Concept of Randomness in Statistics
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Published: Wed, 04 Oct 2017
Part I Introduction
- Introduction on Freshman Seminar
Freshman seminar 1205M offers great opportunities for students to work intimately with professors from the Science faculty on various areas of mathematics. The seminar was targeted to encourage us to open our minds to creative ideas and develop curiosity of influential mathematical theories and various subgroups of contemporary mathematics. In addition to exposure to selected subtopics in contemporary mathematics, we had valuable opportunities to develop our presentation and academic essay writing skills.
1.2 Important roles of Analogy and Intuition
The historical development of mathematics is significantly influenced by intuition acquired from real life experience and analogy quoted from various other areas (Harrison & Treagust, 1993). Analogy is an extraordinary method in developing new concepts in the history of science. In this module, famous topics in the contemporary mathematics, including geometry, number theory, set theory, randomness and game theory have been discussed. Among all topics, our team worked on Analogy and Intuition of Randomness. In this seminar, various creative analogy ideas and intuition/counter-intuition thinking have been presented based on specific cases in modern mathematics.
1.3 Method on Research and Presentation
Our team collected relevant source materials on the randomness, including books, journals, and websites on the Internet. As for presenting applications of randomness, in particular, we focused on the historical development of randomness theory, the simplified key concepts in randomness, the counter-intuitive stories happened, overlapping with other fields in nature, and some significant and influential applications of randomness theory in our daily life. We omitted complicated theories, technical formulas and rigorous proofs. Throughout the whole semester, our team has conducted two informal presentations on randomness. In order to illustrate randomness clearly and intuitively, we adopted various methods: problem solving, in-class quizzes, presentations and attractive stories. Subtopics included: Biology, quantum physics, finance, audio engineering, statistics and so on.
Part II Report on Randomness
2.1 Randomness on Communication Theory
2.1.1 Introduction of Noise in Communication Theory
In statistics, irrelevant or meaningless data is considered noise (random error). Whereas in communication theory, random disturbance in a signal is called “noise”. In essence, noise consists of a large number of disturbances with a statistically randomized time distribution.
It is assumed that noise signals have power spectral density that is proportional to 1/f^β, where f stands for frequencies of noise. For example, the spectral density of white noise is β = 0, while pink noise has β = 1. This special character is widely used for distinguishing among colors of noise.
2.1.2 Laws and Criterions Used to Distinguish Colors and Characteristics of Noise
The color names for noise are derived from an analogy between the spectrum of noise and the equivalent spectrum of lights with different visible colors. For instance, if we translate the sound wave of “white noise” into light waves, the resulting light will be viewed as white color. In electronics, physics, and many other areas, the color of a noise signal is usually understood as some characteristics of its power spectrum. As different colors of noise have significantly different properties. Therefore, each kind of noise requires a specific “color” to match with it.
Start with the most well-known one: “White noise”, people name different noise after colors. This is in analogy with white color light, which has a flat spectrum of power on its frequency range. Other colors, such as violet, blue, red, pink, are then given to different noises with extremely similar spectrum characteristics.
Although most of them have standardized noise patterns with specific disciplines, there are also plenty of noise spectrums with imprecise and informal definitions, like black noise, green noise, brown noise and so on.
These below parts were summarized from Wikipedia terms: Noise (electronics)
2.1.3 Inner Sources of Noise
Thermal noise is generated from the random thermal motion of charges (usually electrons) inside electrical conductors. The amplitude of the signal has a probability density function similar to the Gaussian (Normal) distribution. The amplitude of thermal noise depends on the temperature of the circuit.
Shot noise results from unavoidable random fluctuations when the charges (such as electrons) jump over a gap inside the electric circuits. It sounds rather similar to the noise created by rain falling on a tin roof.
Flicker noise has a frequency spectrum that falls down into the higher frequencies areas steadily.
Burst noise consists of sudden step-like transitions between two or more levels at random and unpredictable times. It sounds like eating popcorn.
2.1.4 Outer sources of Noise
Atmospheric noise is the natural disturbance caused by electricity discharges in thunderstorm and other natural disturbances occurring in nature, like disruptions of high-voltage wires.
Industrial noises are produced by automobiles, aircrafts and so on. The disturbances are produced by the discharge processes in these operations as well, which is similar to the atmospheric noise.
Extraterrestrial noises come from the universe. These noises include: Solar Noise, which is a radiation from the sun due to its intense nuclear reactions and the consequent high temperature, and Cosmic Noise, which are able to transmit its radiation and cosmic rays to almost everywhere.
2.1.5 Classification of Different Colors of Noise
This part was adapted and summarized from an online introductory article: “White, pink, blue and violet: The colors of noise” from the Wired Magazine Science Column, Author: Duncan Geere, Date: Apr. 07, 2011
White noise has a constant power distribution density on its spectrum. It is named after the white color light, which has a flatten frequency everywhere on the spectrum. The term is widely applied in many scientific and technical areas, including physics, audio engineering, telecommunications, statistical forecasting and many other areas. Specifically, White noise is used as a generator for random numbers. In addition, weather forecasting websites also use white noise to generate random digit patterns and simulate real weather.
The power density of pink noise decreases proportionally to 1/f. In the past, the term of flicker noise sometimes refers to pink noise, but it will be more appropriate if we strictly apply it only to electronic circuits. Moreover, Pink noise is also used in analysis of meteorological data and output radiation power of some astronomical bodies.
According to the precise definition, the term Brown noise refers to a noise whose power density decreases inversely proportional to f^2.
The density function can be generated from integrating white noise or via an algorithm of Brownian motion simulation. Brown noise is not named after the color brown spectrum, which is distinct from other noises. It can be used in climatology to describe climate shifts. However, within the scientific community, scientists have been arguing about its value for such purposes for a long time.
The power density of Blue noise is proportional to frequency. Blue noise has an increasing frequency over a finite frequency range. Blue noise is similar to pink noise, but instead of a decreasing spectrum, we observe an increasing one. Sometimes it is mixed up with Violet noise in informal discussion.
Violet noise is also known as the Purple noise. The power density of Violet noise is proportional to f^2, which means it increases in quadratic form. Violet noise is like another version of Brownian noise. Moreover, as Violet noise is the result of differentiating the white noise signal density, so people also call it the “Differentiated White noise”.
Grey noise is a special kind of white noise process with characteristic equal loudness curve. However, it has a higher power density at both ends of the frequency spectrum but very little power near the center. Apparently, this is different from the standard white noise which is equal loud across its power density. However, actually this phenomenon is due to the humans hearing illusion.
2.2 Randomness on Finance
2.2.1 Brief Introduction to Efficient Market Hypothesis
This part was summarized based on an online informal introductory article: “The Efficient Markets Hypothesis”, Authors: Jonathan Clarks, Tomas Jandik, Gershon Mandelker, Website: www.e-m-h.org
In financial fields, the efficient-market hypothesis asserts that stock market prices will evolve with respect to to a random walk. They have the same probability distribution and independent of each other. Random walk states that stocks take a random and unpredictable path. The probability of a stock’s future price going up is equal to going down. Therefore, the past movement (or trend) of a specific stock price or the overall market performance cannot be used as the basis to predict future movements. In addition, it is impossible to outperform the entire market without taking additional risk or putting extra efforts. However, EMH proves that a long-term buy-and-hold strategy is the most efficient, because long term prices will approximately reflect performance of the company very well, whereas short term movements in prices can be only described as a random walk.
2.2.2 Historical Backgrounds of Efficient Market Hypothesis
This part was summarized based on an online nonprofit educational website: www.e-m-h.org and a research paper: History of the Efficient Market Hypothesis, Nov.2004, Author: Martin, Sewell, Publisher: University College London.
Historically, the randomness of stock market prices was firstly modelled by a French broker, Jules Regnault, in 1863. Shortly after, a French mathematician, Louis Bachelier, developed the mathematics of Brownian motion in 1900. In 1923, the famous economist, Keynes clearly stated that investors in financial markets would be rewarded not for knowing better than other participants in the market, but rather for risk taking.
After the WWII, the efficient-market hypothesis emerged as an outstanding theory in the mid-1960s. In the 1960s, Mandelbrot proposed a randomness model for stock pricing. Fama discussed about Mandelbrot’s hypothesis and concluded that the market data confirmed his model. In addition, he defined the so-called “efficient market” for the first time, in his paper “Random Walks in Stock Market Prices”. He explained how random walks in stock market significantly influence individual stock prices. Later, he introduced definitions for three forms of financial market efficiency: weak, semi-strong and strong.
The term was eventually popularized when Burton Malkiel, a Professor of Economics at Princeton University, published his classic and prominent book: “A Random Walk Down Wall Street.”
2.2.3 Three Major Types of Markets: Weak, Semi-Strong and Strong
The three types of EMH were summarized based on an online technical blog: “The Efficient Markets Hypothesis”, Author: Jodi Beggs, Website: About.com
Weak Form of Efficiency
We cannot predict future prices through analyzing prices from the past. And we cannot earn excessive returns by using information based on historical data. In this level, technical analysis is always profitable, as share prices exhibit no dependencies on their past. This implies that future prices depend entirely on performance of companies.
Semi-Strong Form of Efficiency
Information other than market data is released, such as instant news, companies’ management, financial accounting reports, companies’ latest products. Under such condition, share prices will reflect the new information very rapidly. Therefore, investors cannot gain any excess returns by trading on the public information. Semi-strong-form efficiency market implies that neither technical analysis nor fundamental analysis can produce excess returns.
Strong Form of Efficiency
Under such condition, information typically held by corporate insiders is released. Therefore, share prices reflect not only previously public information, but all private information as well. Theoretically, no one can earn excess returns. However, even before major changes are exposed to the public, corporate insiders are able to trade their company’s stocks from abnormal profits. Fortunately, such insider trading is banned by surveillance authorities, like the Securities and Exchange Commission.
2.2.4 Arguments and Critics on Efficiency Market Hypothesis
However, critics blame that the theory’s applications in markets results in financial crisis. In response, proponents of the hypothesis state that the theory is only a simplification model of the world, which means that it may not always hold true under every conditions. Hence, the market is only practically efficient for merely investment purposes in the real world rather than other aims.
2.2.5 Interesting Counter-intuitive Stories on Monkeys
The story was adapted from the Forbes Magazine, Personal Finance Column, Author: Rick Ferri, Date: Dec, 20, 2012
In order to verify the Efficient Market Hypothesis and illustrate the theories explicitly to the public, a group of researchers conducted a monkey experiment. They randomly picked up thirty stocks from a one thousand stocks poll and then let a hundred monkeys throw darts at the stocks printing on newspaper. They kept repeating this experiment for five decades, and tracked the results.
In the end, to their surprise, monkeys’ performance beat the index by 1.7% per year, which indicates that, there is certain situation where traditional technical analysis cannot even beat randomly-selected portfolios. The results have shocked the whole world by how greatly randomness affects the market stock prices.
2.3 Randomness in Physics and Biology
2.3.1 Application of Randomness in Modern Physics
In the early 19th century, physicists use the philosophy of randomness to study motions and behaviors of molecules, and they build models in thermodynamics to explain phenomenon in gas experiments.
In the 20th century, when the era comes for quantum mechanics, microscopic phenomena are considered as completely random. Randomness of things like radioactive decay, photons passing through polarizers, and other bizarre quantum effects cannot be explained and predicted with classical theories in the usual way (Scott, 2009). Therefore, physicists propose a new theory, which claims that in a microscopic world, some of the outcomes appear casual and random. For example, when we describe a radioactive atom, we cannot predict when the atom will decay. What only left for us is the probability of decay during a specific given period. In order to solve this mystery, Einstein postulates the Hidden Variable theory, which states that nature contains irreducible randomness: properties and variables work beyond our scope somehow, but they actually determine the outcomes appear in our world.
2.3.2 Application of Randomness in Biology
The modern evolutionary states that the diversity of life is due to natural selection. Randomness, an essential component of biological diversity, is associated with the growth of biological organization during evolution (Longo & Montevil, 2012). It plays important roles in determining genetic mutation, and the significance of randomness effects appear at different sizes, from microorganisms to large mammals (Bonner, 2013). During this process, a number of random genetic mutations appear in the gene library under both inner and other influences. Although this process is purely random, it indeed systematically leads to a higher chance for survival and reproduction of those individuals who possess these mutations than those without them. This mechanism plays crucial roles in the survivals of animals.
Surprisingly, randomness in biology has remarkable relations to quantum physics. Schrodinger proposes his notion of negative entropy as a form of Gibbs free energy, which also behaves similarly to randomness properties in abstract quantum world (Schrodinger, 1944).
Part III References
Beggs, J. (2014). The Efficient Markets Hypothesis. About. Retrieved Mar 30, 2014 from http://economics.about.com/od/Financial-Markets-Category/a/The-Efficient-Markets-Hypothesis.htm
Bonner, J. (2013). Randomness in Evolution. Princeton University Press. Retrieved Mar 30, 2014 from http://press.princeton.edu/titles/9958.html
Clarke, J. & Jandik, T. (2012). The Efficient Markets Hypothesis. Retrieved Mar 30, 2014 from http://ww.e-m-h.org/ClJM.pdf
Ferri, R. (2012). Any Monkey Can Beat The Market. Forbes. Retrieved Mar 30, 2014 from http://www.forbes.com/sites/rickferri/2012/12/20/any-monkey-can-beat-the-market/
Geere, D. (2011). White, pink, blue and violet: The colors of noise. Wired. Retrieved Mar 30, 2014 from http://www.wired.co.uk/news/archive/2011-04/7/colours-of-noise/viewall
Harrison, A. G., & Treagust, D. F. (1994). Science analogies. The Science Teacher, 61, 40-43.
Longo, G & Montevil, M. (2012). Randomness Increases Order in Biological
Evolution. Retrieved Mar 30, 2014 from http://www.researchgate.net/profile/Giuseppe_Longo2/publication/221350338_Randomness_Increases_Order_in_Biological_Evolution/file/60b7d51544f17cb8d8.pdf
Schrodinger, E.: What Is Life? Cambridge U.P. (1944)
Scoot, J. (2009). Do physicists really believe in true randomness?
Ask a Mathematician. Retrieved Mar 30, 2014 from http://www.askamathematician.com/2009/12/q-do-physicists-really-believe-in-true-randomness/
Sewell, M. (2004). History of the efficient market hypothesis. Retrieved Mar 30, 2014 from http://www.cs.ucl.ac.uk/fileadmin/UCL-CS/images/Research_Student_Information/RN_11_04.pdf
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