Comments on: Log concave functions http://www.johndcook.com/blog/2009/01/09/log-concave-functions/ The blog of John D. Cook Sat, 11 Feb 2012 01:10:06 -0500 http://wordpress.org/?v=2.8.4 hourly 1 By: Shamus http://www.johndcook.com/blog/2009/01/09/log-concave-functions/comment-page-1/#comment-90278 Shamus Tue, 28 Jun 2011 00:47:57 +0000 http://www.johndcook.com/blog/?p=1234#comment-90278 If you would like a guide through Steven Boyd's book, lectures from the two courses he teaches on convex optimization are available on youtube http://www.youtube.com/playlist?p=PL3940DD956CDF0622 and other course material is on the Stanford web site http://see.stanford.edu/see/courseinfo.aspx?coll=2db7ced4-39d1-4fdb-90e8-364129597c87 If you would like a guide through Steven Boyd’s book, lectures from the two courses he teaches on convex optimization are available on youtube http://www.youtube.com/playlist?p=PL3940DD956CDF0622 and other course material is on the Stanford web site http://see.stanford.edu/see/courseinfo.aspx?coll=2db7ced4-39d1-4fdb-90e8-364129597c87

]]> By: wen http://www.johndcook.com/blog/2009/01/09/log-concave-functions/comment-page-1/#comment-48003 wen Wed, 06 Oct 2010 19:49:47 +0000 http://www.johndcook.com/blog/?p=1234#comment-48003 Hi, Thanks! I have a question: is the sum of infinitely many log concave functions a log concave function? Thanks! Hi,

Thanks! I have a question: is the sum of infinitely many log concave functions a log concave function?

Thanks!

]]> By: Brooks http://www.johndcook.com/blog/2009/01/09/log-concave-functions/comment-page-1/#comment-46573 Brooks Fri, 17 Sep 2010 01:58:39 +0000 http://www.johndcook.com/blog/?p=1234#comment-46573 Hi, Q(x) =erf(x); I find a intersting result. When p=1-2p, f =p+(1-2p)Q(x) will be a log concave function. The inequality always holds. %%%%% clear clc close all x=[-10:0.01:10]; y=1/3+1/3.*erf(x); f1= 1/3 *1/(sqrt(2*pi)) *exp(-x.^2/2); f2= -1/3 *1/(sqrt(2*pi)) *exp(-x.^2/2) .* x; L=f2.*y; R=f1.^2; boot=(R>=L); %% (if R>=L, the value will be set to 1, or set to 0) plot(-10:0.01:10,L,'+',-10:0.01:10,R,'*',-10:0.01:10,boot,'s') %%% Thanks. Hi,
Q(x) =erf(x);
I find a intersting result. When p=1-2p, f =p+(1-2p)Q(x) will be a log concave function. The inequality always holds.
%%%%%
clear
clc
close all
x=[-10:0.01:10];
y=1/3+1/3.*erf(x);
f1= 1/3 *1/(sqrt(2*pi)) *exp(-x.^2/2);
f2= -1/3 *1/(sqrt(2*pi)) *exp(-x.^2/2) .* x;
L=f2.*y;
R=f1.^2;
boot=(R>=L); %% (if R>=L, the value will be set to 1, or set to 0)
plot(-10:0.01:10,L,’+',-10:0.01:10,R,’*',-10:0.01:10,boot,’s’)

%%%

Thanks.

]]> By: John http://www.johndcook.com/blog/2009/01/09/log-concave-functions/comment-page-1/#comment-46544 John Thu, 16 Sep 2010 13:33:25 +0000 http://www.johndcook.com/blog/?p=1234#comment-46544 I doubt your conjecture holds. Let f(x) = p + (1 - 2p)Q(x) and test whether f(x) f''(x) ≤ (f'(x))^2 for all p. I believe the inequality does not hold for p = 3/8. I only did a hurried calculation and so I may have made errors, but I believe that approach to resolving the question will work. I doubt your conjecture holds. Let f(x) = p + (1 – 2p)Q(x) and test whether f(x) f”(x) ≤ (f’(x))^2 for all p. I believe the inequality does not hold for p = 3/8.

I only did a hurried calculation and so I may have made errors, but I believe that approach to resolving the question will work.

]]> By: Brooks http://www.johndcook.com/blog/2009/01/09/log-concave-functions/comment-page-1/#comment-46539 Brooks Thu, 16 Sep 2010 12:21:19 +0000 http://www.johndcook.com/blog/?p=1234#comment-46539 Thanks!　I have a question about "log concave". Assume Q(x) is a log concave and it denotes the CDF. I can't be sure that "p+(1-2p)Q(x)" is also a log concave function, if given p (0<p<1). Can you give some analysis for me , please? Thanks!　I have a question about “log concave”.
Assume Q(x) is a log concave and it denotes the CDF.
I can’t be sure that “p+(1-2p)Q(x)” is also a log concave function, if given p (0<p<1).
Can you give some analysis for me , please?

]]> By: KW http://www.johndcook.com/blog/2009/01/09/log-concave-functions/comment-page-1/#comment-25234 KW Mon, 28 Sep 2009 13:46:03 +0000 http://www.johndcook.com/blog/?p=1234#comment-25234 John, A concave function could be increasing or decreasing fast in one segment of the curve and slow in other segments, while the upper half of a circle seems increasing and decreasing constantly. My question is how to interpret this in a mathematics form. Thanks. KW John,

A concave function could be increasing or decreasing fast in one segment of the curve and slow in other segments, while the upper half of a circle seems increasing and decreasing constantly. My question is how to interpret this in a mathematics form.

Thanks.

KW

]]> By: John http://www.johndcook.com/blog/2009/01/09/log-concave-functions/comment-page-1/#comment-21290 John Fri, 17 Jul 2009 00:30:42 +0000 http://www.johndcook.com/blog/?p=1234#comment-21290 seh chun: For a reference, see the book by Stephen Boyd mentioned at the bottom on the post. seh chun: For a reference, see the book by Stephen Boyd mentioned at the bottom on the post.

]]> By: seh chun http://www.johndcook.com/blog/2009/01/09/log-concave-functions/comment-page-1/#comment-21289 seh chun Fri, 17 Jul 2009 00:23:39 +0000 http://www.johndcook.com/blog/?p=1234#comment-21289 Hi, This post is great and useful. About a point you have put, saying "The running average of a log concave function is also log concave". May I know is there any book / reference that I can find more information about that? Thanks. Hi,

This post is great and useful. About a point you have put, saying “The running average of a log concave function is also log concave”. May I know is there any book / reference that I can find more information about that?

Thanks.

]]> By: John http://www.johndcook.com/blog/2009/01/09/log-concave-functions/comment-page-1/#comment-13431 John Mon, 16 Feb 2009 21:43:08 +0000 http://www.johndcook.com/blog/?p=1234#comment-13431 Thanks! You were right: "logic" was a typo for "logit." I've corrected the post. Thanks! You were right: “logic” was a typo for “logit.” I’ve corrected the post.

]]> By: gb http://www.johndcook.com/blog/2009/01/09/log-concave-functions/comment-page-1/#comment-13430 gb Mon, 16 Feb 2009 21:25:27 +0000 http://www.johndcook.com/blog/?p=1234#comment-13430 Great post. One minor issue - I haven't seen the function exp(x)/(1+exp(x)) called the inverse logic function before (which might simply be ignorance on my part). (I have seen it called the <a href="http://en.wikipedia.org/wiki/Logit" rel="nofollow">inverse logit function</a> or the <a href="http://en.wikipedia.org/wiki/Logistic_function" rel="nofollow">logistic function</a>. ) However, given "logit" and "logic" only differ by one character, I did wonder if it might have been a simple typo. Great post.

One minor issue – I haven’t seen the function exp(x)/(1+exp(x)) called the inverse logic function before (which might simply be ignorance on my part).

(I have seen it called the inverse logit function or the logistic function. )

However, given “logit” and “logic” only differ by one character, I did wonder if it might have been a simple typo.

]]> By: Welcome to Carnival of Mathematics 48 = 6!!! « Concrete Nonsense http://www.johndcook.com/blog/2009/01/09/log-concave-functions/comment-page-1/#comment-12585 Welcome to Carnival of Mathematics 48 = 6!!! « Concrete Nonsense Fri, 30 Jan 2009 18:57:28 +0000 http://www.johndcook.com/blog/?p=1234#comment-12585 [...] models the distribution of time customers spend in coffee shops and also gives some properties of log-concave functions (I did not know before that they are closed under [...] [...] models the distribution of time customers spend in coffee shops and also gives some properties of log-concave functions (I did not know before that they are closed under [...]

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