INVERSE of MATRIX - Precission

[wschrabi-disabl]

Dear community,
I tried to invert a Matrix. The Strange is that if I change the digits after the comma in the input the output varies about the factor 10^6!!!

Is this a bug?
Advices are welcome.
It is the same result in MC14 as in MC11. I attached the MC11 sheet.
Walter

[PhilipOakley]

No bug at all. The "condition number" (I used conde() ) for your matrix is 4.6*10^6, so small errors will grow very very rapidly.


Philip Oakley

[wschrabi-disabl]

Thank you very much! I also got the same results with MMA. Did not know about the condition number.
Best regards

[TheodoreM.Bones]

Look, hunt, & peck - this is my uncontrolled
idea of what you wanted.

[TheodoreM.Bones]

The Fisher�s Information Matrix is a NxN matrix of N parameters of a continuous distribution. The idea does not apply to a set of discrete data. A forgery has to be made of discrete data to form a random distribution based on the mean and stdev of the data to get a continuous set. For a single parameter model, this is no problem with methods of minimizing the variance of discrete data such as maximum likelihood statistic (MLS) and minimum Chi-Square statistic are useful based on the forgery while the mean, geometric mean, harmonic mean and a summation vector may be used on discrete data.

An inspection of a histogram of discrete data on a time series will reveal the density of the data and several criteria may be used to trim outliers with low frequency outside of a probability distribution. The more stringent the trim, the fewer data fit at the mean of the entire set. The earlier Behrens stringent test requiring 8 samples minimum to arrive a 50% probability of the mean is needed.

When any or all of these criteria agree, the information is maximized

However, for data with N parameters, the Fisher Information Matrix is needed with the forgery added. Any of the well-known random probability distributions may be used for that.

[wschrabi-disabl]

Dear Theodore,
I try to interpret your statemant. But I though the Fisher INfo Matrix can be used to calculate the std-deviations to get so the tolerances for the obtained system parameters. DO I am wrong with this assumption? Thanks for your reply.

PS: As I can not compute the symbolic derivative with MAthCAd, I used Mathematica for that. Here I got a exact result for the Fishers Info mat. In MathCad I used the differnce-quotient, where I got bad results, as for the fact that the condition number of hte matrix is sooo high. THe matrix is too sensitive. Please see the MMA PDF sheet. I also included the MathCAD 14 sheet with the diff-quotient. Moreover I have attached a printout from "Excel for Chemists", where I could read that the Fisher Info Mat is used for getting the std deviations of the regr coeff.

Moreover I have just read the book Element of Information Theory (see link), wherein a lot about Fisher INformaiton is written, but not much that can be understood by normal-beeings and can be transfered into everyday-works. I think th Stat_of_NLINREG.PDF I have attached is the best fast-transfering reference for non-math-experts. Furthermore I have attached also a paper, where the std deviations of the reg coeff are cited. Finally I have attached the orig work from Johnson. (Here the formular is explained)

[ptc-1368288]

There are two "PseudoInverse" Mathcad ws.
The term is confusing and probably does not mean "PseudoInverse".

jmG

[wschrabi-disabl]

Welcom back, jmG!


Well the data is from a book, I do not know how exact the data is. Yes the blue curve is Insulin and represent like the reaction rate. But see the MC11 (from MC14), the ODE describes the exact system. Your formular is too easy.


Thanks for your reply




I found your french pseudo-invers ws. In MMA they mean the geninv() like that is written in the help. I discoverd, when I make the calculation precision in MMA to 80 then the INVERSE of the mat supplies the same results like the PseudoInvers (= geninv()). Without the setting of the high precision, MMA makes mistakes.



Walter

[ptc-1368288]

Comments:
1. The Mathematica conviviality is 0 with any other
2. Mathematica traceability is 0
Hard or impossible to track your Mathematica material.

The Mathematica "PseudoInverse" is not the Moore-Penrose [rectangular].
The Mathcad geninv(,) is the left pseudo [square]

jmG

[wschrabi-disabl]

On 3/25/2010 11:47:40 PM, jmG wrote:
>Comments:
>1. The Mathematica
>conviviality is 0 with any
>other
>2. Mathematica traceability is
>0
>Hard or impossible to track
>your Mathematica material.
>
>The Mathematica
>"PseudoInverse" is not the
>Moore-Penrose [rectangular].
>The Mathcad geninv(,) is the
>left pseudo [square]
>
Thanks, for that, now it is clear. So, is ist correct that the Moore-Penrose Inv is different from the left pseudo Inv, right?

>jmG

[wschrabi-disabl]

>"The Fisher information is a way of
>measuring the amount of information that
>an observable random variable X carries
>about an unknown parameter � upon which
>the likelihood function of �, L(�) =
>f(X;�), depends. The likelihood function
>is the joint probability of the data,
>the Xs, conditional on the value of �,
>as a function of �. Since the
>expectation of the score is zero, the
>variance is simply the second moment of
>the score, the derivative of the log of
>the likelihood function with respect to
>�. Hence the Fisher information can be
>written as: ..."
>________________________________

jmG, please see:

http://www.shopidea.com.cn/yxsz/ahkl/Teaching/Excel%20for%20Chemists/Ch12.pdf

and there the formular 12-12. Is this a special form the the fisher infromation matrix? (for discrete data?) How can this formular explained?

[TheodoreM.Bones]

That is for a canned solver collection, and is not explained. I did not load all of it.

I contend here that the eye can see my model fit is quite close and can be relied on. See FISHER2.MCD.

[ptc-1368288]

Thanks, for that, now it is clear. So, is it correct that the Moore-Penrose Inv is different from the left pseudo Inv, right?

==> YES: The two most known productive pseudoinverse are
1. Moore-Penrose, generally rectangular
2. the Souriau pseudoinverse square with main application to chemical balance.
...........................

Welcome back, jmG!


Well the data is from a book, I do not know how exact the data is. Yes the blue curve is Insulin and represents like the reaction rate. But see the MC11 (from MC14), the ODE describes the exact system. Your formula is too easy.

Thanks for your reply

==> If the ODE system you propose would be OK it would then "fit �" the data.
It does not, therefore the ODE is incorrect.
Two fits revised: the homograph, a single Lambert DE.
I accept the idea for a method to "clean" the coefficients of a fit. Whoever can proceed with the two attached fits, please do so as a new learning step. If it does reveal so useful, then add to the feature list as an option to the solvers. I have not seen that option in ORIGINLAB .

jmG

[ptc-1368288]

Walter,

Please read more how I do operate my curve fitting forge [at least for this case]. The homograph function will run on simple software like Excel, the Lambert fit needs more advanced software for the polylog. Unless an extra point near the peak is provided, at this stage it's not possible to continue more evaluation.

If you have questions, reply in the work sheet.

jmG

[TheodoreM.Bones]

A hit and run exposition. The old stand by Thiele (Pade?) will operate either of the XY columns with no tweaking. Does anybody know what to do with Walter�s continued request about using the inverse covariance to calculate something?

[ptc-1368288]

A recast of the two fits.

jmG

[ptc-1368288]

The 3 procedures [Paul W., Mathsoft Advisor, jmG]
will end these most interesting project.
However: Lambert is the winner !
proving that the Odesolve system is incompletely specified.

Walter,

Please acknowledge so that I will remove all other posting.

jmG

[wschrabi-disabl]

Thanks a lot! JmG great work.

[ptc-1368288]

On 3/24/2010 11:11:40 AM, wschrabi wrote:
>Dear community,
>I tried to invert a Matrix.
>The Strange is that if I
>change the digits after the
>comma in the input the output
>varies about the factor
>10^6!!!
>
...
>Walter
_______________________________

I don't see that !

jmG

[TheodoreM.Bones]

I searched for a 1-parameter model and obtained a fit to some data Walter gave. I think it is hard to forge a mndel with 5 parameters and compare the solution to the raw data.




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

[wschrabi-disabl]

On 3/25/2010 5:59:56 PM, bones7xx wrote:

>I searched for a 1-parameter

>model and obtained a fit to

>some data Walter gave. I think

>it is hard to forge a mndel

>with 5 parameters and compare

>the solution to the raw data.

>

>

>

>

>------------------------------

>------------------------------

>--------------------



Dear Theodore,

I have some questions in the ws - please would you mind telling me the answers? Thanks.

[TheodoreM.Bones]

Please read the answers on FISHER2.MCD

[wschrabi-disabl]

Dear Theodore,



thanks a lot for your ws. But I nees some time to study your work, as I am completely confused now. Mr Gutman has one time explained that the Fishermatrix is Derive^-1 * Derive and why do you use the N as the function. I thought that would be the derivate? I have not got it yet.





Please help: Do I misunderstand, or did I miss the covariance matrix in my worksheet? Accoring to a paper, where the calculation of the Fisher Mat is described (see PDF) I have forgotten the inverse covariance matrix in my fisher calculation. what is correct?

Theodore, I have some Qeustions in your ws. Please see the yellow text area. in Fisher2b.mcd

[ptc-1368288]

My input stops at Marlett, to be cleaned.
Interesting, will add to my collection.

jmG

[wschrabi-disabl]

Thanks jmG, that is clear. I alos found that there must be a K factor to see the curve otherwise it would be so small.

[ptc-1368288]

On 3/27/2010 2:01:21 PM, wschrabi wrote:
>Thanks jmG, that is clear. I
>alos found that there must be
>a K factor to see the curve
>otherwise it would be so
>small.
______________________________

While you are reading the last posting, fitting a function to data is an extremely difficult task. Some have taken weeks ! A linear fit is for instance the best adaptation between the coefficients of monomials [regress], the monomials have only one form or another as well as whatever model to the Mathcad hyper powerful "linfit". With term that aren't linear like exponential terms and so many other form of terms, then there is "reflexion" [not reflection] ... then for those cases of non linear fitting, no numerical solver can mathematically decide the convenient form of the fit. What it means is that eventually if there is a best and most representative fit, this fit is unlikely to be found. I have discovered that in fitting the Rayleigh distribution [though I'm not sure if it can be called distribution because I know nothing about Rayleigh]. The model to Rayleigh defeats the immediate thinking, at least any theoretical discours. This fit was done in the collab, "partially done" and it toke me several days to redo [recently].

It might be that for the "Insuline", only one point is needed [to be explored]. If so, the all classroom will be empty for a long time. This assumption makes sense as much as a DE that needs only one pair of conditions to determine the only solution to a prescribed system.

Maple sirup is now pissing, sweet.

jmG

[TheodoreM.Bones]

I posted a rather good curve fit & I see no duty in trying to load and then understand various and sundry other platforms beside MathCad. No use going around and drilling holes in every tree you can find because they might not be sugar maples. I will now leave this thread because it is too rough.

[TheodoreM.Bones]

It has been asked, �What does the Fisher Information Statistic do?�

With a few examples here, it is shown that the larger the Information on a single parameter equation, the less the variance and the better the fit, not using least squares. This is simply a means of terminating a line search for the solution of some stated condition within stated limits, etc.
Saves time, that�s all.

[wschrabi-disabl]

On 3/25/2010 4:19:09 PM, jmG wrote:
>On 3/24/2010 11:11:40 AM, wschrabi
>wrote:
>>Dear community,
>>I tried to invert a Matrix.
>>The Strange is that if I
>>change the digits after the
>>comma in the input the output
>>varies about the factor
>>10^6!!!
>>
>...
>>Walter
>_______________________________
>
>I don't see that !
>
It is only for my Matrix, where the conde(Mat)=4.6 * 10^6 !
Here the output change very very rapidly when the input has more digits.

>jmG
>
>
>

[TheodoreM.Bones]

A little more work on this with explanations. The value of the Fisher Information of a single parameter system is the slope at a point with the understanding that the lower the slope, the better the interpolated value there. A bad fit Walter made with ODEs had a Fisher Information of 5.4.