[R] Why two curves and numerical integration look so different?

[C W]

On a side note, is it ok to do?

> which(max(p_x))
and use that instead of numerical integration to get E[X]?

I will try both and report back!  Thank you expeRts

On Fri, Feb 12, 2016 at 11:29 AM, C W <tmrsg11 at gmail.com> wrote:

> Hi Peter,
>
> Great, let me try that and get back to you on my findings in a few hours!
>  :)
>
> On Fri, Feb 12, 2016 at 11:09 AM, peter dalgaard <pdalgd at gmail.com> wrote:
>
>> I don't see here FAQ 7.31 comes in either (for once!...)
>>
>> However, either the density is unnormalized and the integral is not 1, or
>> the integral is 1 and it is normalized. The one in the picture clearly does
>> not integrate to one. You can fit a rectangle of size 0.1 by 1e191 under
>> the curve so the integral should be > 1e190 .
>>
>> As depicted, I don't see why a plain integral from .5 to 1.5 shouldn't
>> work.
>>
>> -pd
>>
>> On 12 Feb 2016, at 16:57 , C W <tmrsg11 at gmail.com> wrote:
>>
>> > Hi Bert,
>> >
>> > Yay fantasyland!
>> >
>> > In all seriousness, You are referring to this?
>> >
>> https://cran.r-project.org/doc/FAQ/R-FAQ.html#Why-doesn_0027t-R-think-these-numbers-are-equal_003f
>> >
>> > In particular, you mean this: .Machine$double.eps ^ 0.5?
>> >
>> > Thanks!
>> >
>> > On Fri, Feb 12, 2016 at 10:53 AM, Bert Gunter <bgunter.4567 at gmail.com>
>> > wrote:
>> >
>> >> You are working in fantasyland. Your density is nonsense.
>> >>
>> >> Please see FAQ 7.31 for links to computer precision of numeric
>> >> calculations.
>> >>
>> >>
>> >> Cheers,
>> >> Bert
>> >> Bert Gunter
>> >>
>> >> "The trouble with having an open mind is that people keep coming along
>> >> and sticking things into it."
>> >> -- Opus (aka Berkeley Breathed in his "Bloom County" comic strip )
>> >>
>> >>
>> >> On Fri, Feb 12, 2016 at 7:36 AM, C W <tmrsg11 at gmail.com> wrote:
>> >>> Hi David,
>> >>>
>> >>> This is the Gaussian looking distribution I am trying to integrate.
>> >>>
>> >>
>> https://drive.google.com/file/d/0B2xN0-A6iTB4NThIZ2tYdGxHc00/view?usp=sharing
>> >>>
>> >>> Notice the unnormalized density goes up as high as 2.5*101^191.
>> >>>
>> >>> I tried to create small intervals like
>> >>>> seq(0.5, 1.3, by = 10^(-8))
>> >>>
>> >>> but that doesn't seem to be good enough, as we know, it should
>> integrate
>> >> to
>> >>> 1.
>> >>>
>> >>> On Thu, Feb 11, 2016 at 3:32 PM, David Winsemius <
>> dwinsemius at comcast.net
>> >>>
>> >>> wrote:
>> >>>
>> >>>>
>> >>>>> On Feb 11, 2016, at 11:30 AM, C W <tmrsg11 at gmail.com> wrote:
>> >>>>>
>> >>>>> Hi David,
>> >>>>>
>> >>>>> My real function is actually a multivariate normal, the simple toy
>> 1-d
>> >>>> normal won't work.
>> >>>>>
>> >>>>> But, you gave me an idea about restricting the bounds, and focus
>> >>>> integrating on that.  I will get back to you if I need any further
>> >>>> assistance.
>> >>>>
>> >>>> You'll probably need to restrict your bounds even more severely than
>> I
>> >> did
>> >>>> in the 1-d case (using 10 SD's on either side of the mean) . You
>> might
>> >> need
>> >>>> adequate representation of points near the center of your
>> >> hyper-rectangles.
>> >>>> At least that's my armchair notion since I expect the densities tail
>> off
>> >>>> rapidly in the corners. You can shoehorn multivariate integration
>> around
>> >>>> the `integrate` function but it's messy and inefficient. There are
>> other
>> >>>> packages that would be better choices. There's an entire section on
>> >>>> numerical differentiation and integration in CRAN Task View:
>> Numerical
>> >>>> Mathematics.
>> >>>>
>> >>>> --
>> >>>> David.
>> >>>>
>> >>>>
>> >>>>>
>> >>>>> Thank you so much!
>> >>>>>
>> >>>>> On Thu, Feb 11, 2016 at 2:06 PM, David Winsemius <
>> >> dwinsemius at comcast.net>
>> >>>> wrote:
>> >>>>>
>> >>>>>> On Feb 11, 2016, at 9:20 AM, C W <tmrsg11 at gmail.com> wrote:
>> >>>>>>
>> >>>>>> I want to do numerical integration w.r.t. mu: P(mu) × N(mu,
>> 0.00001)
>> >>>>>>
>> >>>>>> Because the variance is small, it results in density like:
>> >> 7.978846e+94
>> >>>>>>
>> >>>>>> Is there any good suggestion for this?
>> >>>>>
>> >>>>> So what's the difficulty? It's rather like the Dirac function.
>> >>>>>
>> >>>>>> integrate( function(x) dnorm(x, sd=0.00001), -.0001,0.0001)
>> >>>>> 1 with absolute error < 7.4e-05
>> >>>>>
>> >>>>>
>> >>>>> --
>> >>>>> David.
>> >>>>>
>> >>>>>>
>> >>>>>> Thanks so much!
>> >>>>>>
>> >>>>>>
>> >>>>>> On Thu, Feb 11, 2016 at 9:14 AM, C W <tmrsg11 at gmail.com> wrote:
>> >>>>>>
>> >>>>>>> Wow, thank you, that was very clear.  Let me give it some more
>> runs
>> >>>> and
>> >>>>>>> investigate this.
>> >>>>>>>
>> >>>>>>> On Thu, Feb 11, 2016 at 12:31 AM, William Dunlap <
>> >> wdunlap at tibco.com>
>> >>>>>>> wrote:
>> >>>>>>>
>> >>>>>>>> Most of the mass of that distribution is within 3e-100 of 2.
>> >>>>>>>> You have to be pretty lucky to have a point in sequence
>> >>>>>>>> land there.  (You will get at most one point there because
>> >>>>>>>> the difference between 2 and its nearest neightbors is on
>> >>>>>>>> the order of 1e-16.)
>> >>>>>>>>
>> >>>>>>>> seq(-2,4,len=101), as used by default in curve, does include 2
>> >>>>>>>> but seq(-3,4,len=101) and seq(-2,4,len=100) do not so
>> >>>>>>>> curve(..., -3, 4, 101) and curve(..., -2, 4, 100) will not show
>> >> the
>> >>>> bump.
>> >>>>>>>> The same principal holds for numerical integration.
>> >>>>>>>>
>> >>>>>>>>
>> >>>>>>>> Bill Dunlap
>> >>>>>>>> TIBCO Software
>> >>>>>>>> wdunlap tibco.com
>> >>>>>>>>
>> >>>>>>>> On Wed, Feb 10, 2016 at 6:37 PM, C W <tmrsg11 at gmail.com> wrote:
>> >>>>>>>>
>> >>>>>>>>> Dear R,
>> >>>>>>>>>
>> >>>>>>>>> I am graphing the following normal density curve.  Why does it
>> >> look
>> >>>> so
>> >>>>>>>>> different?
>> >>>>>>>>>
>> >>>>>>>>> # the curves
>> >>>>>>>>> x <- seq(-2, 4, by=0.00001)
>> >>>>>>>>> curve(dnorm(x, 2, 10^(-100)), -4, 4)  #right answer
>> >>>>>>>>> curve(dnorm(x, 2, 10^(-100)), -3, 4)  #changed -4 to -3, I get
>> >> wrong
>> >>>>>>>>> answer
>> >>>>>>>>>
>> >>>>>>>>> Why the second curve is flat?  I just changed it from -4 to -3.
>> >>>> There is
>> >>>>>>>>> no density in that region.
>> >>>>>>>>>
>> >>>>>>>>>
>> >>>>>>>>> Also, I am doing numerical integration.  Why are they so
>> >> different?
>> >>>>>>>>>
>> >>>>>>>>>> x <- seq(-2, 4, by=0.00001)
>> >>>>>>>>>> sum(x*dnorm(x, 2, 10^(-100)))*0.00001
>> >>>>>>>>> [1] 7.978846e+94
>> >>>>>>>>>> x <- seq(-1, 4, by=0.00001) #changed -2 to -1
>> >>>>>>>>>> sum(x*dnorm(x, 2, 10^(-100)))*0.00001
>> >>>>>>>>> [1] 0
>> >>>>>>>>>
>> >>>>>>>>> What is going here?  What a I doing wrong?
>> >>>>>>>>>
>> >>>>>>>>> Thanks so much!
>> >>>>>>>>>
>> >>>>>>>>>       [[alternative HTML version deleted]]
>> >>>>>>>>>
>> >>>>>>>>> ______________________________________________
>> >>>>>>>>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more,
>> >> see
>> >>>>>>>>> https://stat.ethz.ch/mailman/listinfo/r-help
>> >>>>>>>>> PLEASE do read the posting guide
>> >>>>>>>>> http://www.R-project.org/posting-guide.html
>> >>>>>>>>> and provide commented, minimal, self-contained, reproducible
>> >> code.
>> >>>>>>>>>
>> >>>>>>>>
>> >>>>>>>>
>> >>>>>>>
>> >>>>>>
>> >>>>>>      [[alternative HTML version deleted]]
>> >>>>>>
>> >>>>>> ______________________________________________
>> >>>>>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
>> >>>>>> https://stat.ethz.ch/mailman/listinfo/r-help
>> >>>>>> PLEASE do read the posting guide
>> >>>> http://www.R-project.org/posting-guide.html
>> >>>>>> and provide commented, minimal, self-contained, reproducible code.
>> >>>>>
>> >>>>> David Winsemius
>> >>>>> Alameda, CA, USA
>> >>>>>
>> >>>>>
>> >>>>
>> >>>> David Winsemius
>> >>>> Alameda, CA, USA
>> >>>>
>> >>>>
>> >>>
>> >>>        [[alternative HTML version deleted]]
>> >>>
>> >>> ______________________________________________
>> >>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
>> >>> https://stat.ethz.ch/mailman/listinfo/r-help
>> >>> PLEASE do read the posting guide
>> >> http://www.R-project.org/posting-guide.html
>> >>> and provide commented, minimal, self-contained, reproducible code.
>> >>
>> >
>> >       [[alternative HTML version deleted]]
>> >
>> > ______________________________________________
>> > R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
>> > https://stat.ethz.ch/mailman/listinfo/r-help
>> > PLEASE do read the posting guide
>> http://www.R-project.org/posting-guide.html
>> > and provide commented, minimal, self-contained, reproducible code.
>>
>> --
>> Peter Dalgaard, Professor,
>> Center for Statistics, Copenhagen Business School
>> Solbjerg Plads 3, 2000 Frederiksberg, Denmark
>> Phone: (+45)38153501
>> Office: A 4.23
>> Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com
>>
>>
>

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