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Revision as of 09:47, 15 May 2015

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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: Created by a HOUSE OF CARDS: Please edit the explanation below and only mention here why it isn't complete. Do NOT delete this tag too soon.

A 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.

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.

  • 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); if the experimenter wants to find confirmation of this fact, it may try to fit a logarithmic curve.

  • Exponential: f(x) = a*b^x + c
  • Loess: w(x) = (1-|d|^3)^3
  • Linear, No Slope: f(x) = c
  • Logistic: f(x) = L / (1 + e^{-k(x-b)})
  • Confidence Interval: not a type of curve fitting, but a method of depicting the predictive power of a curve
  • 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. 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: 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). See https://en.wikipedia.org/wiki/Cauchy_distribution

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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