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    Re: Is most probable position (MPP) a dangerous misnomer?
    From: Bill Lionheart
    Date: 2019 Feb 5, 22:22 +0000

    I am not a statistician so I only know the bits of statistics I have
    needed as I go along, but here are a couple of not normal
    distributions to think about.
    
    If you take measurements truncated to a certain number of digits and
    then average them, this is the average of  independent identically
    distributed uniform variables.
    
    The resulting distribution is the Bates Distribution
    https://en.wikipedia.org/wiki/Bates_distribution. Obviously the
    central limit theorem tells you it tends towards normal in the limit
    of averaging infinitely many, but for a small number you see the
    kurtosis (it is a short tailed distribution)
    
    Another two of interest to navigators might be the wrapped normal and
    von Mises distributions that are used for angles. Obviously an angle
    cant actually be normally distributed as it wraps around.  Maybe the
    errors in a compass that can spin around wildly if you do some tight
    turns before settling down, observed by a computer so you cant use
    common sense to ignore extreme values.
    
    Bill
    
    On Tue, 5 Feb 2019 at 21:00, Frank Reed  wrote:
    >
    > Dave, you wrote:
    > "One little gem which will delight you I’m sure is Anderson’s ‘Is the Gaussian Distribution Normal?’"
    >
    > The idea that common measurements ("normal" measurements) are not "normally 
    distributed" (pulled from a Gaussian distribution) is well-known. It's 
    introductory statistics. Folks rediscover this periodically and think they've 
    discovered something quite profound. There are other distributions that one 
    can use. But how can we generate simulated data that broadly resembles real 
    navigational data with its relatively high prevalence of outliers?
    >
    > There is a relatively easy way to model the higher prevalence of outliers in 
    sextant observations and other sorts of navigation measurements which allows 
    us to use many of the mathematical properties of Gaussian distributions. We 
    imagine our numbers as being pulled from two bins (both assumed to have mean 
    value equal to zero). One bin, call it bin A, has a relatively "normal" 
    standard deviation, call it s0 (for altitude sights in celestial navigation, 
    s0 might be 0.5 minutes of arc). The other bin, bin B, has a higher standard 
    deviation, call it s1. And s1 might be, in a typical real-world modelling 
    case, three times larger than s0. Number are drawn from bin A some large 
    fraction of the time, e.g. 80% of instances, and drawn from bin B the rest of 
    the time. Of course the numbers aren't labeled with the bin that they came 
    from so all you get in the end is a bunch of numbers with some statistical 
    properties. Those numbers will have a net standard deviation somewhat greater 
    than s0, but the key property is that they, collectively, will not correspond 
    to a Gaussian distribution because there will be more "outliers". The "tails" 
    of the distribution are thicker, more heavily-populated. A statistical 
    measure of this is known by the rather ugly, jargon-y term "kurtosity".
    >
    > So if you're simulating observations, don't use a simpleGaussian. Use a pair 
    of Gaussians, as above. This is a nice, easily-implemented technique for 
    generating model data for navigation simulations, and it can help find cases 
    where the "normal" math might lead you astray.
    >
    > Frank Reed
    >
    > View and reply to this message
    

       
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