Combinatorial Time-Series Measurements
Example measurements of a time-series are presented below for six months of closing price data for General Electric (GE) stock, consisting of time-stamps for the trading day and the associated closing prices (adjusted). The time-series input is plotted in Figure 1. The arithmetic mean (average) of the closing prices is superimposed (dash red curve).
Six-months of Closing Prices for General Electric (GE), plotting only the last three months.
Several scientific measurements are plotted in Figure 2 below for GE’s historical closing prices in Figure 1 above. The measurement service for states counts the number of up or down states based on closing prices. Figure 2a plots the displacement energy, that is, the total energy minus the mechanical energy of a two-state system, equivalently, the energy entering (positive) or exiting (negative) the system. When the displacement energy, E = 0, the system reduces to mechanics and the binary states behave statistically, p+ = p− = 1/2. Figure 2b plots the probabilities, p+(λ,E) (green) and p−(λ,E) (red), that GE stock will go up on the next trading day (green), or, close down on the next trading day (red), respectively. Figure 2c presents a double-sided plot with the time-series measurements for the (entropic) temperature and free-energy (Helmholtz), the energy available to do price movement work. As the free-energy does price movement work by decreasing, the temperature of the system increases. In practice, it is observed that real-world time-series deviate significantly from mechanics and statistical equilibrium, and that systems routinely store and release energy (dissipation of entropic heat). Observe the dissipative structure in the time-series in Figure 2c. When the internal temperature equals the body temperature, the system is in thermal equilibrium. Finally in Figure 2d, we plot the elevated thermal probabilities for the two-state system closing prices, calculated from non-equilibrium thermodynamics [Crooks].
Science-of-Counting Measurements (clockwise from top left): 2a, Energy Displacement; 2b, Next Day Probabilities (NDP, Up or Down Days); 2c, Temperature and Free Energy; and 2d, Thermal Next Day Probabilities (Therm. NDP, Up or Down).
Measurements for Decision Support
Start with repeated decisions, regular in time, to support a goal. For example, how money is spent operationally every week to support Consumer Product Good (CPG) sales which are seasonal in practice. We plot below the units sold every week (counting time-series, black curve), the running mean (green dashed), and the thermodynamically optimal solution (red curve) that gets the best results for the least energy ($$).
Weekly Sales for a CPG from 2018-2022 (MM/10)
In the science-of-counting, causality and forecasting are lost when energy enters or exits the system. This insight shifts the analytical focus from prediction to adaptive decision making—emphasizing control and responsiveness to changes in energy.