The memory center

Przemyslaw Spurek, Jacek Tabor

Abstract: As we know, the length of binary code of a point x∈R<math><mrow is="true"><mi is="true">x</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">R</mi></mrow></math> (with accuracy h > 0) is approximately mh(x)≈log2max1,xh<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">m</mi></mrow><mrow is="true"><mi is="true">h</mi></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">x</mi><mo stretchy="false" is="true">)</mo><mo is="true">≈</mo><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">log</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mrow is="true"><mfenced close=")" open="(" is="true"><mrow is="true"><mi mathvariant="normal" is="true">max</mi><mrow is="true"><mfenced close="}" open="{" is="true"><mrow is="true"><mn is="true">1</mn><mtext is="true">,</mtext><mrow is="true"><mfenced close="|" open="|" is="true"><mrow is="true"><mfrac is="true"><mrow is="true"><mi is="true">x</mi></mrow><mrow is="true"><mi is="true">h</mi></mrow></mfrac></mrow></mfenced></mrow></mrow></mfenced></mrow></mrow></mfenced></mrow></mrow></math>. We will consider the problem where we should translate the origin a of the coordinate system so that the mean amount of bits needed to code a randomly chosen element from a realization of a random variable X is minimal. In other words, we want to find a∈R<math><mrow is="true"><mi is="true">a</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">R</mi></mrow></math> such thatR∋a→E(mh(X-a))<math><mi mathvariant="double-struck" is="true">R</mi><mi is="true">∋</mi><mi is="true">a</mi><mo is="true">→</mo><mi is="true">E</mi><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">m</mi></mrow><mrow is="true"><mi is="true">h</mi></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">X</mi><mo is="true">-</mo><mi is="true">a</mi><mo stretchy="false" is="true">)</mo><mo stretchy="false" is="true">)</mo></math>attains minimum.We show that under reasonable assumptions the choice of a does not depend on h asymptotically. Consequently, we reduce the problem to finding the minimum of functionR∋a→∫Rln(|x-a|)f(x)dx,<math><mi mathvariant="double-struck" is="true">R</mi><mi is="true">∋</mi><mi is="true">a</mi><mo is="true">→</mo><msub is="true"><mo is="true">∫</mo><mrow is="true"><mi mathvariant="double-struck" is="true">R</mi></mrow></msub><mi mathvariant="normal" is="true">ln</mi><mo stretchy="false" is="true">(</mo><mo stretchy="false" is="true">|</mo><mi is="true">x</mi><mo is="true">-</mo><mi is="true">a</mi><mo stretchy="false" is="true">|</mo><mo stretchy="false" is="true">)</mo><mi is="true">f</mi><mo stretchy="false" is="true">(</mo><mi is="true">x</mi><mo stretchy="false" is="true">)</mo><mi mathvariant="italic" is="true">dx</mi><mtext is="true">,</mtext></math>where f is the density distribution of the random variable X. Moreover, we provide constructive approach for determining a.

External IDs:dblp:journals/isci/SpurekT13