0.6 Compression of nonparametric sources  (Page 3/3)

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Because of the AEP  [link] ,

$Pr\left({x}_{1}^{l},:,\left|\frac{-log\left(Pr\left({x}_{1}^{l}\right)\right)}{{l}_{0}},-,H\right|,>,ϵ\right)\to 0.$

Therefore, for a typical input $Pr\left({x}_{1}^{l}\right)>{2}^{-{l}_{0}\left(H+ϵ\right)}$ .

Recall that the interval length is ${l}_{0}=\frac{log\left({n}_{w}\right)}{H+2ϵ},$ and so the probability that an interval cannot be found in the history is

$\frac{{2}^{-{l}_{0}\left(H+2ϵ\right)}}{{2}^{-{l}_{0}\left(H+ϵ\right)}}={2}^{-{l}_{0}ϵ}\to 0.$

For a long enough interval, this probability goes to zero.

Redundancy of parsing schemes

There are many Ziv-Lempel style parsing algorithms  [link] , [link] , [link] , and each of the variantshas different details, but the key idea is to find the longest match in a window of length ${n}_{w}$ . The length of the match is $L$ , where we remind the reader that $L\approx \frac{log\left({n}_{w}\right)}{H}$ .

Now, encoding $L$ requires $log\left({n}_{w}\right)$ bits, and so the per-symbol compression ratio is $\frac{log\left({n}_{w}\right)}{L}$ , which in the limit of large ${n}_{w}$ approaches the entropy rate $H$ .

However, the encoding of $L$ must also desribe its length, and often the symbol that follows the match. These require length $log\left(L\right)\approx log\left(log\left({n}_{w}\right)\right)$ , and the normalized (per-symbol) cost is

$\frac{log\left(log\left({n}_{w}\right)\right)}{L}=O\left(\frac{log\left(log\left({n}_{w}\right)\right)}{log\left({n}_{w}\right)}\right).$

Therefore, the redundancy of Ziv-Lempel style compression algorithms is proportional to $O\left(\frac{log\left(log\left(n\right)\right)}{log\left(n\right)}\right)$ , which is much greater than the $O\left(\frac{log\left(n\right)}{n}\right)$ that we have seen for parametric sources. The fundamental reason why the redundancy is greater is that the class of non-parametric sources is much richer. Detailedredundancy analyses appear in a series of papers by Savari (c.f.  [link] ).

Parsing for lossy compression

The parsing schemes that we have seen can also be adapted to lossy compression. Let us describe several approaches along these lines.

Fixed length: The first scheme, due to Gupta et al.  [link] , constructs a codebook of size $\approx {2}^{LR\left(D\right)}$ codewords, where $L$ is the length of the phrase being matched and $R\left(D\right)$ is the rate distortion function. The algorithm cannot search for perfect matches of the phrase, because this is lossy compression. Instead, it seeks the codeword that matches our input phrase most closely. It turns out that for large $L$ the expected distortion of the lossy match will be approximately $D$ per symbol.

Variable length: Another approach, due to Gioran and Kontoyiannis  [link] , constructs a single long database string, and searches for the longest match whose distortion w.r.t. the input is approximately $D$ ; the location and length of the approximate match are encoded. Seeing that the database is of length $\approx {2}^{LR}$ , encoding the location requires $\approx LR$ bits, and the $D$ -match (a match with distortion $D$ w.r.t. the input string) is typically of length $\approx L$ , giving a per-symbol rate of $\approx R\left(D\right)$ bits.

An advantage of the latter scheme by Gioran and Kontoyiannis  [link] is reduced memory use. The database is a string of length $\approx {2}^{LR\left(D\right)}$ , instead of a codebook comprised of $\approx {2}^{LR\left(D\right)}$ codewords, each of length $L$ . On the other hand, the Gupta et al. algorithm  [link] has better $RD$ performance, because it does not need to spend $\approx log\left(L\right)$ bits per phrase to describe its length. An improved algorithm, dubbed the hybrid algorithm by Gioran and Kontoyiannis, constructs a single database and performs fixed length coding for the best match of length $L$ in the database. Therefore, it combines the memory usage of a single database approach with the $RD$ performance of fixed length coding.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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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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