Hi Chris,

"mixing" is not "adding".

One major point of application is the use of mean values. Consider you measure a variable that is (approximately) normal distributed. The statement you refer to say that the mean value (averages) of several of such values will also be normal distributred. The mean is nothing but the sum of these values, devided be the number of values. From the assumption (ok knowledge) of the normal distribution of the values we can thus make inference about the means we yould get from similar experiments.

A subtraction is also an addition (of a negative value). Subtractions are quite frequent when means of different conditions are compared. The differenece of the mean values if of interest. Again, if the means are normal distributed, then the sum (or difference) of such means is also normal distributed. This again allows us to infer what we can expect from such differences between means in similar experiments.

Hi Chris,

you seem to confuse sum of random variables with mixture of distributions. Consider the following example: The leg length L of males is distributed N(90, 15), the upper body length U (including head) is distributed N(85, 10). Then the full body length B of males is distributed N(175, 25). This is the sum of two normally distributed variables.

A mixture situation would be the body length of male Bm and female Bf. Let's assume that Bf ~ N(165, 20) then the combined distribution is not normally distributed. In an extreme case it would even be two-modal. (Btw. this is also an illustration of a frequent misconception of statistics beginners, who believe that the dependent variable of an ANOVA has to be normally distributed, as opposed to the residuals).

Just in case you are familiar with R, check out the code below.

--Martin

-----------------------------------------------------

## Sum of normal random variables

L

Hi Jochen and Martin,

Thank you very much for your helpful responses. They support what I intuitively felt that there was some difference between mixing the components of two unrelated sets and the conditions under which the sum of the distributions would also be normal. I am a scientist who has used some statistics, but I am not schooled in the subject. What prompted my question is that I have a chemical mixture that is lognormal in the abundances of its components. The mixture components are the result of a very large number of processes which could each be lognormal, but with different N's, loc, and scale. Do you know of a statistical reason for the mixture of these distributions to also be lognormal?

I lost you here: "the mixture components are the result of a very large number of processes which could each be lognormal". Could you specify this, best with a practical example?

Hi Chris,

under certain circumstances a mixture of log-normal distributions will have a multi-modal distribution. So, a mixture of log-normal distributions can never be log-normal in general. However, it can happen that certain realisation look similar to a log-normal. I played a little with the R code below, hoping that this approximates your problem. What I see is that some realizations look similar to a LN distributions, others don't.

CU, Martin.

## Random sample of parameters for ten mixture components

M

Yet another kind of "mixture" :)

a) Sum

b) Union

c) Mixture probability model

You must clearly define what you mean by "mixture".

ok, Jochen and Martin,

Here is the specific situation. I have tested several very complex natural mixtures and find the distributions of their analytical responses to be log-normal. These mixtures were formed by a combination of hundreds or thousands of chemical reactions in the earth or the body. Of course, there are some unknown number of components that are below the analytical detection limit in most of these mixtures. But when the number of the missing components on the low end is estimated by a trick I developed, the remaining component's distribution is lognormal to an amazingly good fit and over many orders of magnitude of response covering both sides of the peak.

I can think of a reason why the abundances of the products of a chemical reaction should be log-normal. But the mixtures I have tested contain the products of lots of reactions. Yet, somehow, when taken as a whole, their abundance distribution is log-normal. It would seem that there is some way in which the distribution of products of each separate reaction are related to the others. For example that the reaction rates of the many reactions have a log-normal distribution.

That is what I am trying to figure out. It is like taking the heights of various number of boys, girls, men, women, cats, frogs and giraffes and finding that the overall distribution of all of them is normal, except that this is log-normal rather than normal. I understand that a log-normal distribution is caused by random multiplicative variances rather than additive.

I wish I understood R, but I was put off by how much I would have to learn just to do some simple things. I have been using Minitab.

I really appreciate your interest in this problem and the help you have already given.

Chris, thank you for the clarification. However, I'm still not sure what you actually are talking about that is (or should be) log-normal distributed. Is it the number of different compounts or the concentration of any one compound? For instance, do you claim that

A) you have n different samples (e.g. some fluid). Each sample contains N[i] different chemical compounds (i = 1...n). The numbers N are log-normal distributed. or

B) you have n different samples, each containing some (more or less large number of) compounds. Taking any one of these compounds, its consentration between the samples is log-normal distributed. or

C) you have some sample (e.g. a vial with some fluid) containing with k different compounds with the concentrations C[j] (j = 1...k). The concentrations C are log-normal distributed.

Or something else?

Hi Chris,

perhaps, you should investigate in the direction of central limit theorems. The common version says that the mean of i.i.d variables converges to the normal distribution (which is therefore called an attractor). It seems that under certain circumstances the log-normal can also be such an attractor, see Wikipedia article on the CLT.

--Martin

Jochen,

It is your case c. In one example, the vial contains crude oil. There are tens of thousands of different chemical compounds in it that can be chemically distinguished and quantified. When I make a list of the analytical responses (proportional to the concentrations of the compounds) The number of compounds in each increment of ln response is log-normal. As I mentioned, I need to add some number of compounds that are below the detection limit to obtain a fit, but when that is done, the fit is excellent over more than 4 orders of magnitude of response. I have done this for complex samples of environmental, plant, and body fluids with the same result so I am working to find a reason for this pattern.

Martin,

Thanks for the tip. I will look into the CLT. Sounds promising!

Chris

I suppose you have multiplicative effects. Each molecule A can react to form x-times the number of molecules B (0 < x < infinity). If the reaction is modeled with normal probability, then the concentrations can be modeled as log-normal.

Normal distribution is a infinitely divisible distribution. In other word, one normal distribution, Z, can be divided into two or more normal distribution (X, Y, etc).

Just a thought: mayby you should think about power laws. Power laws approximate the relative sizes of cities and many other social and natural phenomena. Applications are Gibrat's law and Zipf's law. Your problem seems a bit related.

Jochen, Thank you. That is a line I have been thinking of as well. The products of a chemical reaction are related to the products of the concentrations of the reactants. So there is a multiplicative effect in the formation. The mixture is the result of many such reactions, each of which may be log-normal. Which leads me back to the question of why the combination of many log-normal distributions should be log-normal as well.

Hwan-Jin, How must X and Y be related in order to be subsets of Z?

Mats, You are right. There are many examples of these long-tailed distributions in nature and it looks like the concentrations of compounds in complex natural mixtures is another one. What I am looking for is why that should be so for the mixtures. What aspect of the formation of the mixture makes this so.

Further, and possibly related to Hwan-Jin's point, it looks like if one takes a subset of the mixture, a particular class of compounds, for instance, it is also log-normal, but with a narrower distribution. That is why I am interested in how a number of log-normal distributions could combine to form another, wider log-normal distribution.

Hi Chris,

perhaps, this paper gives you some food for thought. It speaks about general conditions and mechanisms that make complex systems reach a lognormal state. Perhaps, the "greedy agents" is a good approximation of your chemical agents?

http://www.complexity.org.au/ci/vol12/msid48/file.pdf

Martin. Thank you for this reference. It is much to the point. Chemical reactions proceed at a rate and come to an equilibrium concentration in proportion to their concentrations. The higher the concentration of one reactant, the greedier it will be producing the most product. The lower the concentration, the more "difficult" it is to obtain the resource and the lower the equilibrium concentration of product.

One of the problems in applying this distribution to complex chemical mixtures is that a significant fraction of the components are below the detection limit of current analytical techniques. For that reason, the data sets are truncated on the low concentration side. Further, there is no way to know the total number of components in the set.

If you are interested in Lognormal data sets, you may be interested in the work-around I discovered for dealing with such a situation. It is based on the fact that if one plots the log of the frequency for a normal distribution, the curve is parabolic. From the parameters of the parabola, one can derive the parameters of the Gaussian distribution. When the data are fit using these parameters, the number of missing components can be quite accurately estimated. This process is described in my paper, "Undetected Components in Natural Mixtures: How Many? What Concentrations? Do they account for Chemical Noise? What is needed to Detect Them?" in Anal. Chem., 2011, 83, 2539-2546.

Hi Chris,

your solution sounds remotely similar to a solution I one developed for discrete discovery processes [1]. There I used a home grown maximum likelihood function. Currently, I'm planning to re-develop the program with a Bayesian approach.

--Martin

[1] Schmettow, M. Controlling the usability evaluation process under varying defect visibility. BCS-HCI ’09: Proceedings of the 23rd British HCI Group Annual Conference on People and Computers: Celebrating People and Technology, British Computer Society (2009), 188–197.