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Semi-predictive approach

Recall that a context tree source is similar to a Markov source, where the number of states is greatly reduced. Let T be the set of leaves of a context tree source, then the redundancy is

r | T | ( r - 1 ) 2 log n | T | + O ( 1 ) ,

where | T | is the number of leaves, and we have log n | T | instead of log ( n ) , because each state generated n | T | symbols, on average. In contrast, the redundancy for a Markov representation of the tree source T is much larger. Therefore, tree sources are greatly preferable in practice, they offer a significant reductionin redundancy.

How can we compress universally over the parametric class of tree sources? Suppose that we knew T , that is we knew the set of leaves. Then we could process x sequentially, where for each x i we can determine what state its context is in, that is the unique suffix of x 1 i - 1 that belongs to the set of leaf labels in T . Having determined that we are in some state s , Pr ( x i = 0 | s , x 1 i - 1 ) can be computed by examining all previous times that we were in state s and computing the probability with the Krichevsky-Trofimov approach based on the number of times that the following symbol(after s ) was 0 or 1. In fact, we can store symbol counts n x ( s , 0 ) and n x ( s , 1 ) for all s T , update them sequentially as we process x , and compute Pr ( x i = 0 | s , x 1 i - 1 ) efficiently. (The actual translation to bits is performed with an arithmetic encoder.)

While promising, this approach above requires to know T . How do we compute the optimal T * from the data?

Tree pruning in the semi-predictive approach.
Tree pruning in the semi-predictive approach.

Semi-predictive coding : The semi-predictive approach to encoding for context tree sources  [link] is to scan the data twice, where in the first scan we estimate T * and in the second scan we encode x from T * , as described above. Let us describe a procedure for computing the optimal T * among tree sources whose depth is bounded by D . This procedure is visualized in [link] . As suggested above, we count n x ( s , a ) , the number of times that each possible symbol appeared in context s , for all s α D , a α . Having computed all the symbol counts, we process the depth- D tree in a bottom-top fashion, from the leaves to the root, where for each internal node s of the tree (that is, s α d where d < D ), we track T s * , the optimal tree structure rooted at s to encode symbols whose context ends with s , and MDL ( s ) the minimum description lengths (MDL) required for encoding these symbols.

Without loss of generality, consider the simple case of a binary alphabet α = { 0 , 1 } . When processing s we have already computed the symbol counts n x ( 0 s , 0 ) and n x ( 0 s , 1 ) , n x ( 1 s , 0 ) , n x ( 1 s , 1 ) , the optimal trees T 0 s * and T 1 s * , and the minimum description lengths (MDL) MDL ( 0 s ) and MDL ( 1 S ) . We have two options for state s .

  1. Keep T 0 S * and T 1 S * . The coding length required to do so is MDL ( 0 S ) + MDL ( 1 S ) + 1 , where the extra bit is spent to describe the structure of the maximizing tree.
  2. Merge both states (this is also called tree pruning ). The symbol counts will be n x ( s , α ) = n x ( 0 s , α ) + n x ( 1 s , α ) , α { 0 , 1 } , and the coding length will be
    KT ( n x ( s , 0 ) , n x ( s , 1 ) ) + 1 ,
    where KT ( · , · ) is the Krichevsky-Trofimov length  [link] , and we again included an extra bit for the structure of the tree.

Questions & Answers

what is the stm
Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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LITNING
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Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
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Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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Maciej
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Anassong
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Lily
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
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Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Universal algorithms in signal processing and communications. OpenStax CNX. May 16, 2013 Download for free at http://cnx.org/content/col11524/1.1
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