Talk:Logistic function

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[edit] request

can we please have instructions for how to fit a logistic curve's parameters to data? Is there an analytic way to do this; or do you just have to use numerical methods like gradient descent and pray that it converges?

The Levenberg-Marquardt algorithm is a very general method that may be applicable here, depending on what exactly it is that you want to do. --MarkSweep (call me collect) 08:50, 22 March 2006 (UTC)

[edit] Plot

The plot needs labels on the X and Y axis! Targa86 21:25, 28 June 2007 (UTC)

[edit] wishlist entry

Would be neat to have a 1-2 sentence explanation in laymen's terms what is it good for, and why does it fit better or worse a task than other, comparable things/methods/whatevers. See standard deviation for example, where the section Interpretation and application explains in simple terms what is that good for. (Many statistical terms could use such an "interpretation"... most of them mathematically defines the terms but forget to explain it to shoemakers and milkmen. :-) ) --grin ✎ 07:42, 2004 May 6 (UTC)

I agree. As someone researching the diffusion of innovations, which follows an s-curve, I'd like to know what an s-curve really means. I'm not a shoemaker or milkman, but I would like to have a better understanding of what an s-curve is. --Westendgirl 22:59, 3 Jan 2005 (UTC)

This is largely already there; it's just not explicitly labeled as the answer to that question. I'll return to this article soon. Michael Hardy 01:04, 4 Jan 2005 (UTC)
It might be there, but I don't have the math background to understand it. :) My husband, a mathematician, assures me that all I need to know (for my purposes) is that an s-curve is a curve shaped like an S, as a result of data that has been plotted. I recognize that I don't need to understand the formula, but is this s-shape all I (and other lay readers) need to know? --Westendgirl 05:37, 4 Jan 2005 (UTC)
Done.Ancheta Wis 10:29, 8 Apr 2005 (UTC) Cleanup note removed
In effect, the S-curve is the simplest kind of long-term transition you can have where both the starting point and ending point are bounded. The differential equation governing the process is a polynomial of the smallest order (quadratic) that will accomplish the trick. Polynomials of order 0 and 1 won't do the trick.
It also occurs naturally in the study of any chemical process where both sides of a reaction, e.g. 2A <-> B+C are of order 2 or less. The resulting differential equation governing the rate is then quadratic and the solution (if bounded) will be a logistic.
In the study of the world population curve, contrary to the article's assertion, there actually is a good fit -- provided one takes the lower asymptotic to be non-zero. The underlying equation generalizes the Verhulst equation (also called the Malthus-Verhulst equation) by allowing a zero order term on the right, x' = A + Bx + Cx^2 with non-zero A. In fact, what one finds is that for the 1974-2004 range, there is a near-exact symmetry about 1989 (if I recall, it's P(1989-x)+P(1989+x) = 10.386 billion +/- 5 million, according to the international database of the US Census Bureau) and the curve closely fits a logistic with an upper asymptotic of 7.8 billion. The fit is within 6 million over the entire 1974-2004 range with a root-mean-square deviation of around 3 million!
For earler segments of the curve, there is a poor fit to a logistic primarily because (a) there are natural phase boundaries c. 1950 and c. 1850 which are impossible to fit a logistic across, and (b) the 1850-1950 segment does not fit any logistic at all, but rather the limiting case: an exponential with a lower asymptotic of about 1 billion. -- Mark, 26 September 2006

[edit] Standard formula needed

The three pages sigmoid, sigmoid function, and logistic curve seemed rather confusing for readers. So I have moved all info on the "standard" logistic curve 1/(1+exp(-t)) to sigmoid function, and kept here only the stuff about more general logistic curves that are shifted and scaled copied of the latter. I have provided a general formula for these curves, but I am not familiar with this field so I do not know whether the formula and parameters that I picked are OK. If there is a "canonical" formula, would someone please provide it?
Also the plot shows the standard sigmoid function; here one should have a plot of the more general logistic curve, shifted and scaled.
Jorge Stolfi 07:37, 2 Jul 2004 (UTC)

[edit] Logistic curve vs. logistic function

Argh.. I moved logistic curve to logistic function because it seemed more logical, but now I see that most users call it "curve". Profuse apologies. If you can, could you please reverse the move, making logistic function into a redirect? Thanks...
Jorge Stolfi 08:01, 2 Jul 2004 (UTC)

[edit] Typo

I think there is a mistake in the line: "In 1924 during The Great Depression Professor Ray Pearl and Lowell J. Reed used Verhulst's model to predict an upper limit of 2 billion for the world population. This was passed in 1930..." The Roaring 20s were at their height in 1924. Black Tuesday wasn't until 1929.

[edit] Article has wrong focus

The phrasing and structure of the article mistakes the logistic function for one of its applications, namely modeling population growth. Instead, the logistic function should be introduced as a mathematical function. It surely it has mathematical properties. Other important applications are found, for example, in machine learning and statistics. -Pgan002 21:47, 7 February 2006 (UTC)

[edit] Another form of the logistic function

I've only taken one year of calculus, but we learned the logistic function as P(t) = {M \over {Ae^{-kt} + 1}} , not :P(t) = a\frac{1 + m e^{-t/\tau}}{1 + n e^{-t/\tau}} \!. Can someone explain the differences to me? If possible, I'd like to incorporate the derivation of the logistics formula that I added [[1]] into this page. --Alex S

Your formula is a special case of the presented formula with a = M,m = 0,n = A,τ = k − 1

MisterSheik 15:19, 26 June 2006 (UTC)

It's only the same curve moved by translation : P(t) = am/n + a(1-m/n)\frac{1}{1+ne^{-t/\tau}} = am/n + \frac{M}{1+Ae^{-kt}}. It's not an usual presentation but why not. My english is too bad. Sorry, I'm french - HB - 2 August 2007

[edit] Another pov

I came across this curve while examining the qustion: what does it mean to double a probability? Eventually I decided that a nice way to do it is to duble the ratio r = p/(1-p). In effect, this treats the probability as being expressed as gambling odds.

If you take the log of this value ln(p/(1-p)) and flip the x any y axes y=exp(x)/exp(x+1) you get the logistical curve. Pmurray bigpond.com 06:08, 5 June 2006 (UTC)

[edit] proposed merge with logit

The page on logit and logistic function appear to regard the exact same topic, the logistic function. As such, I propose that they be merged. The primary portion of the logit page that I think bears inclusion in this page is the comment at the top regarding logistic regression (in italics). Pdbailey 17:11, 13 June 2007 (UTC)

I am not against a merge, but let's note they are different topics, albeit each other's inverse. Logit would need its own section which would be a redirect target, right? Now that the logit page is cleaned up, it may be doable. Baccyak4H (Yak!) 17:37, 13 June 2007 (UTC)
The more I thought about it, these two articles focus quite clearly on the two functions and little ink (pixels?) would be spared merging. The fact that one is the others inverse doesn't appear to be relevant to either article's point. Pdbailey 23:45, 13 June 2007 (UTC)

[edit] no equation 1

The section on history suggests that equation 1 was first proposed by... but there is no equation 1. I'd link it if I knew which one it was. Pdbailey 17:09, 13 June 2007 (UTC)