<< Chapter < Page Chapter >> Page >
e = F c + ϵ ,

where ϵ is the error in the measurements. Note that if A T is not invertible we can replace A - T with A + T , where A + T is the pseudo inverse of A T , so that F = d i a g ( A + T f ) .

Maximum likelihood estimate

With our problem now rewritten in a statistically more usable form, we can easily apply a method involving the maximum likelihood estimate, described by Calvetti and Somersalo [link] , pages 35-37. Applying this method to our rewritten problem yields the equation

F T Γ - 1 F c = F T Γ - 1 e ¯ ,


e ¯ = 1 N j = 1 N e j ,

and Γ is the covariance matrix of the random variable e . This method assumes that the error is normally distributed, which is a reasonable assumption (see "Distributions" ). Now with this formulation we can solve our problem using statistical knowledge of the problem.

Problems with maximum likelihood estimate approach

The covariance matrix, Γ , comes from the distribution of e j = A x j . Then, Γ ( e ) = A Γ ( x ) A T . In our problem where A R 16 × 14 and Γ ( x ) R 14 × 14 , Γ ( e ) is a singular matrix. In fact in any system where A R m × n where m > n , Γ ( e ) is singular. This presents a problem when solving the system

F T Γ ( e ) - 1 F c = F T Γ ( e ) - 1 e ¯ .

We can avoid the issue of invertibility of Γ ( e ) by instead of solving the problem e = F c , we solve the problem x = F ' c where

F ' = A + F , A + = pseudoinverse of A .

Our problem then is

F ' T Γ ( x ) - 1 F ' c = F ' T Γ ( x ) - 1 x ¯ ,


x ¯ = 1 N j = 1 N x j .

However we see that even if F is nonsingular F ' = A + F is singular, and our problem is not avoided.

Finding optimal force vector

Setting up our problem with the equation e = F c (see "Rewriting the Problem" ) helps us see an important realization. Because F is diagonal,

e = e 1 e 2 e n - 1 e n = F 1 , 1 c 1 F 2 , 2 c 2 F n - 1 , n - 1 c n - 1 F n , n c n .

Thus assuming F i , i 0 ,

c i = e i F i , i or, k i = 1 c i = F i , i e i .

This formulation helps us see another important realization. Namely the importance of F i , i 0 . Recall that F = diag ( A - T f ) . We have control over the f we choose, so it would be wise to choose an f so that A - T f has no zero elements. This happens only if every row of A - T is not orthogonal to f . We would like to choose an f that had as many zero elements as possible, but also is not orthogonal to any of the rows of A - T . We can formulate our search for an optimal f into the optimization problem,

min f T f + γ 1 δ + min ( A - T f ) ,

where γ and δ are adjustable parameters to get the best result. Because of the unsmooth nature of this minimization problem, this is best solved using MATLAB's fminsearch function.

In some networks we may have limitations on the f that we choose. For example in our original network

f 1 = f 6 = f 7 = f 8 = f 10 = f 13 = 0

for every f that we choose. In the case that f j = 0 , we can rewrite our optimization problem to

min f ' T f ' + γ 1 δ + min ( A ' f ' ) ,


A ' = A 1 - T A j - 1 - T A j + 1 - T A n - T , A i - T = the i th column of A - T ,


f ' = f 1 f j - 1 f j + 1 f n .

Using these methods, we found optimal f 's using MATLAB's fminsearch function on simulated data from small simple spring networks. In every network the A matrix was square and invertible. We tested networks having 2, 4, 6, and 8 springs. In each case our initial force vector was a perfectly reasonable guess that gave incomplete solutions for the spring constants. However when we inputted that initial force vector into fminsearch , the output was a force vector that made physical sense and gave accurate solutions for the spring constants.

Further research

There are two main areas of further research that directly follow the research presented in this paper. One is applying the maximum likelihood estimate approach to spring systems having a singular A matrix. We saw in "Problems with Maximum Likelihood Estimate Approach" the problems we ran into when we applied the statistical technique to a system with a singular A matrix. But this problem is expected due to the underdetermined nature of the problem (see "An Inverse Problem" ). We dealt with that problem by stacking originally, so further research could be done in using the statistical approach and stacking.

The other area for further research is in optimizing the f vector (see "Finding Optimal Force Vector" ). Further work could be done on tweaking the minimization problem so that the optimal f can be found for larger networks and networks with more restrictions on f . Also, if our system is underdetermined then even an optimal f won't give us a complete solution without stacking. So more research could be done in trying to find a set of optimal f 's that, when stacked, give the most accurate and complete solutions.


We would like to thank Dr. Steven Cox and Dr. Mark Embree for their guidance, as well as Dr. Derek Hansen and Jeffrey Hokanson for their help and support.

This Connexions module describes work conducted as part of Rice University's VIGRE program, supported by National Science Foundationgrant DMS–0739420.

See also

Pfieffer, P. Pfieffer Applied Probability. Connexions.org.

Cox, S. CAAM 335: Course Notes. www.caam.rice.edu.http://www.caam.rice.edu/ cox/main.pdf

Cox, S., Embree, M.,&Hokanson, J. CAAM 335: Lab Manual.

www.caam.rice.edu.http://www.caam.rice.edu/ caam335lab/labman.pdf

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The art of the pfug' conversation and receive update notifications?