"Scatterplots are the preferred medium for adding smooth curves to show a causal functional relationship or an association […] However, despite the advantage of the scatterplot for seeing some types of patterns, the linked micromap design adds geographic location to the information displayed and so enables searches for geographic patterns that the scatterplot omits." (Daniel B Carr & Linda W Pickle, "Visualizing Data Patterns with Micromaps", 2010)
"[…] according to the bell-shaped curve the likelihood of a very-large-deviation event (a major outlier) located in the striped region appears to be very unlikely, essentially zero. The same event, though, is several thousand times more likely if it comes from a set of events obeying a fat-tailed distribution instead of the bell-shaped one." (John L Casti, "X-Events: The Collapse of Everything", 2012)
"As geometers study shape, the student of calculus examines change: the mathematics of how an object transforms from one state into another, as when describing the motion of a ball or bullet through space, is rendered pictorial in its graphs’ curves." (Daniel Tammet, "Thinking in Numbers" , 2012)
"Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values" (the bifurcation parameters) of a system causes a sudden"qualitative" or topological change in its behavior. Bifurcations can occur in both continuous systems" (described by ODEs, DDEs, or PDEs) and discrete systems" (described by maps)." (Tianshou Zhou, "Bifurcation", 2013)
"To make progress, every field of science needs to have data commensurate with the complexity of the phenomena it studies. [...] With big data and machine learning, you can understand much more complex phenomena than before. In most fields, scientists have traditionally used only very limited kinds of models, like linear regression, where the curve you fit to the data is always a straight line. Unfortunately, most phenomena in the world are nonlinear. [...] Machine learning opens up a vast new world of nonlinear models." (Pedro Domingos, "The Master Algorithm", 2015)
"Underfitting is when a model doesn’t take into account enough information to accurately model real life. For example, if we observed only two points on an exponential curve, we would probably assert that there is a linear relationship there. But there may not be a pattern, because there are only two points to reference. [...] It seems that the best way to mitigate underfitting a model is to give it more information, but this actually can be a problem as well. More data can mean more noise and more problems. Using too much data and too complex of a model will yield something that works for that particular data set and nothing else." (Matthew Kirk, "Thoughtful Machine Learning", 2015)
"When memorization happens, you may have the illusion that everything is working well because your machine learning algorithm seems to have fitted the in sample data so well. Instead, problems can quickly become evident when you start having it work with out-of-sample data and you notice that it produces errors in its predictions as well as errors that actually change a lot when you relearn from the same data with a slightly different approach. Overfitting occurs when your algorithm has learned too much from your data, up to the point of mapping curve shapes and rules that do not exist [...]. Any slight change in the procedure or in the training data produces erratic predictions." (John P Mueller & Luca Massaron, "Machine Learning for Dummies", 2016)
"Bias is error from incorrect assumptions built into the model, such as restricting an interpolating function to be linear instead of a higher-order curve. [...] Errors of bias produce underfit models. They do not fit the training data as tightly as possible, were they allowed the freedom to do so. In popular discourse, I associate the word 'bias' with prejudice, and the correspondence is fairly apt: an apriori assumption that one group is inferior to another will result in less accurate predictions than an unbiased one. Models that perform lousy on both training and testing data are underfit." (Steven S Skiena, "The Data Science Design Manual", 2017)
"With time series though, there is absolutely no substitute for plotting. The pertinent pattern might end up being a sharp spike followed by a gentle taper down. Or, maybe there are weird plateaus. There could be noisy spikes that have to be filtered out. A good way to look at it is this: means and standard deviations are based on the naïve assumption that data follows pretty bell curves, but there is no corresponding 'default' assumption for time series data" (at least, not one that works well with any frequency), so you always have to look at the data to get a sense of what’s normal. [...] Along the lines of figuring out what patterns to expect, when you are exploring time series data, it is immensely useful to be able to zoom in and out." (Field Cady, "The Data Science Handbook", 2017)
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