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Curve-Fitting
Cauchy-Lorentz: "Something alarmingly mathematical is happening, and you should probably pause to Google my name and check what field I originally worked in."
Title text: Cauchy-Lorentz: "Something alarmingly mathematical is happening, and you should probably pause to Google my name and check what field I originally worked in."

Explanation

Ambox notice.png This explanation may be incomplete or incorrect: Please edit the explanation below and only mention here why it isn't complete. Do NOT delete this tag too soon.

An illustration of several plots of the same data with curves fitted to the points, paired with conclusions that you might draw about the person who made them.

When modeling a phenomenon statistically, it is common to search for trends, and fitted curves can help reveal these trends. Much of the work of a data scientist or statistician is knowing which fitting method to use for the data in question. Here we see various hypothetical scientists or statisticians each applying their own interpretations, 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.

This comic is similar to 977: Map Projections which also uses a scientific method not commonly thought about by the general public to determine specific characteristics of one's personality and approach to science.

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

Linear

f(x) = mx + b

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 derivable function are locally well approximated by a straight line, it's usually the first and most trivial attempt of fit.

Quadratic

f(x) = ax^2 + bx + c

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 stab at fitting the data.

Logarithmic

f(x) = a*\log_b(x) + c

A logarithmic curve is typical of a phenomenon whose growth gets slower and slower as time passes (indeed, its derivative - i.e. its growth rate - is \propto \frac{1}{x} \rightarrow 0 for x \rightarrow +\infty), but still grows without bound rather than approaching a horizontal asymptote. (If it did approach a horizontal asymptote, then one of the other models subtracted from a constant would probably be better, e.g. f(x) = a - \frac{b}{x} or f(x) = a - b^{-cx}.) If the experimenter wants to find confirmation of this fact, they may try to fit a logarithmic curve.

Exponential

f(x) = a*b^x + c

An 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.

  • The logarithmic and exponential interpretations could very easily be fudged or engineered by a researcher with an agenda (such as by taking a misleading subset or even outright lying about the regression), which the comic mocks by juxtaposing them side-by-side on the same set of data.

LOESS

w(x) = (1-|d|^3)^3 (notice: this is just the function used for the weights, not the actually fitted curve formula, as it's a piecewise polynomial)

A 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 differently data points as they get further from it) and patching them together

As it has much 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 well the data points, with a good degree of rejection from outliers.

Linear, No Slope

f(x) = c

Apparently, the person making this line figured out pretty early on that their data analysis was turning into a scatter plot, and wanted to escape their personal stigma of scatter plots by drawing an obviously false regression line on top of it. Alternatively, they were hoping the data would be flat, and are trying to pretend that there's no real trend to the data by drawing a horizontal trend line.

Logistic

f(x) = L / (1 + e^{-k(x-b)})

A logistic curve provides a smooth, S-shaped transition curve between two flat intervals; indeed the caption says that the experimenter just wants to find a mathematically-respectable way to link two flat lines.

Confidence Interval

Not a type of curve fitting, but a method of depicting the predictive power of a curve.

Providing a confidence interval over the graph shows the uncertainty of the acquired data, thus acknowledging the uncertain results of the experiment, and showing the will not to "cheat" with "easy" regression curves.

Piecewise

Mapping different curves to different segments of the data. This is a legitimate strategy, but the different segments should be meaningful, such as if they were pulled from different populations.

Connecting lines

Not useful whatsoever, but it looks nice!

Ad-Hoc Filter

Drawing a bunch of different lines by hand, keeping in only the data points perceived as "good". Also not useful.

House of Cards

Not a real method, but a common consequence of mis-application 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.

Cauchy-Lorentz

Cauchy-Lorentz is a continuous probability distribution which does not have an expected value or a defined variance. This means that the law of large numbers does not hold and that estimating e.g. the sample mean will diverge (be all over the place) the more data points you have. Hence very troublesome (mathematically alarming).

Since so many different models can fit this data set at first glance, Randall may be making a point about how if a data set is sufficiently messy, you can read any trend you want into it, and the trend that is chosen may say more about the researcher than about the data. This is a similar sentiment to 1725: Linear Regression, which also pokes fun at dubious trend lines on scatterplots.

Transcript

Ambox notice.png This transcript is incomplete. Please help editing it! Thanks.
Curve-Fitting Methods
and the messages they send
[In a single frame twelve scatter plots with unlabeled x- and y-axes are shown. Each plot consists of the same data-set of approximately thirty points located all over the plot but slightly more distributed around the diagonal. Every plot shows in red a different fitting method which is labeled on top in gray.]
[The first plot shows a line starting at the left bottom above the x-axis rising towards the points to the right.]
Linear
"Hey, I did a regression."
[The second plot shows a curve falling slightly down and then rising up to the right.]
Quadratic
"I wanted a curved line, so I made one with Math."
[At the third plot the curve starts near the left bottom and increases more and more less to the right.]
Logarithmic
"Look, it's tapering off!"
[The fourth plot shows a curve starting near the left bottom and increases more and more steeper to the right.]
Exponential
"Look, it's growing uncontrollably!"
[The fifth plot uses a fitting to match many points. It starts at the left bottom, increases, then decreases, then rapidly increasing again, and finally reaching a plateau.]
LOESS
"I'm sophisticated, not like those bumbling polynomial people."
[The sixth plot simply shows a line above but parallel to the x-axis.]
Linear, no slope
"I'm making a scatter plot but I don't want to."
[At plot #7 starts at a plateau above the x-axis, then increases, and finally reaches a higher plateau.]
Logistic
"I need to connect these two lines, but my first idea didn't have enough Math."
[Plot #8 shows two red lines embedding most points and the area between is painted as a red shadow.]
Confidence interval
"Listen, science is hard. But I'm a serious person doing my best."
[Plot #9 shows two not connected lines, one at the lower left half, and one higher at the right. Both have smaller curved lines in light red above and below.]
Piecewise
"I have a theory, and this is the only data I could find."
[The plot at the left bottom shows a line connecting all points from left to right, resulting in a curve going many times up and down.]
Connecting lines
"I clicked 'Smooth Lines' in Excel."
[The next to last plot shows a echelon form, connecting a few real and some imaginary points.]
Ad-Hoc filter
"I had an idea for how to clean up the data. What do you think?"
[The last plot shows a wave with increasing peak values.]
House of Cards
"As you can see, this model smoothly fits the- wait no no don't extend it AAAAAA!!"


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