Editing 2048: Curve-Fitting

An illustration of several plots of the same data with {{w|Curve fitting|curves fitted}} to the points, paired with conclusions that you might draw about the person who made them. These data, when plotted on an X/Y graph, appear to have a general upward trend, but the data is far too noisy, with too few data points, to clearly suggest any specific growth pattern. In such a case, many different mathematical and statistical models ''could'' be presented as roughly fitting the data, but none of them fits well enough to compellingly represent the data.  

When modeling such a problem statistically, much of the work of a data scientist or statistician is knowing which fitting method is most appropriate for the data in question. Here we see various hypothetical scientists or statisticians each applying their own interpretations to the exact same data, and the comic mocks each of them for their various personal biases or other assorted excuses. In general, the researcher will specify the form of an equation for the line to be drawn, and an algorithm will produce the actual line.

{{w|Linear regression}} is the most basic form of regression; it tries to find the straight line that best approximates the data. As it's the simplest, most widely taught form of regression, and in general differentiable functions are locally well approximated by a straight line, it's usually the first and most trivial attempt of fit.

The picture to the right shows how totally different data sets can result in the same line. It's obvious that some more basics about the nature of the data must be used to understand if this simple line really does make sense.

{{w|Polynomial regression|Quadratic fit}} (i.e. fitting a parabola through the data) is the lowest grade polynomial that can be used to fit data through a curved line; if the data exhibits clearly "curved" behavior (or if the experimenter feels that its growth should be more than linear), a parabola is often the first, easiest, stab at fitting the data.

The comment below the graph ''"I wanted a curved line, so I made one with math."'' suggests that a quadratic regression is used when straight lines no longer satisfy the researcher, but they still want to use simple math expression. Quadratic correlations like this are mathematically valid and one of the simplest kind of curve in math, but this curve doesn't appear to satisfy the data any better than does simple, linear regression.

<math>f(x) = a\log_b(x)</math>

A {{w|Logarithm|logarithmic}} curve grows slower on higher values, but still grows without bound to infinity rather than approaching a horizontal {{w|asymptote}}. The small ''b'' in the formula represents the base which is in most cases ''{{w|e (mathematical constant)|e}}'', 10, or 2. If the data presumably does approach a horizontal asymptote then this fit isn't an effective method to explain the nature of the data.

<math>f(x) = a\cdot b^x</math>

An {{w|Exponential growth|exponential curve}}, on the contrary, is typical of a phenomenon whose growth gets rapidly faster and faster - a common case is a process that generates stuff that contributes to the process itself; think bacteria growth or compound interest.

A {{w|Local regression|LOESS fit}} doesn't use a single formula to fit all the data, but approximates data points locally using different polynomials for each "zone" (weighting data points differently as they get further from it) and patching them together. As it has many more degrees of freedom compared to a single polynomial, it generally "fits better" to any data set, although it is generally impossible to derive any strong, "clean" mathematical correlation from it - it is just a nice smooth line that approximates the data points well, with a good degree of rejection from outliers.

The comment below the graph ''"I'm sophisticated, not like those bumbling polynomial people."'' emphasises this more complicated interpretation, but without a simple mathematical description it's not very helpful to find informative interpretations of the underlying data.

The comment below the graph ''"I'm making a scatter plot but I don't want to."'' is probably done by a student who isn't happy with their choice of field of study.

The comment below the graph ''"I need to connect these two lines, but my first idea didn't have enough math."'' implies the experimenter just wants to find a mathematically-respectable way to link two flat lines.

The comment below the graph ''"I have a theory, and this is the only data I could find."'' is a bit ambiguous because there are many data points ignored. Without an explanation why only a subset of the data is used this isn't a useful interpretation at all. As a matter of fact, with the extra degrees of freedom offered by the piecewise regression, it could indicate that the researcher is trying to fit the data to confirm their theory, rather than building their theory off of the data.

The comment below the graph ''"I clicked 'Smooth Lines' in {{w|Microsoft Excel|Excel}}."'' refers to the well known spreadsheet application from {{w|Microsoft Office}}. Like other spreadsheet applications it has the feature to visualize data from a table into a graph by many ways. "Smooth Lines" is a setting meant for use on a {{w|line graph}}, a graph in which one axis represents time; as it simply joins up every point using bezier (or similar) curves as necessary to pass through every point (rather than finding a more sensible line that accepts some minimal but non-zero acceptible level of error in the datapoints), it is not suitable for regression.

Not a real method, but a common consequence of misapplication of statistical methods: a curve can be generated that fits the data extremely well, but immediately becomes absurd as soon as one glances outside the training data sample range, and your analysis comes crashing down "like a house of cards". This is a type of ''overfitting''. In other words, the model may do quite well for (approximately) {{w|Interpolation|interpolating}} between values in the sample range, but not extend at all well to {{w|Extrapolation|extrapolating}} values outside that range.

''Note:'' Exact polynomial fitting, a fit which gives the unique <math>(n-1)</math>th degree polynomial through <math>n</math> points, often display this kind of behaviour.

A brief Google search reveals that Augustin-Louis Cauchy originally worked as a junior engineer in a managerial position. Upon his acceptance to the Académie des Sciences in March 1816, many of his peers expressed outrage. Despite his early work in "mere" engineering, Cauchy is widely regarded as one of the founding influences in the rigorous study of calculus & accompanying proofs.  Notably, his later work included theoretical physics, and Lorentz was also a well-known physicist.  Therefore, the title-text may be referring back to [[793: Physicists]].

 

Alternately, the title-text could be implying that the person who applied the Cauchy-Lorentz curve-fitting method may not be well qualified to the task assigned.

:"I wanted a curved line, so I made one with math."

*Regressions have been the subject of several previous comics. [[1725: Linear Regression]] was about linear regressions on uncorrelated or poorly correlated data. [[1007: Sustainable]], [[1204: Detail]] and [[1281: Minifigs]] depict linear regressions on data that was actually logistic, leading to bizarre extrapolations. [[605: Extrapolating]] shows a line extrapolating from just two data points.