A Picture's Worth

"The drawing shows me at a glance what would be spread over ten pages in a book." (Ivan Turgenev, 1862) [2]

"Sometimes, half a dozen figures will reveal, as with a lighting-flash, the importance of a subject which ten thousand labored words with the same purpose in view, had left at last but dim and uncertain." (Mark Twain, "Life on the Mississippi", 1883) 

"One good picture is worth many pages of written description." (William Sproston Caine, 1891) [2]

"One look is worth a thousand words" (Kathleen Caffyn, 1903) 

"Use a picture. It's worth a thousand words." (Arthur Brisbane, The Post-Standard, 1911)

"One Look Is Worth A Thousand Words" ([advertisement] 1913)

"A picture is worth ten thousand words. If you can’t see the truth in these pictures you are among the vast majority that must learn only by experience." (Arthur Brisbane, 1915)

"One picture is worth ten thousand words." (Frederick R Barnard, Printer’s Ink, 1921)

"One Picture Worth Ten Thousand Words" ([Chinese proverb] 1927)

"In many instances, a picture is indeed worth a thousand words. To make this true in more diverse circumstances, much more creative effort is needed to pictorialize the output from data analysis. Naive pictures are often extremely helpful, but more sophisticated pictures can be both simple and even more informative." (John W Tukey & Martin B Wilk, "Data Analysis and Statistics: An Expository Overview", 1966)

"One word is worth a thousand pictures. If it's the right word." (Edward Abbey, "Beyond the Wall: Essays from the Outside", 1984)

"A picture may be worth a thousand words, a formula is worth a thousand pictures." (Edsger Dijkstra, [conference at ETH Zurich] 1994)

"A magnificent picture is never worth a thousand perfect words." (John Dunning, "The Bookman's Wake", 1995)

"A picture tells a thousand words. But you get a thousand pictures from someone's voice." (Paul Fleischman, "Seek", 2001)

"If a picture is worth a thousand words, a metaphor is worth a thousand pictures." (Daniel H Pink, "A Whole New Mind: Why Right-Brainers Will Rule the Future", 2005)

"A picture may be worth a thousand words, but not all pictures are readable, interpretable, meaningful, or relevant." (Kristen Sosulski, "Data Visualization Made Simple: Insights into Becoming Visual", 2018)

"A good metaphor is worth a thousand pictures." (Anon) 

"As the Chinese say, 1001 words is worth more than a picture." (John McCarthy [source]) 

References:
[1] Wikipedia (2024) A picture is worth a thousand words [link]
[2] Quote Investigator (2022) A Picture Is Worth Ten Thousand Words [link] 
[3] SQL-Troubles (2024) Charts vs. Thousand Words [link]

David G Green - Collected Quotes

"Although it might be intuitively apparent that a system is complex, defining complexity has proved difficult to pin down with numerous definitions on record. As yet there is no agreed theory of complexity. Much of the mathematics is intractable and computer simulation plays a major part." (Terry R J Bossomaier & David G Green, 2000) 

"Although many natural phenomena may result from the interaction of complex entities, the details of the components may be unimportant. In the discussion of neural networks, the individual neuron turns out to be a highly sophisticated biological system. But the collective properties of neurons may be captured by spin-glass models, in which the neuron is simplified to a binary quantity […] " (Terry R J Bossomaier & David G Green, 2000) 

"Interaction: the other major source of complexity is the interaction of many autonomous, adaptive agents. Again, there are many questions to ask about the agents, the nature of the interaction and the circumstances in which complex surface phenomena result. (Terry R J Bossomaier & David G. Green, 2000)

"Iteration: fractals and chaos result from repetition of simple operations. These generating rules produce complex phenomena. There are many interesting questions to ask about how to describe the processes, how to measure the resulting complexity, whether we can work backwards from the reult to the rules and so on. (Terry R J Bossomaier & David G. Green, 2000)

"Of course what we would all like to see is a general theory of complex systems or complexity. Despite several promising candidates the selection process is still under way. Maybe there is no universal theory, but there are certainly common paradigms and methods which have proved to be useful across a wide area." (Terry R J Bossomaier & David G Green, 2000) 

The evident power of simple heuristics […] teaches us the important lesson that global behavior patterns, and social organization, can emerge out of local interactions. Organisms do not necessarily need to have an over-riding plan nor do they require awareness of the large-scale. Grand patterns and processes can emerge as the nett effect of small-scale, local behavior.

"If entropy must increase, then how is it possible (say) for all the variety of the living world to persist? The usual answer to the above question is that living systems are open systems, not closed, so the law does not apply locally. However this answer is somewhat unsatisfying. In effect all systems are open systems, since everything interacts with its surroundings to some degree." (David G Green, 2000) 

"The really crucial question in multi-object systems is whether local interactions do grow into large-scale patterns." (David G Green, 2000) 

"The self-similarity on different scales arises because growth often involves iteration of simple, discrete processes (e.g. branching). These repetitive processes can often be summarized as sets of simple rules." (David G Green, 2000) 

Paul Klee - Collected Quotes

"Our initial perplexity before nature is explained by our seeing at first the small outer branches and not penetrating to the main branches or the trunk. But once this is realized, one will perceive a repetition of the whole law even in the outermost leaf and turn it to good use." (Paul Klee, [diary entry] 1904)

"When looking at any significant work of art, remember that a more significant one probably has had to be sacrificed." (Paul Klee, [diary entry] 1904)

"The beautiful, which is perhaps inseparable from art, is not after all tied to the subject, but to the pictorial representation. In this way and in no other does art overcome the ugly without avoiding it." (Paul Klee, [diary entry] 1905)

"Things are not quite so simple with 'pure' art as it is dogmatically claimed. In the final analysis, a drawing simply is no longer a drawing, no matter how self-sufficient its execution may be. It is a symbol, and the more profoundly the imaginary lines of projection meet higher dimensions, the better. In this sense I shall never be a pure artist as the dogma defines him. We higher creatures are also mechanically produced children of God, and yet intellect and soul operate within us in completely different dimensions." (Paul Klee, [diary entry] 1905)

"Nature can afford to be prodigal in everything, the artist must be frugal down to the last detail."  Paul Klee, [diary entry] 1909)

"First of all, the art of living; then as my ideal profession, poetry and philosophy, and as my real profession, plastic arts; in the last resort, for lack of income, illustrations." (Paul Klee, cca. 1910

"Graphic work as the expressive movement of the hand holding the recording pencil.... is so fundamentally different from dealing with tone and color that one can use this technique quite well in the dark, even in the blackest night. On the other hand, tone (movement from light to dark) presupposes some light, and color presupposes a great deal of light." (Paul Klee, 1912)

"We document, explain, justify, construct, organize: these are good things, but we do not succeed in coming to the whole [...]. But we may as well calm down: construction is not absolute. Our virtue is this: by cultivating the exact we have laid the foundations for a science of art, including the unknown X." (Paul Klee, "Statement of 1917"

"A tendency toward the abstract is inherent in linear expression: graphic imagery being confined to outlines has a fairy-like quality and at the same time can achieve great precision." (Paul Klee, "Creative Credo", 1920)

"Things appear to assume a broader and more diversified meaning, often seemingly contradicting the rational experience of yesterday. There is a striving to emphasize the essential character of the accidental." (Paul Klee, "Creative Credo", 1920)

"For the artist communication with nature remains the most essential condition. The artist is human; himself nature; part of nature within natural space." (Paul Klee, 1923)

"It is possible that a picture will move far away from Nature and yet find its way back to reality. The faculty of memory, experience at a distance produces pictorial associations." (Paul Klee, cca. 1925)

"Thought is the medially between earth and world. The broader the magnitude of his reach, the more painful man's tragic limitation. Thought is the medially between earth and world. The broader the magnitude of his reach, the more painful man's tragic limitation. To get where motion is interminate." ( Paul Klee, "Pedagogical Sketch Book, 1925)

"The longer a line, the more of the time element it contains. Distance is time whereas a surface is apprehended more in terms of the moment." (Paul Klee, "Exact Experiments in the Realm of Art", 1927)

"What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze." (Paul Klee, "Bauhaus prospectus", 1929)

"Art should be like a holiday: something to give a man the opportunity to see things differently and to change his point of view." (Paul Klee)

"It is interesting to observe how real the object remains, in spite of all abstractions." (Paul Klee)

Richard B Braithwaite - Collected Quotes

"It has been a fortunate fact in the modern history of physical science that the scientist constructing a new theoretical system has nearly always found that the mathematics [...] required [...] had already been worked out by pure mathematicians for their own amusement [...] The moral for statesmen would seem to be that, for proper scientific 'planning' , pure mathematics should be endowed fifty years ahead of scientists." (Richard B Braithwaite, "Scientific Explanation: A Study of the Function of Theory, Probability and Law in Science", 1953)

"[...] no batch of observations, however large, either definitively rejects or definitively fails to reject the hypothesis H0." (Richard B Braithwaite, "Scientific Explanation: A Study of the Function of Theory, Probability and Law in Science", 1953)

"The peaks of science may appear to be floating in the clouds, but their foundations are in the hard facts of experience." (Richard B Braithwaite, "Scientific Explanation: A Study of the Function of Theory, Probability and Law in Science", 1953)

"The peculiarity of [...] statistical hypotheses is that they are not conclusively refutable by any experience." (Richard B Braithwaite, "Scientific Explanation: A Study of the Function of Theory, Probability and Law in Science", 1953)

"The ultimate justification for any scientific belief will depend upon the main purpose for which we think scientifically - that of predicting and thereby controlling the future." (Richard B Braithwaite, "Scientific Explanation: A Study of the Function of Theory, Probability and Law in Science", 1953)

"The world is not made up of empirical facts with the addition of the laws of nature: what we call the laws of nature are conceptual devices by which we organize our empirical knowledge and predict the future." (Richard B Braithwaite, "Scientific Explanation: A Study of the Function of Theory, Probability and Law in Science", 1953)

Howard Wainer - Collected Quotes

"Although arguments can be made that high data density does not imply that a graphic will be good, nor one with low density bad, it does reflect on the efficiency of the transmission of information. Obviously, if we hold clarity and accuracy constant, more information is better than less. One of the great assets of graphical techniques is that they can convey large amounts of information in a small space." (Howard Wainer, "How to Display Data Badly", The American Statistician Vol. 38(2), 1984) 

"The essence of a graphic display is that a set of numbers having both magnitudes and an order are represented by an appropriate visual metaphor - the magnitude and order of the metaphorical representation match the numbers. We can display data badly by ignoring or distorting this concept." (Howard Wainer, "How to Display Data Badly", The American Statistician Vol. 38(2), 1984)

"The standard error of most statistics is proportional to 1 over the square root of the sample size. God did this, and there is nothing we can do to change it." (Howard Wainer, "Improving Tabular Displays, With NAEP Tables as Examples and Inspirations", Journal of Educational and Behavioral Statistics Vol 22 (1), 1997)

"[…] a graph is nothing but a visual metaphor. To be truthful, it must correspond closely to the phenomena it depicts: longer bars or bigger pie slices must correspond to more, a rising line must correspond to an increasing amount. If a graphical depiction of data does not faithfully follow this principle, it is almost sure to be misleading. But the metaphoric attachment of a graphic goes farther than this. The character of the depiction ism a necessary and sufficient condition for the character of the data. When the data change, so too must their depiction; but when the depiction changes very little, we assume that the data, likewise, are relatively unchanging. If this convention is not followed, we are usually misled." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"A graphic display has many purposes, but it achieves its highest value when it forces us to see what we were not expecting." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Nothing that had been produced before was even close. Even today, after more than two centuries of graphical experience, Playfair’s graphs remain exemplary standards for clearcommunication of quantitative phenomena. […] Graphical forms were available before Playfair, but they were rarely used to plot empirical information." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Oftentimes a statistical graphic provides the evidence for a plausible story, and the evidence, though perhaps only circumstantial, can be quite convincing. […] But such graphical arguments are not always valid. Knowledge of the underlying phenomena and additional facts may be required." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Placing a fact within a context increases its value greatly. […] . An efficacious way to add context to statistical facts is by embedding them in a graphic. Sometimes the most helpful context is geographical, and shaded maps come to mind as examples. Sometimes the most helpful context is temporal, and time-based line graphs are the obvious choice. But how much time? The ending date (today) is usually clear, but where do you start? The starting point determines the scale. […] The starting point and hence the scale are determined by the questions that we expect the graph to answer." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Simpson’s Paradox can occur whenever data are aggregated. If data are collapsed across a subclassification (such as grades, race, or age), the overall difference observed may not represent what is going on. Standardization can help correct this, but nothing short of random assignment of individuals to groups will prevent the possibility of yet another subclassificatiion, as yet unidentified, changing things around again. But I believe that knowing of the possibility helps us, so that we can contain the enthusiasm of our impulsive first inferences." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"The appearance, and hence the perception, of any statistical graphic is massively influenced by the choice of scale. If the scale of the vertical axis is too narrow relative to the scale of the horizontal axis, random meanders look meaningful." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"The difficult task of properly setting the scale of a graph remains difficult but not mysterious. There is agreement among experts spanning two hundred years. The default option should be to choose a scale that fills the plot with data. We can deviate from this under circumstances when it is better not to fill the plot with data, but those circumstances are usually clear. It is important to remember that the sin of using too small a scale is venial; the viewer can correct it. The sin of using too large a scale cannot be corrected without access to the original data; it can be mortal." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Usually the effectiveness of a good display increases with the complexity of the data. When there are only a few points, almost anything will do; even a pie chart with only three or four categories is usually comprehensible." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"Thus when we look at, or prepare, a time-based statistical graphic, it is important to ask what is the right time scale, the right context, for the questions of greatest interest. The answer to this question is sometimes complex, but the very act of asking it provides us with some protection against surprises." (Howard Wainer, "Graphic Discovery: A trout in the milk and other visuals" 2nd, 2008)

"The only thing we know for sure about a missing data point is that it is not there, and there is nothing that the magic of statistics can do change that. The best that can be managed is to estimate the extent to which missing data have influenced the inferences we wish to draw." (Howard Wainer, "14 Conversations About Three Things", Journal of Educational and Behavioral Statistics Vol. 35(1, 2010)

"For an analyst to willfully avoid learning about the science is akin to malfeasance. Of course, it is likely that a deep understanding both of the science and of data analytic methods does not reside in the same person. When it does not, data analysis should be done jointly. It is my understanding that data mining is not often done as a team. This is unfortunate, for then it is too easy to miss what might have been found." (Howard Wainer, Comment, Journal of Computational and Graphical Statistics Vol. 20(1), 2011)

"Too often there is a disconnect between the people who run a study and those who do the data analysis. This is as predictable as it is unfortunate. If data are gathered with particular hypotheses in mind, too often they (the data) are passed on to someone who is tasked with testing those hypotheses and who has only marginal knowledge of the subject matter. Graphical displays, if prepared at all, are just summaries or tests of the assumptions underlying the tests being done. Broader displays, that have the potential of showing us things that we had not expected, are either not done at all, or their message is not able to be fully appreciated by the data analyst." (Howard Wainer, Comment, Journal of Computational and Graphical Statistics Vol. 20(1), 2011)

Clay Helberg - Collected Quotes

"Another key element in making informative graphs is to avoid confounding design variation with data variation. This means that changes in the scale of the graphic should always correspond to changes in the data being represented." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"Another trouble spot with graphs is multidimensional variation. This occurs where two-dimensional figures are used to represent one-dimensional values. What often happens is that the size of the graphic is scaled both horizontally and vertically according to the value being graphed. However, this results in the area of the graphic varying with the square of the underlying data, causing the eye to read an exaggerated effect in the graph." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"It may be helpful to consider some aspects of statistical thought which might lead many people to be distrustful of it. First of all, statistics requires the ability to consider things from a probabilistic perspective, employing quantitative technical concepts such as 'confidence', 'reliability', 'significance'. This is in contrast to the way non-mathematicians often cast problems: logical, concrete, often dichotomous conceptualizations are the norm: right or wrong, large or small, this or that." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"[...] many non-mathematicians hold quantitative data in a sort of awe. They have been lead to believe that numbers are, or at least should be, unquestionably correct." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"Most statistical models assume error free measurement, at least of independent (predictor) variables. However, as we all know, measurements are seldom if ever perfect. Particularly when dealing with noisy data such as questionnaire responses or processes which are difficult to measure precisely, we need to pay close attention to the effects of measurement errors. Two characteristics of measurement which are particularly important in psychological measurement are reliability and validity." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

"Remember that a p-value merely indicates the probability of a particular set of data being generated by the null model - it has little to say about the size of a deviation from that model (especially in the tails of the distribution, where large changes in effect size cause only small changes in p-values)." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995)

"There are a number of ways that statistical techniques can be misapplied to problems in the real world. Three of the most common hazards are designing experiments with insufficient power, ignoring measurement error, and performing multiple comparisons." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995)

"We can consider three broad classes of statistical pitfalls. The first involves sources of bias. These are conditions or circumstances which affect the external validity of statistical results. The second category is errors in methodology, which can lead to inaccurate or invalid results. The third class of problems concerns interpretation of results, or how statistical results are applied (or misapplied) to real world issues." (Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995) 

References:
[1] Clay Helberg, "Pitfalls of Data Analysis (or How to Avoid Lies and Damned Lies)", 1995 [link]

On Probability (2000 - )

"In the laws of probability theory, likelihood distributions are fixed properties of a hypothesis. In the art of rationality, to explain is to anticipate. To anticipate is to explain." (Eliezer S. Yudkowsky, "A Technical Explanation of Technical Explanation", 2005)

"I have always thought that statistical design and sampling from populations should be the first courses taught, but all elementary courses I know of start with statistical methods or probability. To me, this is putting the cart before the horse!" (Walter Federer, "A Conversation with Walter T Federer", Statistical Science Vol 20, 2005)

"For some scientific data the true value cannot be given by a constant or some straightforward mathematical function but by a probability distribution or an expectation value. Such data are called probabilistic. Even so, their true value does not change with time or place, making them distinctly different from  most statistical data of everyday life." (Manfred Drosg, "Dealing with Uncertainties: A Guide to Error Analysis", 2007)

"In fact, H [entropy] measures the amount of uncertainty that exists in the phenomenon. If there were only one event, its probability would be equal to 1, and H would be equal to 0 - that is, there is no uncertainty about what will happen in a phenomenon with a single event because we always know what is going to occur. The more events that a phenomenon possesses, the more uncertainty there is about the state of the phenomenon. In other words, the more entropy, the more information." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"The four questions of data analysis are the questions of description, probability, inference, and homogeneity. [...] Descriptive statistics are built on the assumption that we can use a single value to characterize a single property for a single universe. […] Probability theory is focused on what happens to samples drawn from a known universe. If the data happen to come from different sources, then there are multiple universes with different probability models.  [...] Statistical inference assumes that you have a sample that is known to have come from one universe." (Donald J Wheeler," Myths About Data Analysis", International Lean & Six Sigma Conference, 2012)

"When statisticians, trained in math and probability theory, try to assess likely outcomes, they demand a plethora of data points. Even then, they recognize that unless it’s a very simple and controlled action such as flipping a coin, unforeseen variables can exert significant influence." (Zachary Karabell, "The Leading Indicators: A short history of the numbers that rule our world", 2014)

"Entropy is a measure of amount of uncertainty or disorder present in the system within the possible probability distribution. The entropy and amount of unpredictability are directly proportional to each other." (G Suseela & Y Asnath V Phamila, "Security Framework for Smart Visual Sensor Networks", 2019)

On Probability (1975 - 1999)

"Of course, we know the laws of trial and error, of large numbers and probabilities. We know that these laws are part of the mathematical and mechanical fabric of the universe, and that they are also at play in biological processes. But, in the name of the experimental method and out of our poor knowledge, are we really entitled to claim that everything happens by chance, to the exclusion of all other possibilities?" (Albert Claude, "The Coming of Age of the Cell", Science, 1975)

"We often use the ideas of chance, likelihood, or probability in everyday language. For example, 'It is unlikely to rain today', 'The black horse will probably win the next race', or 'A playing card selected at random from a pack is unlikely to be the ace of spades' . Each of these remarks, if accepted at face value, is likely to reflect the speaker's expectation based on experience gained in the same position, or similar positions, on many  previous occasions. In order to be quantitative about probability, we focus on this aspect of repeatable situations." (Peter Lancaster, "Mathematics: Models of the Real World", 1976)

"The theory of probability is the only mathematical tool available to help map the unknown and the uncontrollable. It is fortunate that this tool, while tricky, is extraordinarily powerful and convenient." (Benoit Mandelbrot, "The Fractal Geometry of Nature", 1977)

"In decision theory, mathematical analysis shows that once the sampling distribution, loss function, and sample are specified, the only remaining basis for a choice among different admissible decisions lies in the prior probabilities. Therefore, the logical foundations of decision theory cannot be put in fully satisfactory form until the old problem of arbitrariness (sometimes called 'subjectiveness') in assigning prior probabilities is resolved." (Edwin T Jaynes, "Prior Probabilities", 1978)

"Another reason for the applied statistician to care about Bayesian inference is that consumers of statistical answers, at least interval estimates, commonly interpret them as probability statements about the possible values of parameters. Consequently, the answers statisticians provide to consumers should be capable of being interpreted as approximate Bayesian statements." (Donald B Rubin, "Bayesianly justifiable and relevant frequency calculations for the applied statistician", Annals of Statistics 12(4), 1984)

"In the path-integral formulation, the essence of quantum physics may be summarized with two fundamental rules: (1). The classical action determines the probability amplitude for a specific chain of events to occur, and (2) the probability that either one or the other chain of events occurs is determined by the probability amplitudes corresponding to the two chains of events. Finding these rules represents a stunning achievement by the founders of quantum physics." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)

"In the design of experiments, one has to use some informal prior knowledge. How does one construct blocks in a block design problem for instance? It is stupid to think that use is not made of a prior. But knowing that this prior is utterly casual, it seems ludicrous to go through a lot of integration, etc., to obtain ‘exact’ posterior probabilities resulting from this prior. So, I believe the situation with respect to Bayesian inference and with respect to inference, in general, has not made progress. Well, Bayesian statistics has led to a great deal of theoretical research. But I don’t see any real utilizations in applications, you know. Now no one, as far as I know, has examined the question of whether the inferences that are obtained are, in fact, realized in the predictions that they are used to make." (Oscar Kempthorne, "A conversation with Oscar Kempthorne", Statistical Science vol. 10, 1995)

"Events may appear to us to be random, but this could be attributed to human ignorance about the details of the processes involved." (Brain S Everitt, "Chance Rules", 1999)

On Convergence IV

"Once more, an invariably-recurring lesson of geological history, at whatever point its study is taken up: the lesson of the almost infinite slowness of the modification of living forms. The lines of the pedigrees of living things break off almost before they begin to converge." (Thomas H Huxley, On the Formation of Coal, 1870)

"Analytic functions are those that can be represented by a power series, convergent within a certain region bounded by the so-called circle of convergence. Outside of this region the analytic function is not regarded as given a priori ; its continuation into wider regions remains a matter of special investigation and may give very different results, according to the particular case considered." (Felix Klein, "Sophus Lie", [lecture] 1893)

"Particular landforms or surface morphologies may be generated, in some cases, by several different processes, sets of environmental controls, or developmental histories. This convergence to similar forms despite variations in processes and controls is called equifinality." (Jonathan Phillips, "Simplexity and the Reinvention of Equifinality", Geographical Analysis Vol. 29 (1), 1997)

"The underlying reason for convergence seems to be that all organisms are under constant scrutiny of natural selection and are also subject to the constraints of the physical and chemical factors that severely limit the action of all inhabitants of the biosphere. Put simply, convergence shows that in a real world not all things are possible." (Simon C Morris, "The Crucible of Creation", 1998)

"Equifinality is the principle which states that morphology alone cannot be used to reconstruct the mode of origin of a landform on the grounds that identical landforms can be produced by a number of alternative processes, process assemblages or process histories. Different processes may lead to an apparent similarity in the forms produced. For example, sea-level change, tectonic uplift, climatic change, change in source of sediment or water or change in storage may all lead to river incision and a convergence of form." (Olav Slaymaker, "Equifinality", 2004)

"Convergence is, in my opinion, not only deeply fascinating but, curiously, it is as often overlooked. More importantly, it hints at the existence of a deeper structure to biology. It helps us to delineate a metaphorical map across which evolution must navigate. In this sense the Darwinian mechanisms and the organic substrate we call life are really a search engine to discover particular solutions, including intelligence and - risky thought - perhaps deeper realities?" (Simon C Morris,  "Aliens like us?", Astronomy and Geophysics Vol. 46 (4), 2005)

"Sometimes the most important fit statistic you can get is ‘convergence not met’ - it can tell you something is wrong with your model." (Oliver Schabenberger, "Applied Statistics in Agriculture Conference", 2006)

On Regression IV

"One feature [...] which requires much more justification than is usually given, is the setting up of unplausible null hypotheses. For example, a statistician may set out a test to see whether two drugs have exactly the same effect, or whether a regression line is exactly straight. These hypotheses can scarcely be taken literally." (Cedric A B Smith, "Book review of Norman T. J. Bailey: Statistical Methods in Biology", Applied Statistics 9, 1960)

"Stepwise regression is probably the most abused computerized statistical technique ever devised. If you think you need stepwise regression to solve a particular problem you have, it is almost certain that you do not. Professional statisticians rarely use automated stepwise regression." (Leland Wilkinson, "SYSTAT", 1984)

"Someone has characterized the user of stepwise regression as a person who checks his or her brain at the entrance of the computer center." (Dick R Wittink, "The application of regression analysis", 1988)

"Multiple regression, like all statistical techniques based on correlation, has a severe limitation due to the fact that correlation doesn't prove causation. And no amount of measuring of 'control' variables can untangle the web of causality. What nature hath joined together, multiple regression cannot put asunder." (Richard Nisbett, "2014 : What scientific idea is ready for retirement?", 2013)

"Regression does not describe changes in ability that happen as time passes […]. Regression is caused by performances fluctuating about ability, so that performances far from the mean reflect abilities that are closer to the mean." (Gary Smith, "Standard Deviations", 2014)

"We encounter regression in many contexts - pretty much whenever we see an imperfect measure of what we are trying to measure. Standardized tests are obviously an imperfect measure of ability. [...] Each experimental score is an imperfect measure of “ability,” the benefits from the layout. To the extent there is randomness in this experiment - and there surely is - the prospective benefits from the layout that has the highest score are probably closer to the mean than was the score." (Gary Smith, "Standard Deviations", 2014)

"When a trait, such as academic or athletic ability, is measured imperfectly, the observed differences in performance exaggerate the actual differences in ability. Those who perform the best are probably not as far above average as they seem. Nor are those who perform the worst as far below average as they seem. Their subsequent performances will consequently regress to the mean." (Gary Smith, "Standard Deviations", 2014)

"Regression describes the relationship between an exploratory variable (i.e., independent) and a response variable (i.e., dependent). Exploratory variables are also referred to as predictors and can have a frequency of more than 1. Regression is being used within the realm of predictions and forecasting. Regression determines the change in response variable when one exploratory variable is varied while the other independent variables are kept constant. This is done to understand the relationship that each of those exploratory variables exhibits." (Danish Haroon, "Python Machine Learning Case Studies", 2017)