Smoluchowski Model

[TheodoreM.Bones]

When studying raw data from a time series stochastic process, it is difficult to determine if matters are getting better or worse with time. Statistics gathering the dats about the MLE, minimum chi-square, or mean gradually approach those values as outliers are trimmed and nothing is gained. The skewness parameter is not always revealing.Here the one-dimensional
Smoluchowski model is used to arrange the data as if Brownian movements were affecting the process and thud afford a conclusion.

[TheodoreM.Bones]

Expanding this thread, some arbitrary raw data from a stochastic time series is studied for a conclusion that Smoluchowski�s Theorem (Wikepedia �Brownian Movements,� aptly shows the activities below and above the mean. Then the Theorem is proved by reverse study of the data, with slight errors occurring with mixed variables. Essential statistics is not used. The worksheet is wordy.

[TheodoreM.Bones]

A third revision of the original worksheet to incorporate corrections of several errors and separate the two experimental verifications of Smoluchowslki�s lemma. Excellent and clear results are obtained. Other raw data may be loaded, like a long file made with white noise integers about an integer mean. A maximum of about 120 points is set by the MathCad limits on the factorial (Gamma) function. There is some doubt about the normalizing factor 2^n in his lemma, which may be changed to 2^(n-1) at times.

[ptc-1368288]

In the work sheet, only the text above.
All other material = total scrap, nothing to see.

jmG

[TheodoreM.Bones]

Did you mean only the Summary and the worksheet itself was only crap? This is not a curve fit but a process of preparing data and then letting the lemma prove itself. It won't prove if the equilibru point is not the mean of the data. This is Einstein stuff, dated about 1909 AD. I don't hsve access to Smoluchowsli's paper. I had to experiment.

[TheodoreM.Bones]

My kind of due diligence takes longer than a minute long computer check. I was skeptical of a university math professor's advertised intention of modeling a medical cell growth diffusion provess in several dimension space. The ad was written by the university's footbsll team.

[ptc-1368288]

I was able to get M back and the other regions not affected and paste all in a blank sheet. Will look at it later, thanks Theodore.

jmG

[TheodoreM.Bones]

No amount of Minerr or Find will enable the Smoluchowski lemma to partition a raw data file. Even very close initial guesses are needed to run Minerr on a factorial fraction.

[TheodoreM.Bones]

The Smoluchowski lemma for diffusion in one dimension was studied at some length and it was found that it could be programmed to give the number of events equal and below the mean of the entire data and those above the mean. The only parameter needed is the length of the file (data), which may be any kind in any order. The partition shows the possible data that meet the lemma law. The two halves cannot become
equal. Roundoff errors may affect the results one or two events. The proof is an actual eye witness count of the data entered.

[TheodoreM.Bones]

The preceeding worksheet illustrated the response to "whie noise" data, which is not a real stochastic time series. Here is actual data obtsined from real observastions at random times in series. The "white noise" data could never get to a fixed mean.

[TheodoreM.Bones]

The preceeding worksheet illustrated the response to "white noise" data, which is not a real stochastic time series. Here is actual data obtsined from real observastions at random times in series. The "white noise" data could never get to a fixed mean.

[TheodoreM.Bones]

Much pharmo -kinetics work has been done recently in several dimensional space, and I have been anxious to prove Smoluchowski�s Theorem numerically in one dimension only with real stochastic (random) data so that it can be explained that openness data is not a random time series process by using the Theorem on it. Using white noise about a mean does not quality as stochastic. Verbatim data from any sources is not always such either. The idea of Brownian Movements coming from scattered or selected directions can be ignored by calculating the half-life of the incipient. That is not done here, just proving the normalized binomial coefficient (n/k) verifies whether the data is really random or not. Smoluchowski had a good enough reputation. good as Einstein�s.

[TheodoreM.Bones]

I may have been putting the cart before the horse.

Here the Smoluchowski lemma for one-dimensional diffusion (n/k) is given first to predict the partition of a file of length (L) and then a rack of random functions is given from which the file is loaded and the lemma is verified by actual count of the load and its partition. Due to the use of roundoff, there may be slight errors on comparing the prediction with actual count of the loaded random data. Then there may be skepticism whether the random data loaded is stochastic or not.

[TheodoreM.Bones]

Is the Smoluchowski lemma for one-dimensional diffusion correct or not? What kind of random data may be used to find out? Thiele himself wrote his first interpolation theorem as part of particle motion studies. How do theories that the lemma is actually an exponential function appear? After all, the lemma is a rational fraction of Gamma functions, which Smoluchowski
discovered was the binomial coefficient (n/k). After this, it is easier to use the posting of RANDOM10.MCD (qv) on this thread.

[ptc-1368288]

>After this, it is easier to use the posting of RANDOM10.MCD (qv) on this thread.<<br> ________________________

RANDOM10.MCD added, work sheet organised a bit.
Saved in My Stat file.

Thanks Theodore.

jmG