The perturbation bound for the Perron vector of a transition probability tensor. Issue 6 (28th May 2013)

• Record Type: Journal Article
• Authors: Li, Wen; Cui, Lu‐Bin; Ng, Michael K.; Lim, Lek‐Heng; Ng, Michael K.; Qi, Liqun
• Is Part Of: Numerical linear algebra with applications. Volume 20:Issue 6(2013:Nov.)
• Page Start: 985
• Page End: 1000
• Publication Date: 2013-05-28
• Subjects: Algebras, Linear -- Periodicals; 512.5
• DOI: 10.1002/nla.1886
• Languages: English
• ISSNs: 1070-5325

Abstract

<abstract abstract-type="main" id="nla1886-abs-0001"> <title>SUMMARY</title> <p id="nla1886-para-0001">In this paper, we study the perturbation bound for the Perron vector of an <italic>m</italic>th‐order <italic>n</italic>‐dimensional transition probability tensor <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3xcn8kqk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10705325:media:nla1886:nla1886-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">P</mml:mi><mml:mo class="MathClass-rel">=</mml:mo><mml:mrow><mml:mo class="MathClass-open">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo class="MathClass-punc">, </mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo class="MathClass-punc">, </mml:mo><mml:mo class="MathClass-op">…</mml:mo><mml:mo class="MathClass-punc">, </mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub></mml:mrow><mml:mo class="MathClass-close">)</mml:mo></mml:mrow></mml:math></alternatives> with <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3xcn8kp1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10705325:media:nla1886:nla1886-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo class="MathClass-punc">, </mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo class="MathClass-punc">, </mml:mo><mml:mo class="MathClass-op">…</mml:mo><mml:mo class="MathClass-punc">, </mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mi>⩾</mml:mi><mml:mn>0</mml:mn></mml:math></alternatives> and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3xcn8k7v" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10705325:media:nla1886:nla1886-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mo class="MathClass-op">∑</mml:mo></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo class="MathClass-rel">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo class="MathClass-punc">, </mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo class="MathClass-punc">, </mml:mo><mml:mo class="MathClass-op">…</mml:mo><mml:mo class="MathClass-punc">, </mml:mo><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msub><mml:mo class="MathClass-rel">=</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives>. The Perron vector <bold>x</bold> associated to the largest <italic>Z</italic>‐eigenvalue 1 of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3xcn8k8d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10705325:media:nla1886:nla1886-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">P</mml:mi></mml:math></alternatives>, satisfies <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3xcn8k9z" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10705325:media:nla1886:nla1886-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">P</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="bold-italic">x</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi><mml:mo class="MathClass-bin">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo class="MathClass-rel">=</mml:mo><mml:mi mathvariant="bold-italic">x</mml:mi></mml:math></alternatives> where the entries <italic>x</italic><sub><italic>i</italic></sub> of <bold>x</bold> are non‐negative and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3xcn8kbh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10705325:media:nla1886:nla1886-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mo class="MathClass-op">∑</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo class="MathClass-rel">=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msubsup><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub><mml:mo class="MathClass-rel">=</mml:mo><mml:mn>1</mml:mn></mml:math></alternatives>. The main contribution of this paper is to show that when <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3xcn8kc2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10705325:media:nla1886:nla1886-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">P</mml:mi></mml:math></alternatives> is perturbed to an another transition probability tensor <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3xcn8kdm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10705325:media:nla1886:nla1886-math-0008" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">P</mml:mi></mml:mrow><mml:mo class="MathClass-op">̃</mml:mo></mml:mover></mml:math></alternatives> by <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3xcn8kf5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10705325:media:nla1886:nla1886-math-0009" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi><mml:mi mathvariant="script">P</mml:mi></mml:math></alternatives>, the 1‐norm error between <bold>x</bold> and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3xcn8kh8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10705325:media:nla1886:nla1886-math-0010" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="bold-italic">x</mml:mi></mml:mrow><mml:mo class="MathClass-op">̃</mml:mo></mml:mover></mml:math></alternatives> is bounded by <italic>m</italic>, <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3xcn8kgq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10705325:media:nla1886:nla1886-math-0011" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi><mml:mi mathvariant="script">P</mml:mi></mml:math></alternatives>, and the computable quantity related to the uniqueness condition for the Perron vector <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3xcn8kkc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10705325:media:nla1886:nla1886-math-0012" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="bold-italic">x</mml:mi></mml:mrow><mml:mo class="MathClass-op">̃</mml:mo></mml:mover></mml:math></alternatives> of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3xcn8kjt" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="block" altimg="urn:x-wiley:10705325:media:nla1886:nla1886-math-0013" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="script">P</mml:mi></mml:mrow><mml:mo class="MathClass-op">̃</mml:mo></mml:mover></mml:math></alternatives>. Based on our analysis, we can derive a new perturbation bound for the Perron vector of a transition probability matrix which refers to the case of <italic>m</italic> = 2. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis. Copyright © 2013 John Wiley & Sons, Ltd.</p> </abstract>