exercise:6fff21e1d0: Difference between revisions
(Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> In his book, ''Wahrscheinlichkeitsrechnung und Statistik,''<ref group="Notes" >A. Engle, ''Wahrscheinlichkeitsrechnung und Statistik,'' vol. 2 (Stuttgart: Klett Verlag, 1976).</ref> A. Engle proposes an algorithm for finding the fixed vector for...") |
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\newcommand{\mathds}{\mathbb}</math></div> In his book, ''Wahrscheinlichkeitsrechnung und | \newcommand{\mathds}{\mathbb}</math></div> In his book, ''Wahrscheinlichkeitsrechnung und Statistik,''<ref group="Notes" >A. Engle, ''Wahrscheinlichkeitsrechnung und Statistik,'' vol. 2 (Stuttgart: Klett Verlag, 1976).</ref> A. Engle proposes an algorithm for finding the fixed vector for an ergodic Markov chain when the transition probabilities are rational numbers. Here is his algorithm: For each state <math>i</math>, let <math>a_i</math> be the least common multiple of the denominators of the non-zero entries in the <math>i</math>th row. Engle describes his algorithm in terms of moving chips around on the states---indeed, for small examples, he recommends implementing the algorithm this way. Start by putting <math>a_i</math> chips on state <math>i</math> for all <math>i</math>. Then, at each state, redistribute the <math>a_i</math> chips, sending <math>a_ip_{ij}</math> to state <math>j</math>. The number of chips at state <math>i</math> after this redistribution need not be a multiple of <math>a_i</math>. For each state <math>i</math>, add just enough chips to bring the number of chips at state <math>i</math> up to a multiple of <math>a_i</math>. Then redistribute the chips in the same manner. This process will eventually reach a point where the number of chips at each state, after the redistribution, is the same as before redistribution. At this point, we have found a fixed vector. Here is an example: | ||
Statistik,''<ref group="Notes" >A. Engle, ''Wahrscheinlichkeitsrechnung und | |||
Statistik,'' vol. 2 | |||
(Stuttgart: Klett Verlag, 1976).</ref> A. Engle proposes an | |||
algorithm for finding the fixed | |||
vector for an ergodic Markov chain when the transition probabilities are | |||
rational numbers. | |||
Here is his algorithm: For each state <math>i</math>, let <math>a_i</math> be the least common | |||
multiple of the | |||
denominators of the non-zero entries in the <math>i</math>th row. Engle describes his | |||
algorithm in | |||
terms of moving chips around on the states---indeed, for small examples, he | |||
recommends | |||
implementing the algorithm this way. Start by putting <math>a_i</math> chips on state <math>i</math> | |||
for | |||
all <math>i</math>. Then, at each state, redistribute the <math>a_i</math> chips, sending <math> | |||
state <math>j</math>. The number of chips at state <math>i</math> after this redistribution need not | |||
be a | |||
multiple of <math>a_i</math>. For each state <math>i</math>, add just enough chips to bring the | |||
number of chips at state <math>i</math> up to a multiple of <math>a_i</math>. Then redistribute the | |||
chips in the same manner. This process will eventually reach a point where the | |||
number of chips at each state, after the redistribution, is the same as before | |||
redistribution. At this point, we have found a fixed vector. Here is an | |||
example: | |||
<math display="block"> | <div class="d-flex justify-content-center"> | ||
\mat {P} = \ | {| | ||
|- | |||
| || || <span style="width:40px;text-align:center;display:inline-block;margin-left:5px;"><math>1</math></span> <span style="width:30px;;text-align:center;display:inline-block;"><math>2</math></span><span style="width:45px;;text-align:center;display:inline-block;"><math>3</math></span> | |||
|- | |||
| <math>\mat{P}=\,\,</math> | |||
</math> | | style= "padding:0px" |<math>\begin{array}{c c c c} | ||
We start with <math>\mat {a} = (4,2,4)</math>. The chips after successive redistributions | 1 \\ | ||
are | 2\\ | ||
shown in [[ | 3 | ||
\end{array}</math> || <math>\begin{pmatrix} | |||
1 & 2 & 3 \\ | |||
1/2 & 1/4 & 1/4 \\ | |||
1/2 & 0 & 1/2 \\ | |||
1/2 & 1/4 & 1/4\\ | |||
\end{pmatrix};</math> | |||
|} | |||
</div> | |||
We start with <math>\mat {a} = (4,2,4)</math>. The chips after successive redistributions are shown in [[#table 11.4 |Table]]. | |||
< | <span id="table 11.4 "/> | ||
<div class="d-flex justify-content-center"> | |||
<math> | |||
\begin{array}{lll} | \begin{array}{lll} | ||
(4 & 2\;\; & 4)\\ | (4 & 2\;\; & 4)\\ | ||
| Line 66: | Line 53: | ||
\end{array} | \end{array} | ||
</math> | </math> | ||
</div> | |||
We find that <math>\mat {a} = (20,8,12)</math> is a fixed vector. | We find that <math>\mat {a} = (20,8,12)</math> is a fixed vector. | ||
<ul><li> Write a computer program to implement this algorithm. | <ul style="list-style-type:lower-alpha"><li> Write a computer program to implement this algorithm. | ||
</li> | </li> | ||
<li> Prove that the algorithm will stop. '' Hint'': Let <math>\mat{b}</math> be a | <li> Prove that the algorithm will stop. '' Hint'': Let <math>\mat{b}</math> be a | ||
Latest revision as of 22:36, 17 June 2024
In his book, Wahrscheinlichkeitsrechnung und Statistik,[Notes 1] A. Engle proposes an algorithm for finding the fixed vector for an ergodic Markov chain when the transition probabilities are rational numbers. Here is his algorithm: For each state [math]i[/math], let [math]a_i[/math] be the least common multiple of the denominators of the non-zero entries in the [math]i[/math]th row. Engle describes his algorithm in terms of moving chips around on the states---indeed, for small examples, he recommends implementing the algorithm this way. Start by putting [math]a_i[/math] chips on state [math]i[/math] for all [math]i[/math]. Then, at each state, redistribute the [math]a_i[/math] chips, sending [math]a_ip_{ij}[/math] to state [math]j[/math]. The number of chips at state [math]i[/math] after this redistribution need not be a multiple of [math]a_i[/math]. For each state [math]i[/math], add just enough chips to bring the number of chips at state [math]i[/math] up to a multiple of [math]a_i[/math]. Then redistribute the chips in the same manner. This process will eventually reach a point where the number of chips at each state, after the redistribution, is the same as before redistribution. At this point, we have found a fixed vector. Here is an example:
| [math]1[/math] [math]2[/math][math]3[/math] | ||
| [math]\mat{P}=\,\,[/math] | [math]\begin{array}{c c c c} 1 \\ 2\\ 3 \end{array}[/math] | [math]\begin{pmatrix} 1 & 2 & 3 \\ 1/2 & 1/4 & 1/4 \\ 1/2 & 0 & 1/2 \\ 1/2 & 1/4 & 1/4\\ \end{pmatrix};[/math] |
We start with [math]\mat {a} = (4,2,4)[/math]. The chips after successive redistributions are shown in Table.
[math] \begin{array}{lll} (4 & 2\;\; & 4)\\ (5 & 2 & 3)\\ (8 & 2 & 4)\\ (7 & 3 & 4)\\ (8 & 4 & 4)\\ (8 & 3 & 5)\\ (8 & 4 & 8)\\ (10 & 4 & 6)\\ (12 & 4 & 8)\\ (12 & 5 & 7)\\ (12 & 6 & 8)\\ (13 & 5 & 8)\\ (16 & 6 & 8)\\ (15 & 6 & 9)\\ (16 & 6 & 12)\\ (17 & 7 & 10)\\ (20 & 8 & 12)\\ (20 & 8 & 12)\ . \end{array} [/math]
We find that [math]\mat {a} = (20,8,12)[/math] is a fixed vector.
- Write a computer program to implement this algorithm.
- Prove that the algorithm will stop. Hint: Let [math]\mat{b}[/math] be a vector with integer components that is a fixed vector for [math]\mat{P}[/math] and such that each coordinate of the starting vector [math]\mat{a}[/math] is less than or equal to the corresponding component of [math]\mat{b}[/math]. Show that, in the iteration, the components of the vectors are always increasing, and always less than or equal to the corresponding component of [math]\mat{b}[/math].
Notes