Predictability

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Predictability is the degree to which a correct prediction or forecast of a system's state can be made, either qualitatively or quantitatively.

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Predictability and causality

Causal determinism has a strong relationship with predictability. Perfect predictability implies strict determinism, but lack of predictability does not necessarily imply lack of determinism. Limitations on predictability could be caused by factors such as a lack of information or excessive complexity.

In experimental physics, there are always observational errors determining variables such as positions and velocities. So perfect prediction is practically impossible. Moreover, in modern quantum mechanics, Werner Heisenberg's indeterminacy principle puts limits on the accuracy with which such quantities can be known. So such perfect predictability is also theoretically impossible.

Laplace's demon

Laplace's demon is a supreme intelligence who could completely predict the one possible future given the Newtonian dynamical laws of classical physics and perfect knowledge of the positions and velocities of all the particles in the world. In other words, if it were possible to have every piece of data on every atom in the universe from the beginning of time, it would be possible to predict the behavior of every atom into the future. Laplace's determinism is usually thought to be based on his mechanics, but he could not prove mathematically that mechanics is deterministic. Rather, his determinism is based on general philosophical principles, specifically on the principle of sufficient reason and the law of continuity. [1]

In statistical physics

Although the second law of thermodynamics can determine the equilibrium state that a system will evolve to, and steady states in dissipative systems can sometimes be predicted, there exists no general rule to predict the time evolution of systems distanced from equilibrium, e.g. chaotic systems, if they do not approach an equilibrium state. Their predictability usually deteriorates with time and to quantify predictability, the rate of divergence of system trajectories in phase space can be measured (Kolmogorov–Sinai entropy, Lyapunov exponents).

In mathematics

In stochastic analysis a random process is a predictable process if it is possible to know the next state from the present time.

The branch of mathematics known as Chaos Theory focuses on the behavior of systems that are highly sensitive to initial conditions. It suggests that a small change in an initial condition can completely alter the progression of a system. This phenomenon is known as the butterfly effect, which claims that a butterfly flapping its wings in Brazil can cause a tornado in Texas. The nature of chaos theory suggests that the predictability of any system is limited because it is impossible to know all of the minutiae of a system at the present time. In principle, the deterministic systems that chaos theory attempts to analyze can be predicted, but uncertainty in a forecast increases exponentially with elapsed time. [2]

As documented in, [3] three major kinds of butterfly effects within Lorenz studies include: the sensitive dependence on initial conditions, [4] [5] the ability of a tiny perturbation to create an organized circulation at large distances, [6] and the hypothetical role of small-scale processes in contributing to finite predictability. [7] [8] [9] The three kinds of butterfly effects are not exactly the same.

In human–computer interaction

In the study of human–computer interaction, predictability is the property to forecast the consequences of a user action given the current state of the system.

A contemporary example of human-computer interaction manifests in the development of computer vision algorithms for collision-avoidance software in self-driving cars. Researchers at NVIDIA Corporation, [10] Princeton University, [11] and other institutions are leveraging deep learning to teach computers to anticipate subsequent road scenarios based on visual information about current and previous states.

Another example of human-computer interaction are computer simulations meant to predict human behavior based on algorithms. For example, MIT has recently developed an incredibly accurate algorithm to predict the behavior of humans. When tested against television shows, the algorithm was able to predict with great accuracy the subsequent actions of characters. Algorithms and computer simulations like these show great promise for the future of artificial intelligence. [12]

In human sentence processing

Linguistic prediction is a phenomenon in psycholinguistics occurring whenever information about a word or other linguistic unit is activated before that unit is actually encountered. Evidence from eyetracking, event-related potentials, and other experimental methods indicates that in addition to integrating each subsequent word into the context formed by previously encountered words, language users may, under certain conditions, try to predict upcoming words. Predictability has been shown to affect both text and speech processing, as well as speech production. Further, predictability has been shown to have an effect on syntactic, semantic and pragmatic comprehension.

In biology

In the study of biology – particularly genetics and neuroscience – predictability relates to the prediction of biological developments and behaviors based on inherited genes and past experiences.

Significant debate exists in the scientific community over whether or not a person's behavior is completely predictable based on their genetics. Studies such as the one in Israel, which showed that judges were more likely to give a lighter sentence if they had eaten more recently. [13] In addition to cases like this, it has been proven that individuals smell better to someone with complementary immunity genes, leading to more physical attraction. [14] Genetics can be examined to determine if an individual is predisposed to any diseases, and behavioral disorders can most often be explained by analyzing defects in genetic code. Scientist who focus on examples like these argue that human behavior is entirely predictable. Those on the other side of the debate argue that genetics can only provide a predisposition to act a certain way and that, ultimately, humans possess the free will to choose whether or not to act.

Animals have significantly more predictable behavior than humans. Driven by natural selection, animals develop mating calls, predator warnings, and communicative dances. One example of these engrained behaviors is the Belding's ground squirrel, which developed a specific set of calls that warn nearby squirrels about predators. If a ground squirrel sees a predator on land it will elicit a trill after it gets to safety, which signals to nearby squirrels that they should stand up on their hind legs and attempt to locate the predator. When a predator is seen in the air, a ground squirrel will immediately call out a long whistle, putting himself in danger but signaling for nearby squirrels to run for cover. Through experimentation and examination scientists have been able to chart behaviors like this and very accurately predict how animals behave in certain situations. [15]

The study of predictability often sparks debate between those who believe humans maintain complete control over their free-will and those who believe our actions are predetermined. However, it is likely that neither Newton nor Laplace saw the study of predictability as relating to determinism. [16]

In weather and climate

As climate change and other weather phenomenon become more common, the predictability of climate systems becomes more important. The IPCC notes that our ability to predict future detailed climate interactions is difficult, however, long term climate forecasts are possible. [17] [18]

The dual nature with distinct predictability

Over 50 years since Lorenz's 1963 study and a follow-up presentation in 1972, the statement “weather is chaotic” has been well accepted. [4] [5] Such a view turns our attention from regularity associated with Laplace's view of determinism to irregularity associated with chaos. In contrast to single-type chaotic solutions, recent studies using a generalized Lorenz model [19] have focused on the coexistence of chaotic and regular solutions that appear within the same model using the same modeling configurations but different initial conditions. [20] [21] The results, with attractor coexistence, suggest that the entirety of weather possesses a dual nature of chaos and order with distinct predictability. [22]

Using a slowly varying, periodic heating parameter within a generalized Lorenz model, Shen and his co-authors suggested a revised view: “The atmosphere possesses chaos and order; it includes, as examples, emerging organized systems (such as tornadoes) and time varying forcing from recurrent seasons”. [23]

Spring predictability barrier

The spring predictability barrier refers to a period of time early in the year when making summer weather predictions about the El Niño–Southern Oscillation is difficult. It is unknown why it is difficult, although many theories have been proposed. There is some thought that the cause is due to the ENSO transition where conditions are more rapidly shifting. [24]

In macroeconomics

Predictability in macroeconomics refers most frequently to the degree to which an economic model accurately reflects quarterly data and the degree to which one might successfully identify the internal propagation mechanisms of models. Examples of US macroeconomic series of interest include but are not limited to Consumption, Investment, Real GNP, and Capital Stock. Factors that are involved in the predictability of an economic system include the range of the forecast (is the forecast two years "out" or twenty) and the variability of estimates. Mathematical processes for assessing the predictability of macroeconomic trends are still in development. [25]

See also

Related Research Articles

<span class="mw-page-title-main">Butterfly effect</span> Idea that small causes can have large effects

In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state.

<span class="mw-page-title-main">Chaos theory</span> Field of mathematics and science based on non-linear systems and initial conditions

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state. A metaphor for this behavior is that a butterfly flapping its wings in Texas can cause a tornado in Brazil.

A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication systems, complex software and electronic systems, social and economic organizations, an ecosystem, a living cell, and ultimately the entire universe.

<span class="mw-page-title-main">Determinism</span> Philosophical view that events are pre-determined

Determinism is the philosophical view that events are completely determined by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and considerations. Like eternalism, determinism focuses on particular events rather than the future as a concept. The opposite of determinism is indeterminism, or the view that events are not deterministically caused but rather occur due to chance. Determinism is often contrasted with free will, although some philosophers claim that the two are compatible. 

<span class="mw-page-title-main">Lyapunov exponent</span> The rate of separation of infinitesimally close trajectories

In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector diverge at a rate given by

A deterministic system is a conceptual model of the philosophical doctrine of determinism applied to a system for understanding everything that has and will occur in the system, based on the physical outcomes of causality. In a deterministic system, every action, or cause, produces a reaction, or effect, and every reaction, in turn, becomes the cause of subsequent reactions. The totality of these cascading events can theoretically show exactly how the system will exist at any moment in time.

<span class="mw-page-title-main">Laplace's demon</span> Hypothetical all-predicting intellect

In the history of science, Laplace's demon was a notable published articulation of causal determinism on a scientific basis by Pierre-Simon Laplace in 1814. According to determinism, if someone knows the precise location and momentum of every atom in the universe, their past and future values for any given time are entailed; they can be calculated from the laws of classical mechanics.

<span class="mw-page-title-main">Hénon map</span> Discrete-time dynamical system

In mathematics, the Hénon map, sometimes called Hénon–Pomeau attractor/map, is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Hénon map takes a point (xn, yn) in the plane and maps it to a new point

<span class="mw-page-title-main">Economic model</span> Simplified representation of economic reality

An economic model is a theoretical construct representing economic processes by a set of variables and a set of logical and/or quantitative relationships between them. The economic model is a simplified, often mathematical, framework designed to illustrate complex processes. Frequently, economic models posit structural parameters. A model may have various exogenous variables, and those variables may change to create various responses by economic variables. Methodological uses of models include investigation, theorizing, and fitting theories to the world.

<span class="mw-page-title-main">Dynamical systems theory</span> Area of mathematics used to describe the behavior of complex dynamical systems

Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a Cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.

<span class="mw-page-title-main">Numerical weather prediction</span> Weather prediction using mathematical models of the atmosphere and oceans

Numerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in the 1950s that numerical weather predictions produced realistic results. A number of global and regional forecast models are run in different countries worldwide, using current weather observations relayed from radiosondes, weather satellites and other observing systems as inputs.

<span class="mw-page-title-main">Edward Norton Lorenz</span> American mathematician

Edward Norton Lorenz was an American mathematician and meteorologist who established the theoretical basis of weather and climate predictability, as well as the basis for computer-aided atmospheric physics and meteorology. He is best known as the founder of modern chaos theory, a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions.

<span class="mw-page-title-main">J. Doyne Farmer</span> American physicist and entrepreneur (b.1952)

J. Doyne Farmer is an American complex systems scientist and entrepreneur with interests in chaos theory, complexity and econophysics. He is Baillie Gifford Professor of Complex Systems Science at the Smith School of Enterprise and the Environment, Oxford University, where he is also director of the Complexity Economics programme at the Institute for New Economic Thinking at the Oxford Martin School. Additionally he is an external professor at the Santa Fe Institute. His current research is on complexity economics, focusing on systemic risk in financial markets and technological progress. During his career he has made important contributions to complex systems, chaos, artificial life, theoretical biology, time series forecasting and econophysics. He co-founded Prediction Company, one of the first companies to do fully automated quantitative trading. While a graduate student he led a group that called itself Eudaemonic Enterprises and built the first wearable digital computer, which was used to beat the game of roulette.

<span class="mw-page-title-main">Ensemble forecasting</span> Multiple simulation method for weather forecasting

Ensemble forecasting is a method used in or within numerical weather prediction. Instead of making a single forecast of the most likely weather, a set of forecasts is produced. This set of forecasts aims to give an indication of the range of possible future states of the atmosphere. Ensemble forecasting is a form of Monte Carlo analysis. The multiple simulations are conducted to account for the two usual sources of uncertainty in forecast models: (1) the errors introduced by the use of imperfect initial conditions, amplified by the chaotic nature of the evolution equations of the atmosphere, which is often referred to as sensitive dependence on initial conditions; and (2) errors introduced because of imperfections in the model formulation, such as the approximate mathematical methods to solve the equations. Ideally, the verified future atmospheric state should fall within the predicted ensemble spread, and the amount of spread should be related to the uncertainty (error) of the forecast. In general, this approach can be used to make probabilistic forecasts of any dynamical system, and not just for weather prediction.

<span class="mw-page-title-main">Chua's circuit</span>

Chua's circuit is a simple electronic circuit that exhibits classic chaotic behavior. This means roughly that it is a "nonperiodic oscillator"; it produces an oscillating waveform that, unlike an ordinary electronic oscillator, never "repeats". It was invented in 1983 by Leon O. Chua, who was a visitor at Waseda University in Japan at that time. The ease of construction of the circuit has made it a ubiquitous real-world example of a chaotic system, leading some to declare it "a paradigm for chaos".

<span class="mw-page-title-main">Lorenz system</span> System of ordinary differential equations with chaotic solutions

The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. In popular media the "butterfly effect" stems from the real-world implications of the Lorenz attractor, namely that several different initial chaotic conditions evolve in phase space in a way that never repeats, so all chaos is unpredictable. This underscores that chaotic systems can be completely deterministic and yet still be inherently unpredictable over long periods of time. Because chaos continually increases in systems, we cannot predict the future of systems well. E.g., even the small flap of a butterfly’s wings could set the world on a vastly different trajectory, such as by causing a hurricane. The shape of the Lorenz attractor itself, when plotted in phase space, may also be seen to resemble a butterfly.

<span class="mw-page-title-main">Echo state network</span> Type of reservoir computer

An echo state network (ESN) is a type of reservoir computer that uses a recurrent neural network with a sparsely connected hidden layer. The connectivity and weights of hidden neurons are fixed and randomly assigned. The weights of output neurons can be learned so that the network can produce or reproduce specific temporal patterns. The main interest of this network is that although its behaviour is non-linear, the only weights that are modified during training are for the synapses that connect the hidden neurons to output neurons. Thus, the error function is quadratic with respect to the parameter vector and can be differentiated easily to a linear system.

The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable when historically observed, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ways.

<span class="mw-page-title-main">Edward Ott</span> American physicist

Edward Ott is an American physicist most noted for his contributions to the development of chaos theory.

References

  1. ↑ van Strien, Marij (2014-03-01). "On the origins and foundations of Laplacian determinism" (PDF). Studies in History and Philosophy of Science Part A. 45 (Supplement C): 24–31. Bibcode:2014SHPSA..45...24V. doi:10.1016/j.shpsa.2013.12.003. PMID   24984446.
  2. ↑ Sync: The Emerging Science of Spontaneous Order, Steven Strogatz, Hyperion, New York, 2003, pages 189-190.
  3. ↑ Shen, Bo-Wen; Pielke, Roger A.; Zeng, Xubin; Cui, Jialin; Faghih-Naini, Sara; Paxson, Wei; Atlas, Robert (2022-07-04). "Three Kinds of Butterfly Effects within Lorenz Models". Encyclopedia. 2 (3): 1250–1259. doi: 10.3390/encyclopedia2030084 . ISSN   2673-8392.
  4. 1 2 Lorenz, Edward N. (1963-03-01). "Deterministic Nonperiodic Flow". Journal of the Atmospheric Sciences. 20 (2): 130–141. Bibcode:1963JAtS...20..130L. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 . ISSN   0022-4928.
  5. 1 2 Lorenz, Edward (1993). The Essence of Chaos. Seattle, WA, USA: University of Washington Press. pp. 227p.
  6. ↑ Lorenz, Edward (2022-08-17). "Predictability: Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?" (PDF). MIT.
  7. ↑ Lorenz, Edward N. (1969-01-01). "The predictability of a flow which possesses many scales of motion". Tellus. 21 (3): 289–307. Bibcode:1969Tell...21..289L. doi: 10.3402/tellusa.v21i3.10086 . ISSN   0040-2826.
  8. ↑ Palmer, T N; Döring, A; Seregin, G (2014-08-19). "The real butterfly effect". Nonlinearity. 27 (9): R123–R141. Bibcode:2014Nonli..27R.123P. doi:10.1088/0951-7715/27/9/r123. ISSN   0951-7715. S2CID   122339502.
  9. ↑ Shen, Bo-Wen; Pielke, Roger A.; Zeng, Xubin (2022-05-07). "One Saddle Point and Two Types of Sensitivities within the Lorenz 1963 and 1969 Models". Atmosphere. 13 (5): 753. Bibcode:2022Atmos..13..753S. doi: 10.3390/atmos13050753 . ISSN   2073-4433.
  10. ↑ "The AI Car Computer for Autonomous Driving". NVIDIA. Retrieved 27 September 2017.
  11. ↑ Chen, Chenyi. "Deep Learning for Self -driving Car" (PDF). Princeton University. Retrieved 27 September 2017.
  12. ↑ "Teaching machines to predict the future". 21 June 2016.
  13. ↑ "Justice is served, but more so after lunch: How food-breaks sway the decisions of judges".
  14. ↑ "Gene research finds opposites do attract". TheGuardian.com . 24 May 2009.
  15. ↑ Sherman, Paul W (1985). "Alarm calls of Belding's ground squirrels to aerial predators: Nepotism or self-preservation?". Behavioral Ecology and Sociobiology. 17 (4): 313–323. doi:10.1007/BF00293209. S2CID   206774065.
  16. ↑ "Predictability".
  17. ↑ "Predictability of the Climate System". Working Group I: The Scientific Basis. IPCC. Retrieved 26 September 2017.
  18. ↑ Solomon, S., D. Qin, M. Manning, Z. Chen, M. Marquis, K. Averyt, M. Tignor, and H. L. Miller Jr., Eds (2007). Climate Change 2007: The Physical Science Basis. Cambridge, United Kingdom and New York, NY, USA: Cambridge University Press. p. 996.{{cite book}}: CS1 maint: multiple names: authors list (link)
  19. ↑ Shen, Bo-Wen (2019-03-01). "Aggregated Negative Feedback in a Generalized Lorenz Model". International Journal of Bifurcation and Chaos. 29 (3): 1950037–1950091. Bibcode:2019IJBC...2950037S. doi: 10.1142/S0218127419500378 . ISSN   0218-1274. S2CID   132494234.
  20. ↑ Yorke, James A.; Yorke, Ellen D. (1979-09-01). "Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model". Journal of Statistical Physics. 21 (3): 263–277. Bibcode:1979JSP....21..263Y. doi:10.1007/BF01011469. ISSN   1572-9613. S2CID   12172750.
  21. ↑ Shen, Bo-Wen; Pielke Sr., R. A.; Zeng, X.; Baik, J.-J.; Faghih-Naini, S.; Cui, J.; Atlas, R.; Reyes, T. A. L. (2021). "Is Weather Chaotic? Coexisting Chaotic and Non-chaotic Attractors within Lorenz Models". In Skiadas, Christos H.; Dimotikalis, Yiannis (eds.). 13th Chaotic Modeling and Simulation International Conference. Springer Proceedings in Complexity. Cham: Springer International Publishing. pp. 805–825. doi:10.1007/978-3-030-70795-8_57. ISBN   978-3-030-70795-8. S2CID   245197840.
  22. ↑ Shen, Bo-Wen; Pielke, Roger A.; Zeng, Xubin; Baik, Jong-Jin; Faghih-Naini, Sara; Cui, Jialin; Atlas, Robert (2021-01-01). "Is Weather Chaotic?: Coexistence of Chaos and Order within a Generalized Lorenz Model". Bulletin of the American Meteorological Society. 102 (1): E148–E158. Bibcode:2021BAMS..102E.148S. doi: 10.1175/BAMS-D-19-0165.1 . ISSN   0003-0007. S2CID   208369617. CC-BY icon.svg Text was derived from this source, which is available under a Creative Commons Attribution 4.0 International License.
  23. ↑ Shen, Bo-Wen; Pielke, Roger; Zeng, Xubin; Cui, Jialin; Faghih-Naini, Sara; Paxson, Wei; Kesarkar, Amit; Zeng, Xiping; Atlas, Robert (2022-11-12). "The Dual Nature of Chaos and Order in the Atmosphere". Atmosphere. 13 (11): 1892. Bibcode:2022Atmos..13.1892S. doi: 10.3390/atmos13111892 . ISSN   2073-4433.
  24. ↑ L'Heureux, Michelle. "The Spring Predictability Barrier: we'd rather be on Spring Break". Climate.gov. NOAA. Retrieved 26 September 2017.
  25. ↑ Diebold, Francis X. (2001). "Measuring Predictability: Theory and Macroeconomic Applications" (PDF). Journal of Applied Econometrics. 16 (6): 657–669. doi:10.1002/jae.619. JSTOR   2678520. S2CID   16040363.
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