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M. Temple-Raston, PhD, Decision Machine/ Precision Alpha
Businesses rely on the fact that people know how to count and tell time, and, subject to clear counting rules, the results are independent of the person counting.
Businesses are also subject to chance, probability, and contingency. To calculate a probability, a set of appropriate states (mutually exclusive and exhaustive) must be defined initially for the problem at hand. Usually the states are obvious: up or not up, etc., so that with this selection of states the question becomes: what is the probability that the next measurement will be “up”? And what of the probability after that? And so on. Forecasting.
In many real-world problems probabilities can be solved exactly (mathematically and with closed-form expressions) through the Science of Counting formulated as modern machine learning.
You would think that this would be better known. To be fair, Data Science is a very young field. It is not surprising that everything we know how to do scientifically has yet to be incorporated. Note that the Science of Counting is part of the undergraduate physics major. More recently E.T. Jaynes teaches the Logic of Science in his superb book on Probability Theory, proving that Human Intelligence can still outshine Artificial Intelligence.
The next section presents the Science of Counting as Machine Learning, and plots some output for discussion. Dissipative structures are important because they are associated with repeatable behavior in non-equilibrium systems (self-regulating).
Section three develops forecasting for the Science of Counting.
Section four discusses contemporary operational challenges to conventional ML that the Science of Counting addresses. Presented as web services, the Science of Counting provides tools for data scientists and business analysts that focus on understanding the reasons for the dynamics of their data, not on model building.
Machine Learning, in general, enforces constraints on maximum entropy as a variational problem. Science itself is based on a faith statement: I believe what I measure. Without this human belief, science can’t exist. By using our measurements as constraints on maximum entropy, Machine Learning and Science are united.
In general, numerical estimates, regressions, simulations, or mathematical bounds can be employed to estimate a probability.
The simplest, non-trivial example of scientific machine learning is exactly solvable and is known as the Science of Counting.
In the science of counting the input data (events, states, or units) are enumerated by humans exactly; so that no error bars are present on the input data (see Figure 1a below for a two state example: counting heads and tails on coin tosses, n+ and n-, respectively, as measurements). By keeping track of the heads and tails on each toss, a path in the grid of all possibilities is determined. Notice that only the intersection points of the grid are relevant, nothing in between the intersection points is relevant. Therefore, there are no error bars on the measurement data because we know that humans know how to count.
In Figure 1b, the path of heads and tails measured in Figure 1a is used to evaluate the energy time-series, so that the energy also has no error bars! The energy of course does not have integer values like the input data, but still has no error bars. Therefore, the science of counting presents precise scientific measurements that can be plotted for better understanding.
As stated above, the science of counting is solved uniquely. Therefore, unlike AI, there is no model or even room to introduce model parameters. The science of counting is the math that implements: “I believe what I measure.”
As we shall now show, the Science of Counting contradicts a modern dogma that science and emotions are in conflict. Emotions, in fact, are integral to science and measurable from the time-series data.
We change the coordinates in Figure 1b to make the plots more expressive. In the two-state system in Figures 1a and 1b, equilibrium occurs when the probability of heads equals the probability of tails (unbiased coin). Equilibrium means that it is just as likely to toss a heads as a tails. Now by subtracting off the equilibrium energy from the total energy, we are left with the displacement energy from equilibrium.
The displacement energy is the net energy entering or exiting the system. When the displacement energy is equal to zero (∆E=0, x-axis in Figure 1b), then there is no energy entering or exiting the system, the total energy is constant (conserved), and therefore defines a mechanical system. We can also say that the system is “objective”. Objectivity states that measurements must be the same for every observer regardless of time — -so no net energy is allowed into or out of the system by any observer. Objective implies mechanics, and mechanics implies objective.
When the displacement energy does not vanish (∆E≠0), the dynamics is neither mechanical nor objective. If something is not objective, then it is subjective. Therefore, there are two natural components to the total energy: a mechanical/objective component, and a non-mechanical/subjective component. When this subjective energy is due to human behavior, then the displacement energy is called emotions, and non-mechanical becomes aesthetics. And with that social emotions can be measured exactly.
In the plots in Figure 2 we observe financial market time-series data. The input data is six months of closing prices for Pepsi (PEP), where we plot only the last three months (Figure 2a). Like the coin tosses, we will count two states: how many days PEP has gone up and how many days PEP has not gone up at each time-step.
Figure 2b is the displacement energy from equilibrium. Clearly the system plotted in Figure 2b most of the time is not in equilibrium (∆E≠0). We see energy flowing into the system in November, and then energy flowing out at the beginning of the year.
In Figure 2c, two sets of next day probabilities are plotted: mechanical (short) and thermal (tall, Crooke). Our derived measurements for financial market time-series indicate that markets are rarely in equilibrium, and usually far from equilibrium (E=0). Away from equilibrium, thermal probabilities routinely exceed 0.65 and account for large price movements.
Finally in Figure 2d below, two important derived measurements for the symbol GE are plotted: the free energy (blue) and temperature (red). Free energy is that part of the energy available to do price work, and the temperature increases as price work is done. The plot of free energy and temperature in Figure 2d also measures the temperature at thermal equilibrium (rectangle and arrow). Non-equilibrium oscillations around thermal equilibrium are also observed in Figure 2d — also known as dissipative structure, or a heat engine. Non-equilibrium thermal oscillations in financial markets are not unusual. In January 2022, a systematic analysis of both the NYSE and NASD was undertaken, and approximately 1/3 of the symbols in both exchanges presented overt non-equilibrium thermal oscillations.
We summarize what we have learned from our analysis so far that could improve forecasting:
- Exact probability should lead to improved forecasting over one that has only estimated probability,
- Dissipative structures have elevated thermal probability that can dominate the system dynamics and should improve the forecast,
- The temperature of the environment, TR, in which the system operates should improve a forecast over one that discards the information.
Field testing indicates that the thermal probability is largest when TR is set to the correct value. TR is a critical value for forecasting. The previous section showed how to measure TR.
With a value for TR, the equations of motions for the Science of Counting can also be derived for all the quantities in Table A. The exact forecasted probability based on learning can be put to immediate use: the exact probability is fed to a state machine to produce an output state. Now, add the output state to the historical record and repeat the process. From this an individual forecast is produced for any finite horizon. An ensemble forecast is created by generating multiple individual forecasts.
In Figure 3 below, the science of counting produces an individual forecast for the same six months of ZIM closing prices as in previous sections. We plot the last 30 days and then the individual forecast to a horizon of 30 days (total of 60). The dotted vertical line is where the individual forecast begins. The state machine with the probability computed by the science of counting appears to behave consistently as we move from past to future.
The consistency and transparency of counting on which business relies, is lost in conventional formulations of machine learning.
The science of counting returns transparency to machine learning by restricting to counting measurements.
With the transparency of math, other problems with conventional machine learning vanish. Notably, in Science of Counting there are:
- No model parameters to estimate,
- No model biases to be concerned about,
- No iterative model development processes to design, implement and accelerate,
simply because there is no model! Table 1 summarizes the key benefits realized from the Science of Counting.
There is still bias to defend against, however, but the bias is entirely a “data bias”. Data biases may well be in the time-series data itself and depend on what and how the time-series data was acquired. In short, the tasks that remain for the data scientist and analyst using the Science of Counting resembles the world of the experimental scientist, rather than the theoretician (model builder).
[1] E.T. Jaynes, Probability Theory: The Logic of Science, Cambridge University Press (2003).
[2] G.E. Crooks, Phys. Rev. E 60, 2721–2726.
[3] D. Kondepudi, I. Prigogine, Modern Thermodynamics: From Heat Engines to Dissipative Structures, Wiley (2015).