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    27 Nov '16 15:181 edit
    Originally posted by DeepThought

    Your resu the distribution so plausibly your program is trying to sample too small a subset of the domain.
    i will also investigate that next.
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    27 Nov '16 15:281 edit
    Arrr I think I might have misunderstood your formula. I will investigate that next and then come back to you.
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    27 Nov '16 16:082 edits
    Originally posted by humy
    Arrr I think I might have misunderstood your formula. I will investigate that next and then come back to you.
    Arr I HAVE misunderstood your formula. Correct me if I am wrong but you mean;

    I = K sqrt(2π/m) exp( -mσ^2 / (2*K^2))

    (and NOT I = K sqrt(2π/m) exp( (-mσ^2 / 2)*K^2) )

    where σ^2 = ( ( ∑ [n=1, m] (h{n})^2 ) / m ) - ( ( ∑ [n=1, m] h{n} ) / m )^2

    + I have just spotted another big error I made so NOW I think I might be on the right track. I will come back to you.
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    27 Nov '16 18:118 edits
    DeepThought

    SUCCESS! (at last)
    Your formula is exactly correct and it was me who was in error (as usual).
    The reason why I was getting nonsense of my program was simply because I kept entering the wrong formula (rubbish-in, rubbish-out), not because of anything wrong with the behavior with the iteration of my software I especially developed to test such formulas.
    As usual, once I got the formula correct and tested it with my software,my software gave a result showing the formula to be accurate to many significant figures of accuracy -how many just depends on what inputs I use.
    In fact, I found that with none-extreme input values, my software often gave an amazing level of accuracy of a massive 14SF or often 13DP -sometimes even more if I really push it to its limits! I am so impressed with it that I think perhaps one day I can sell this software over the net for a profit; but first I must finish my book.

    Incidentally, using my new systematic naming protocols I have especially invented for naming the hundreds of new probability distributions I have discovered analyzed and defined (some of which are infinite sets of distributions), this particular maths is for what I call the "mean-wild norm poly-gmav" distribution, which is related to the normal distribution and based on random walks but with an unknown mean-average parameter value.
  5. Standard memberDeepThought
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    28 Nov '16 01:04
    Originally posted by humy
    DeepThought

    SUCCESS! (at last)
    Your formula is exactly correct and it was me who was in error (as usual).
    The reason why I was getting nonsense of my program was simply because I kept entering the wrong formula (rubbish-in, rubbish-out), not because of anything wrong with the behavior with the iteration of my software I especially developed to test such fo ...[text shortened]... normal distribution and based on random walks but with an unknown mean-average parameter value.
    Well, I misunderstood the entire problem in the first place and then fouled up substituting into my own formula. We're suffering a little from having to use a text system to write maths rather than LaTeX or some such, and my attempt to clarify generated even more confusion...

    My initial misunderstanding was due to looking at your formula in a morning aftery kind of way. It could possibly have been avoided by alternating square and round brackets to indicate nesting, so instead of:

    ∫[–∞, ∞] 1 / e^( ( 0.5/K^2 ) * ∑[n=1, m] ( x – h{n} )^2 ) dx

    you would type in:

    ∫{–∞, ∞} 1 / exp( [0.5/K^2 ] * ∑{n=1, m} [x – h(n)]^2 ) dx

    so I'm suggesting a system where curly brackets denote sets or limits of integrations and square and round brackets are alternated to group terms in our expressions. It just makes it easier to look at. My formula would then be:

    I = K sqrt(2π/m) exp( -[mσ^2] / [2K^2])

    which is easier to look at than:

    I = K sqrt(2π/m) exp(-(mσ^2)/(2K^2))

    Regarding your numerical integration program; if it's behaving on every problem you give it then don't worry, but I think that there are functions where there is an advantage in trying to sample the regions where it is varying quickly more finely than the regions where it's not varying quickly (those being the regions where a linear or quadratic approximation will work well over large intervals).
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    28 Nov '16 07:148 edits
    -or perhaps instead of writing just

    ∑[n=1, m] ( x – h{n} )^2

    for absolute clarity, write that as;

    ∑[n=1, m] ( ( x – h{n} )^2 )

    or, perhaps even better, write that as;

    ∑[n=1, m] [ ( x – h{n} )^2 ]

    then surely there cannot be any easy misunderstanding?

    And if you wanted to change the meaning of that from 'the sum of the squares of the differences' to 'the sum of the differences then square it', perhaps you can write that different meaning as;

    ( ∑[n=1, m] [ x – h{n} ] )^2

    so, as the rule, you simply put everything to be summed for a particular summation entirely within a pair of square brackets?
    I am now considering adopting this notation permanently and esp in my book.
  7. Subscribersonhouse
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    28 Nov '16 13:59
    Originally posted by humy
    -or perhaps instead of writing just

    ∑[n=1, m] ( x – h{n} )^2

    for absolute clarity, write that as;

    ∑[n=1, m] ( ( x – h{n} )^2 )

    or, perhaps even better, write that as;

    ∑[n=1, m] [ ( x – h{n} )^2 ]

    then surely there cannot be any easy misunderstanding?

    And if you wanted to change the meaning of that from 'the sum of the squares of the diff ...[text shortened]... of square brackets?
    I am now considering adopting this notation permanently and esp in my book.
    Hum, is this a non-academic work or is it part of a thesis?
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    28 Nov '16 19:036 edits
    Originally posted by sonhouse
    Hum, is this a non-academic work or is it part of a thesis?
    sort-of non-academic work;
    it is a possible recommendation to put in my book to all readers for a new notation for summations to help reduce potential misinterpretations and which hopefully will be widely adopted after publication of my book.
    I hope my book does a lot more than put forward a thesis (although it puts forward many theorems with maths proofs) and is written to, among several other important things, change both maths/statistics terminology and notation generally used in a significant way.
    + revolutionize much of philosophy and science + completely revolutionize A.I (eventually; it would take a very long time to apply it to A.I) and the whole of probability theory and the science of statistical analysis.
  9. Standard memberDeepThought
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    28 Nov '16 23:51
    Originally posted by humy
    sort-of non-academic work;
    it is a possible recommendation to put in my book to all readers for a new notation for summations to help reduce potential misinterpretations and which hopefully will be widely adopted after publication of my book.
    I hope my book does a lot more than put forward a thesis (although it puts forward many theorems with maths proofs) an ...[text shortened]... to apply it to A.I) and the whole of probability theory and the science of statistical analysis.
    Just for clarity, in what I wrote above I was referring to posts in this forum. In your book you should use standard notation unless you have an overriding reason not to.

    The confusion in the first few posts was because you typed in:

    1/exp((1/Q)*(P))

    where P and Q are placeholders for the sum and the factor of 2K^2 respectively, which I mistook for:

    P/exp(1/Q).

    I would have written your expression as: exp(-P/Q) or possibly exp(-(P)/(Q)) which I think would have been clearer. That was the root of my misunderstanding - had the parentheses made the nesting of brackets clearer we might have avoided it - but most readers will be expecting your numerators to be on the left of your denominators when writing inline formulae (in other words where writing it above the denominator is an option such as in these forums). Also 1/x^y = x^-y and the latter notation is preferable when you are writing inline formulae. What I'm saying is that my comments about brackets are relevant to this forum, where typesetting possibilities are limited to unicode, and not to media where decent typesetting is available.
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    29 Nov '16 07:323 edits
    Originally posted by DeepThought
    Just for clarity, in what I wrote above I was referring to posts in this forum. In your book you should use standard notation unless you have an overriding reason not to.

    The confusion in the first few posts was because you typed in:

    1/exp((1/Q)*(P))

    where P and Q are placeholders for the sum and the factor of 2K^2 respectively, which I mistook ...[text shortened]... ng possibilities are limited to unicode, and not to media where decent typesetting is available.
    I am struggling to see how you can misinterpret 1/exp((1/Q)*(P)) as P/exp(1/Q).
    It is my understanding that you must do the maths operations in the brackets first and always to the maths operations within the inner most pair of brackets before the outer most pair?
    With that being correct, how can it make any sense to write 1/exp((1/Q)*(P)) to mean P/exp(1/Q)? There doesn't seem to be any function for the outer brackets for that meaning and it would make much more sense (although not a good idea because of possible confusion with order of precedence of the / and * operators) to write 1/exp(1/Q)*(P) to mean P/exp(1/Q) although I think it would be better to write (1/exp(1/Q))*(P) than to write 1/exp(1/Q)*(P).
    What meaning did you attach to the outer brackets to 1/exp((1/Q)*(P)) ?

    But I will take your advice including "...1/x^y = x^-y and the latter notation is preferable when you are writing inline formula"
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    29 Nov '16 11:481 edit
    I just thought of such a simple way to avoid confusion I don't understand why I didn't think of it before.
    with expressions like;

    ∫[–∞, ∞] 1 / e^( ( 0.5/K^2 ) * ∑[n=1, m] ( x – h{n} )^2 ) dx

    I could simply use a "where statement" and rewrite that like this;

    ∫[–∞, ∞] 1 / e^( sum /(2*K^2) ) dx
    where sum = ∑[n=1, m] (( x – h{n} )^2 )

    Problem solved. I will for now on use that strategy in my book.
  12. Joined
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    29 Nov '16 19:18
    Originally posted by humy
    I just thought of such a simple way to avoid confusion I don't understand why I didn't think of it before.
    with expressions like;

    ∫[–∞, ∞] 1 / e^( ( 0.5/K^2 ) * ∑[n=1, m] ( x – h{n} )^2 ) dx

    I could simply use a "where statement" and rewrite that like this;

    ∫[–∞, ∞] 1 / e^( sum /(2*K^2) ) dx
    where sum = ∑[n=1, m] (( x – h{n} )^2 )

    Problem solved. I will for now on use that strategy in my book.
    actually, I think we have agreed that it would be better to write that as;

    ∫[–∞, ∞] e^( –sum / (2*K^2) ) dx
    where sum = ∑[n=1, m] (( x – h{n} )^2 )
  13. Standard memberDeepThought
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    29 Nov '16 21:31
    Originally posted by humy
    I am struggling to see how you can misinterpret 1/exp((1/Q)*(P)) as P/exp(1/Q).
    It is my understanding that you must do the maths operations in the brackets first and always to the maths operations within the inner most pair of brackets before the outer most pair?
    With that being correct, how can it make any sense to write 1/exp((1/Q)*(P)) to mean P/exp(1/Q) ...[text shortened]... ing "...1/x^y = x^-y and the latter notation is preferable when you are writing inline formula"
    I am struggling to see how you can misinterpret 1/exp((1/Q)*(P)) as P/exp(1/Q).

    Easy, there's a complex expression, lots of brackets, and when I read it I misread the nesting of the brackets - it was early in the morning and I need new glasses. I thought you had:

    1/(exp(1/Q))*(P)

    Problem solved. I will for now on use that strategy in my book.

    Good, for really complex expressions it can be helpful to use placeholders, but I'd recommend not defining subsidiary functions as a blanket policy in your book, you can render it unreadable. Just use it when you think it will make it more readable. In this forum it would be helpful, but assuming you have some sort of typesetting program (e.g. LaTeX, or even the Maths writing functionality in Word) for the relatively simple formula:

    ∫[–∞, ∞] exp( -∑[n=1, m] ( x – h{n} )^2 / (2*K^2) ) dx

    There is no need to write:

    ∫[–∞, ∞] e^(- S(x) / (2*K^2) ) dx
    where,
    S(x) = ∑[n=1, m] (( x – h{n} )^2 )

    the typeset equation (with the ^2 in the sum as an actual superscript) will read perfectly well.
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