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Asymptotic Property of Universal Lossless Coding for Independent Piecewise Identically Distributed Sources Tota Suko Toshiyasu Matsushima Shigeichi Hirasawa Waseda University, JAPAN
Introduction ,[object Object],[object Object],[object Object],We prove that the Bayes codes minimizes the mean redundancy asymptotically. result
[object Object],Source coding Sources encoder decoder channel 01001011 ・・・ data sequence If  is known ⇒ We can achieve the theoretical limitation of compression  using Arithmetic codes or Huffman codes. And, code length is given by  . code word
[object Object],[object Object],[object Object],[object Object],[object Object],Source coding coding probability code length is given by
Independent piecewise identically distributed (i.p.i.d.) sources ,[object Object],[object Object],[object Object]
Definition of i.p.i.d. sources … 1st transition 2nd transition ( |T m | -1)th transition i.i.d. i.i.d. i.i.d. i.i.d. i.i.d. parameters transition at time  . : a source symbol : a data sequence : a transition pattern : a set of transition times
Definition of i.p.i.d. sources the probability of a sequence … 1st transition 2nd transition ( |T m | -1)th transition i.i.d. i.i.d. i.i.d. i.i.d. i.i.d. parameters transition at time  .
Definition of i.p.i.d. sources j  th transition j +1 th transition length   l j   length  l j +1   ・・・ ・・・ : the number of stationary segments ( number of transitions + 1 ) : the length of a stationary segment stationary segments
Previous researchs  ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],: the number of stationary segments ( number of transitions + 1 ) : the length of a stationary segment Previous researchs propose some codes, and evaluate the proposed codes.
Previous researches Table. Restrictions for transition patterns We evaluate the Bayes codes under this condition. l j =O (log  N ) this study      : known      :unknown [Shamir99] [Shamir00] [Merhav93] [Willems96]      :unknown     is constant
Criteria of universal source codes ,[object Object],[object Object],[object Object],[object Object],: true
Restriction for transition patterns ,[object Object],[object Object],[object Object],[object Object],[object Object],ex)  Parameter transitions are occurred by Bernoulli trials.
the Bayes Codes ,[object Object],[object Object],The Bayes coding probability minimizes the Bayes redundancy for any  N .
Assumption of prior distribution ,[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],Class of codes
main result : the mean redundancy of the Bayes codes  ,[object Object],[object Object],[object Object],the Bayes Codes minimize the  mean redundancy  asymptotically.   the mean redundancy of the Bayes codes
[object Object],[object Object],[object Object],[object Object],Outline of the proof
Outline of the proof ,[object Object],[object Object],⇒ We prove that the mean redundancy doesn’t depend on  asymptotically. the mean redundncy the Bayes redundancy
Outline of the proof B A
Outline of the proof Part A From Assumption 1 and law of large numbers, It does not depend on  .
Outline of the proof Similarly,  we have It does not depend on  . Part B Therefor, the Teorem 1 is proved.
Conclusion ,[object Object],[object Object],[object Object],[object Object]

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ma92008id393

  • 1. Asymptotic Property of Universal Lossless Coding for Independent Piecewise Identically Distributed Sources Tota Suko Toshiyasu Matsushima Shigeichi Hirasawa Waseda University, JAPAN
  • 2.
  • 3.
  • 4.
  • 5.
  • 6. Definition of i.p.i.d. sources … 1st transition 2nd transition ( |T m | -1)th transition i.i.d. i.i.d. i.i.d. i.i.d. i.i.d. parameters transition at time . : a source symbol : a data sequence : a transition pattern : a set of transition times
  • 7. Definition of i.p.i.d. sources the probability of a sequence … 1st transition 2nd transition ( |T m | -1)th transition i.i.d. i.i.d. i.i.d. i.i.d. i.i.d. parameters transition at time .
  • 8. Definition of i.p.i.d. sources j th transition j +1 th transition length l j length l j +1 ・・・ ・・・ : the number of stationary segments ( number of transitions + 1 ) : the length of a stationary segment stationary segments
  • 9.
  • 10. Previous researches Table. Restrictions for transition patterns We evaluate the Bayes codes under this condition. l j =O (log N ) this study      : known      :unknown [Shamir99] [Shamir00] [Merhav93] [Willems96]      :unknown     is constant
  • 11.
  • 12.
  • 13.
  • 14.
  • 15.
  • 16.
  • 17.
  • 18.
  • 19. Outline of the proof B A
  • 20. Outline of the proof Part A From Assumption 1 and law of large numbers, It does not depend on .
  • 21. Outline of the proof Similarly, we have It does not depend on . Part B Therefor, the Teorem 1 is proved.
  • 22.