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Intuitive explanations for Gaussian distribution function and mahalanobis distance 
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#1
Dec1812, 09:59 AM

P: 3

Hello
I was wondering If anyone could give intuitive explanations for the multivariate Gaussian distribution function and mahalanobis distance? My professor didn't explain these in probability class, they were only defined... Where did the formula come from? Why is the Gaussian function the way it is? Is there a way to intuitively explain mahalanobis distance? Thank you for any support 


#2
Dec1812, 10:51 AM

Sci Advisor
P: 3,247

I myself don't have a good understanding of why the 1dimensional Gaussian distribution has its particular formula  to get that intuition, one would need to study why the gaussian shape is a limiting shape for binomial distributions You could get intuition in the way that a person who has inutition for algebra and calculus gets inutition. As to the maninthestreet type of intuition, have you seen those demonstrations where balls are dropped into a pyramid of dividers and land in a bell shaped pattern? 


#3
Dec1812, 11:02 AM

P: 3

Answers to both of your questions are: NO
I know what Zscore is "= How many standard deviations from the mean" but that's all I know about it. And no I have not seen the pyramid divider, I'll try to look for it. Maybe you know where to look? Thank you! =) 


#4
Dec1812, 01:35 PM

Sci Advisor
P: 3,247

Intuitive explanations for Gaussian distribution function and mahalanobis distance
The video of a conceptual normal distribution machine http://www.youtube.com/watch?v=PM7z_03o_kk is clearer than the videos of the physical machines that I found.



#5
Dec1812, 02:12 PM

Sci Advisor
P: 3,247

Think about what you must do to convert two histograms to the "same scale". If one is measuring dollars on the xaxis and the other is measuring Ohms, there is no law of economics or physics that establishes a definite relation between dollars and Ohms. So you can't convert the measurements to a common unit. Even if distribution were both measuring dollars on the xaxis, there is no law of that tells you that $1000 dollars and $5000 dollars are measurements "a long ways" apart. If you're talking about car prices, they might be. If you're talking about the national GNP, they aren't. Next, think about how you would convert two 2dimensional histograms to the same scale. Suppose one histogram has measurement of ordered pairs (weight of person, blood glucose level of person) and the other has measurements of (mileage on car, price of car). 


#6
Dec1812, 11:47 PM

P: 8

The roots of the gaussian distribution lies in the method of least squares, used widely in the 17th and 18th century for navigation, astronomy. To explain the method of least squares, arises the Gaussian distribution.



#7
Dec1912, 10:36 PM

P: 570

A Gaussian dist arises when you have the sum of a large number of independent factors, all of which are small. It is the limit as the number of factors become infinitely large, and the size of each factor becomes infinitely small. In actual practice many situations converge to this limit quite quickly. The basic Gaussian is one dimensional, but it is possible to have any number of dimensions. You can think of a large number of tiny insects flying around at random in an empty space. 


#8
Dec2012, 08:25 PM

P: 615

You can see that [itex]f_{\mathbf x}(x_1,\ldots,x_k)\, = \frac{1}{(2\pi)^{k/2}\boldsymbol\Sigma^{1/2}} \exp\left(\frac{1}{2}({\mathbf x}{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{1}({\mathbf x}{\boldsymbol\mu}) \right)[/itex] becomes [itex]f(x) = \frac{1}{\sigma \sqrt{2\pi}}\exp\left(\frac{1}{2} (\frac{x\mu}{\sigma})^2\right)[/itex] if Ʃ is a real number (1 X 1 matrix). Mahalanobis distance is Mahalanobis distance. Tautology is always intuitive. 


#9
Dec2112, 06:07 AM

P: 570




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