# [R] How to find maximum values on the density function of arandom variable

guox at ucalgary.ca guox at ucalgary.ca
Fri Mar 13 18:38:33 CET 2009

```Thanks for your ideas.
I would like to find all possible maximums - "mountains" on a graph of a
given density function but I have no ideas. In calculus, there is a
general approach for a given function f(x): Find derivative of f(x) and
estimate all zeros of f'(x). These zeros give us locations of mountains if
f(x) is differentiable. If f(x) is not differentiable (f(x) is not
continuous in this case), we cannot use this approach. Bur in discrete
case, we can still talk about local maximums, I guess. Your ideas are very
-james
> If you are trying to build your own function then presumably you do
> not want the global maximum, since that is trivially returned by max.
> So what do you really want? Is this a programming question or just a
> general statistics question?
>
> If you want to search along a (specific) sequence for local maxima,
> then you could try programming with "second differences" looking for
> shifts in sign from positive to negative.
>
> vec <- c(1:5, 6, 5:1)
>
>  > diff(diff(vec))
> [1]  0  0  0  0 -2  0  0  0  0
>
> It is a bit ambiguous what you would do with this vector:
>
>  > vec2 <- c(1:5,rep(6,3),5:1)
>  > diff(diff(vec2))
>   [1]  0  0  0  0 -1  0 -1  0  0  0  0
>
> --
> David Winsemius
>
>
>
> On Mar 13, 2009, at 12:48 PM, guox at ucalgary.ca wrote:
>
>> Yes, a random variable, discrete or continuous one, should associate
>> with
>> a probability space and a measurable space.
>> I thought that graph of density(rv) below could give us an example
>> of a
>> density function. I am very sorry for confusing you.
>>
>> My question is how to find/estimate maximum values of a given density
>> function (even any given function within a given domain).
>> The number of these maximum values might be > 1 but the global one
>> is unique.
>> Any ideas and references?
>> Thanks,
>> -james
>>> There is some considerable confusion in both the question and the
>>>
>>> rv is **not**  a random variable. It is an (iid) sample from (i.e. a
>>> "realization" of) a random variable. It has *no* "density function"
>>> and
>>> the
>>> density() function is simply a procedure to **estimate** the
>>> density of
>>> the
>>> underlying random variable from which rv was sampled at a finite
>>> number
>>> of
>>> points. The result of density()and the max given in the reply will
>>> depend
>>> on
>>> the particular parameters given to density()(see ?density for
>>> details),
>>> as
>>> well as the data. In other words, both the question and answer
>>> posted are
>>> nonsense.
>>>
>>> Now let me contradict what I just said. **If** you consider rv a
>>> finite,
>>> discrete distribution (i.e. the whole population), then, in fact,
>>> it does
>>> have a discrete density, with point mass j(i)/n at each unique sample
>>> value
>>> i, where n is the total sample size (= 10000 in the example) and
>>> j(i) is
>>> the
>>> number of samples values == i, which would probably be 1 for all i.
>>> Then,
>>> of
>>> course, one can talk about the density of this finite distribution
>>> in the
>>> obvious way and its maximum or maxima, occur at those i for which
>>> n(i) is
>>> largest.
>>>
>>> But of course that's not what the poster really meant, so that
>>> brings us
>>> back to the nonsense question and answer. What James probably meant
>>> to
>>> was: "How can the maximum of the underlying population density
>>> function
>>> be
>>> estimated?" Well, that's a complicated issue. One could, of course,
>>> use
>>> some
>>> sort of density estimate -- there are tons -- and find its max;
>>> that was
>>> the
>>> approach taken in the answer, but it's not so simple as it appears
>>> because
>>> of the need to choose the **appropriate** estimate (including the
>>> parameters
>>> of the statistical algorithm doing the estimating ). This is the
>>> sort of
>>> thing that actually requires some careful thought and statistical
>>> expertise.
>>> You will find, I believe, that the prescription for finding the max
>>> suggested below can give quite different answers depending on the
>>> parameters
>>> chosen for this estimate, and on the estimate used. So if you need
>>> to do
>>> this right, may I suggest consulting the literature on density
>>> estimation
>>> or
>>> perhaps talking with your local statistician?
>>>
>>> -- Bert Gunter
>>> Genentech Nonclinical Statistics
>>>
>>> -----Original Message-----
>>> From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org
>>> ]
>>> On
>>> Behalf Of Mike Lawrence
>>> Sent: Thursday, March 12, 2009 5:40 PM
>>> To: guox at ucalgary.ca
>>> Cc: r-help at r-project.org
>>> Subject: Re: [R] How to find maximum values on the density function
>>> of
>>> arandom variable
>>>
>>> rv <- rbinom(10000,1,0.1) + rnorm(10000)
>>>
>>> d.rv = density(rv)
>>> d.x = d.rv\$x
>>> d.y = d.rv\$y
>>>
>>> d.rv.max = d.rv\$x[which.max(d.rv\$y)]
>>>
>>> plot(d.rv)
>>> abline(v=d.rv.max)
>>>
>>> #that what you want?
>>>
>>> On Thu, Mar 12, 2009 at 6:28 PM,  <guox at ucalgary.ca> wrote:
>>>> I would like to find the maximum values on the density function of a
>>>> random variable. For example, I have a random variable
>>>>
>>>> rv <- rbinom(10000,1,0.1) + rnorm(10000)
>>>>
>>>> Its density function is given by density(rv) and can be displayed by
>>>> plot(density(rv)). How to calculate its maximum values?
>>>> A density function may have a few (global and local) maximum values.
>>>> -james
>>>>
>>>> ______________________________________________
>>>> R-help at r-project.org mailing list
>>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>> http://www.R-project.org/posting-guide.html
>>>> and provide commented, minimal, self-contained, reproducible code.
>>>>
>>>
>>>
>>>
>>> --
>>> Mike Lawrence
>>> Department of Psychology
>>> Dalhousie University
>>>
>>> Looking to arrange a meeting? Check my public calendar:
>>> http://tinyurl.com/mikes-public-calendar
>>>
>>> ~ Certainty is folly... I think. ~
>>>
>>> ______________________________________________
>>> R-help at r-project.org mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>> http://www.R-project.org/posting-guide.html
>>> and provide commented, minimal, self-contained, reproducible code.
>>>
>>>
>>>
>>
>> ______________________________________________
>> R-help at r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>
> David Winsemius, MD
> Heritage Laboratories
> West Hartford, CT
>
>
>

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