Statistics & Formulas
Quick overview of the relevant statistics before we discuss the ALE algorithm.
TODO: Improve <code> tag CSS: white-space: pre, inline-block and pretty block. Use JQMath?
Gaussian Function
A Gaussian function, named after Carl Friedrich Gauss, is a bell curve in the form:
ƒ(x,μ,σ) = a * e (x - μ)²/-2σ² + b
where:
- μ is the mean or expected value and sets the center of the peak
- σ is the standard deviation, which determines the width of the curve
- a is the height at the peak of the curve
- b is the value the curve approaches at the tails (usually 0)
The multiplier 'a' can be used to normalize the curve such that the area under the curve sums to 1.
a = 1/(σ * √2π) In this special case, the curve is also
a probability density function of a normal distribution. Probability density functions
are very useful when estimating the likelihood ... TODO: finish this thought
Full-width Half-max
An alternate unit to discuss the width of the curve at half the maximum height.
FWHM can vary, giving tall narrow curves for small FWHM and broad and short curves for large ones
Union vs Conjunction
Union is cumulative probability, the likelihood of either probabilities. U = p1 + (1 - p1)*p2. ∪ Show graph of two curves and the whole area shaded.
Maximum is not cumulative - slightly more conservative than union. M = max(p1, p2). Show graph of two curves with max - overlaid with previous? Overlay all of them?
The conjunction or intersection is the similarities only, the likelihood of both probabilities. C = min(p1, p2). ∩ Show graph with just overlap shaded.
...(someone with insane javascript skills should write an interactive gaussian graph that show the three options overlaid over two gaussians where you can adjust the distance between the two curves. that'd be neat.)
