctor myself about. What kind of surprise in empirical reality made me behave in this squid-like way, i.e. release a cloud of ink? By experience, I know that what makes me doctoring is cognitive dissonance, which, in turn, pokes its head out of my mind when I experience too much incoherence in the facts of life. When I have smeared the jam of my understanding over too big a toast of reality, I feel like adding more jam on the toast.

As I am wrestling with those shares of renewable energies in the total consumption of energy, and in the primary generation of electricity, what I encounter are very different social environments, with very different shares of renewables in their local cocktails of energy, and those shares seem not to be exactly scalable on kind of big, respectable socio-economic traits, like GDP per capita or capital stock per capita. These idiosyncrasies go as far as looking as paradoxes, in some instances. In Europe, we have practically no off-grid electricity from renewable sources. In Africa or in Asia, they have plenty. Building a power source off-grid means, basically, that the operator of the power grid doesn’t give a s*** about it and you are financially on your own. Hence, what you need is capital. Logically, there should be more off-grid power systems in regions with lots of capital per capita, and with a reasonably high density of population. Lots of capital per capita times lots of capita per square kilometre gives lots of money to finance any project. Besides, lots of capital per capita is usually correlated with better an education in the average capita, so with better an understanding of how important it is to have reliable and clean, local sources of energy. Still, it is exactly the opposite that happens: those off-grid, green power systems tend to pop up where there is much less capital per capita and where the average capita has much poorer an access to education.

At the brutal bottom line, it seems that what drives people to install solar farms or windfarms in their vicinity is the lack of access to electricity from power grids – so the actual lack and need of electricity – much more than the fact of being wealthy and well educated. Let’s name it honestly: poverty makes people figure out, and carry out, much more new things than wealth does. I already have in my database one variable, very closely related to poverty: it is food deficit, at the very core of being poor. Dropping food deficit in a model related to very nearly any socio-economic phenomenon instantaneously makes those R2’s ramp up. Still, a paradox emerges: when I put food deficit, or any other variable reflecting true poverty, into a quantitative model, I can test it only on those cases, where this variable takes a non-null value. Where food deficit is zero, I have a null value associated with non-null values in other variables, and such observations are automatically cut out of my sample. With food deficit in an equation, empirical tests yield their results only regarding those countries and years, where and when local populations actually starved. I can test with Ethiopia, but I cannot test with Belgium. What can I do in such case? Well, this is where I can use that tool called ‘control variable’. If dropping a variable into an equation proves kind of awkward, I can find a way around it by keeping that variable out of the equation but kind of close to. This is exactly what I did when I tested some of my regressions in various classes of food deficit (see, for example ‘Cases of moderate deprivation’ ).

Good, so I have that control variable, and different versions of my basic model, according to the interval of values in said control variable. I kind of have two or more special cases inside a general theoretical framework. The theory I can make out of it is basically that there are some irreducible idiosyncrasies in my reality. Going 100% green, in a local community in Africa or in Asia is so different from going the same way inside European Union that it risks being awkward to mix those two in the same cauldron. If I want that SEAP, or Sustainable Energy Action plan (see the website of the Global Covenant of Mayors for more information ), and I want it to be truly good a SEAP, it has to be based on different socio-economic assumptions according the way local communities work. One SEAP for those, who starve more or less, and have problems with basic access to electricity. Another SEAP for the wealthy and well-educated ones, whose strive for going 100% green is driven by cultural constructs rather than by bare needs.

Right, it is time to be a bit selfish, thus to focus on my social environment, i.e. Poland and Europe in general, where no food deficit is officially reported at the national scale. I take that variable from the World Bank –  the percentage of renewable energy in the primary production of electricity (https://data.worldbank.org/indicator/EG.ELC.RNEW.ZS ) – and I name it ‘%RenEl’, and I am building a model of its occurrence. It is quite hard to pin down the correlates of this variable as such. There seems to be a lot of history in the belt of each country as for their power systems and therefore it is hard to capture those big, macroeconomic connections. Interestingly, its strongest correlation is with that other metric of energy structure, namely the percentage of renewables in the final consumption of energy (https://data.worldbank.org/indicator/EG.FEC.RNEW.ZS ), or ‘%Ren’ in acronym. This is logical: the ways we produce energy are linked to the ways we consume it. Still, I have a basic intuition that in relatively non-starving societies people have energy to think, so they have labs, and they make inventions in those labs, and they kind of speed up their technological change with those inventions. Another intuition that I have regarding my home continent is that we have big governments, with lots of liquid assets in their governmental accounts. Those liquid public assets are technically the residual difference between gross public debt and net public debt. Hence, I take the same formal hypothesis I made in ‘Those new SUVs are visibly purchased with some capital rent’ and I pepper it with the natural logarithms of, respectively, the number of patent applications per million people (‘PatApp/Pop’), and the share of liquid public assets in the GDP (‘LPA/GDP’). That makes me state formally that

ln(%RenEl) = a1*ln(GDP per capita) + a2*ln(Pop) + a3*ln(Labsh) + a4*ln(Capital stock per capita) + a5*ln(%Ren) + a6*ln(LPA/GDP) + a7*ln(PatApp/Pop) + residual

When I test this equation in my database, I can see an interesting phenomenon. The fact of adding the ln(PatApp/Pop) factor to my equation, taken individually, adds some 2% to the overall explanatory power of my model. Without the ln(PatApp/Pop), my equation is being tested on n = 2 068 observations and yields a coefficient of determination equal to R2 = 0,492. Then, I drop a pinch of ln(PatApp/Pop) into my soup, it reduces my sample to n = 1 089 observations, but pumps up the coefficient of determination to R2 = 0,515. Yet, the ln(PatApp/Pop) is not really robustly correlated with my ln(%RenEl): the p-value attached to this correlation is p = 0,503. It means that for any given value of ln(PatApp/Pop), my ln(%RenEl) can be found anywhere in one entire half of the normal distribution. This is one of those cases when I can see a pattern, but I cannot guess what is it exactly what I see.

If I want to remain bluntly quantitative, which sometimes pays off, I take those patent applications out of the equation and park it close by, as a control variable. I make classes in it, or rather it is my software, Wizard for MacOS that does, and I test my equation without ln(PatApp/Pop)[1] in it in those various classes and I look at the values my R2 coefficient takes in each of those classes. Here are the results:

Class #1: no patent applications observed >> 979 observations yield R2 = 0,490

Class #2: less than 3,527 patent applications per million people >> 108 observations yield R2 = 0,729

Class #3: between 3,527 and 23,519 patent applications per million people >> 198 observations and R2 = 0,427

Class #4: 23,519 < PatApp/Pop < 77,675   >> 267 observations and R2 = 0,625

Class #5: 77,675 < PatApp/Pop < 160,682  >> 166 observations and R2 = 0,697

Class #6: 160,682 < PatApp/Pop < 290,87  >> 204 observations and R2 = 0,508

Class #7: 290,87 < PatApp/Pop < 3 276,584 >> 146 observations and R2 = 0,965

Now, I can see there are two sub-samples in my sample – countries with really low rate of invention and those with an extremely high one – where the equation really works much stronger than anywhere else (much higher an R2 than in other classes). This is the job a control variable can do: it can serve to define special cases and to refine my hypotheses. Now, I can say, for example, that when the local rate of patentable invention in a society is really high, I can make a very plausible model of them going 100% green in their electricity output.

[1] So it is ln(%RenEl) = a1*ln(GDP per capita) + a2*ln(Pop) + a3*ln(Labsh) + a4*ln(Capital stock per capita) + a5*ln(%Ren) + a6*ln(LPA/GDP) + residual