Commit f1fde8ee authored by Jim Hefferon's avatar Jim Hefferon

edits of markov

parent e7f1023e
This diff is collapsed.
......@@ -19574,8 +19574,7 @@
produces the identity matrix
so there is no need for column-swapping operations
to end with a partial-identity.
\partsitem The reduction is expressed in matrix multiplication
as
\partsitem In matrix multiplication the reduction is
\begin{equation*}
\begin{mat}[r]
1 &-1 \\
......@@ -19592,7 +19591,7 @@
H
=I
\end{equation*}
(note that composition of the Gaussian operations is performed
(note that composition of the Gaussian operations is
from right to left).
\partsitem Taking inverses
\begin{equation*}
......@@ -19823,7 +19822,7 @@
\mapsto
\colvec{x_1}
\end{equation*}
is indeed expressible as a composition of swaps\Dash as zero swaps.
is expressible as a composition of swaps\Dash as zero swaps.
For the inductive step we assume that the map induced by
any permutation of fewer than
$n$ numbers can be expressed with swaps only, and we consider the map
......@@ -20066,14 +20065,14 @@ octave:6> B100*[0;1;0;0;0;0]
\end{equation*}
We get this transition matrix.
\begin{equation*}
\begin{pmatrix}
\begin{mat}
1/6 &0 &0 &0 &0 &0 \\
1/6 &2/6 &0 &0 &0 &0 \\
1/6 &1/6 &3/6 &0 &0 &0 \\
1/6 &1/6 &1/6 &4/6 &0 &0 \\
1/6 &1/6 &1/6 &1/6 &5/6 &0 \\
1/6 &1/6 &1/6 &1/6 &1/6 &6/6
\end{pmatrix}
\end{mat}
\end{equation*}
\partsitem This is the Octave session,
with outputs edited out and condensed into the table at the end.
......@@ -20158,9 +20157,9 @@ octave:7> v5=F*v4
have little influence.
That is, while a company may move or stay because of where it is,
it is unlikely to move or stay because of where it was.
\partsitem This Octave session has been edited, with the outputs
\partsitem This is the Octave session, slightly edited, with the outputs
put together in a table at the end.
\begin{computercode}
\begin{lstlisting}
octave:1> M=[.787,0,0,.111,.102;
> 0,.966,.034,0,0;
> 0,.063,.937,0,0;
......@@ -20177,8 +20176,8 @@ octave:3> v1=M*v0
octave:4> v2=M*v1
octave:5> v3=M*v2
octave:6> v4=M*v3
\end{computercode}
is summarized in this table.
\end{lstlisting}
This table summarizes.
\begin{center}
\begin{tabular}{c|cccc}
$\vec{p}_0$ &$\vec{p}_1$ &$\vec{p}_2$
......@@ -20222,13 +20221,13 @@ octave:6> v4=M*v3
\end{center}
\partsitem This is a continuation of the Octave session from
the prior item.
\begin{computercode}
\begin{lstlisting}
octave:7> p0=[.0000;.6522;.3478;.0000;.0000]
octave:8> p1=M*p0
octave:9> p2=M*p1
octave:10> p3=M*p2
octave:11> p4=M*p3
\end{computercode}
\end{lstlisting}
This summarizes the output.
\begin{center}
\begin{tabular}{c|cccc}
......@@ -20272,7 +20271,7 @@ octave:11> p4=M*p3
\end{tabular}
\end{center}
\partsitem This is more of the same Octave session.
\begin{computercode}
\begin{lstlisting}
octave:12> M50=M**50
M50 =
0.03992 0.33666 0.20318 0.02198 0.37332
......@@ -20294,7 +20293,7 @@ p51 =
0.54442
0.33091
0.29076
\end{computercode}
\end{lstlisting}
This is close to a steady state.
\end{exparts}
......@@ -20332,7 +20331,7 @@ p51 =
\\ s_{A}(n+1) \\ s_{B}(n+1)}
\end{equation*}
\partsitem This is the Octave code, with the output removed.
\begin{computercode}
\begin{lstlisting}
octave:1> T=[.5,.25,.25,0,0;
> .25,.5,0,0,0;
> .25,0,.5,0,0;
......@@ -20350,7 +20349,7 @@ octave:4> p2=T*p1
octave:5> p3=T*p2
octave:6> p4=T*p3
octave:7> p5=T*p4
\end{computercode}
\end{lstlisting}
Here is the output.
The probability of ending at $s_A$ is about $0.23$.
\begin{equation*}
......@@ -20410,7 +20409,7 @@ octave:7> p5=T*p4
\end{array}
\end{equation*}
\partsitem With this file as \texttt{learn.m}
\begin{computercode}
\begin{lstlisting}
# Octave script file for learning model.
function w = learn(p)
T = [1-2*p,p, p, 0, 0;
......@@ -20422,11 +20421,11 @@ function w = learn(p)
p5 = T5*[1;0;0;0;0];
w = p5(4);
endfunction
\end{computercode}
\end{lstlisting}
issuing the command \texttt{octave:1> learn(.20)} yields
\texttt{ans = 0.17664}.
\partsitem This Octave session
\begin{computercode}
\begin{lstlisting}
octave:1> x=(.01:.01:.50)';
octave:2> y=(.01:.01:.50)';
octave:3> for i=.01:.01:.50
......@@ -20434,7 +20433,7 @@ octave:3> for i=.01:.01:.50
> endfor
octave:4> z=[x, y];
octave:5> gplot z
\end{computercode}
\end{lstlisting}
yields this plot.
There is no threshold value \Dash no probability above which the
curve rises sharply.
......@@ -20516,7 +20515,7 @@ octave:5> gplot z
\begin{ans}{6}
These are the $p=.55$ vectors,
\begin{center}\small
\begin{tabular}{@{}rl|lllllll@{}}
\begin{tabular}{@{}rc|ccccccc@{}}
&$n=0$ &$n=1$ &$n=2$ &$n=3$ &$n=4$ &$n=5$ &$n=6$ &$n=7$ \\
\hline
\begin{tabular}{@{}c@{}}
......@@ -20757,7 +20756,7 @@ octave:5> gplot z
\end{center}
and these are the $p=.60$ vectors.
\begin{center}\small
\begin{tabular}{@{}rl|lllllll@{}}
\begin{tabular}{@{}rc|ccccccc@{}}
&$n=0$ &$n=1$ &$n=2$ &$n=3$ &$n=4$ &$n=5$ &$n=6$ &$n=7$ \\
\hline
\begin{tabular}{@{}c@{}}
......@@ -20997,9 +20996,8 @@ octave:5> gplot z
\end{tabular}
\end{center}
\begin{exparts}
\partsitem The script from the computer code section can be easily
adapted.
\begin{computercode}
\partsitem We can adapt the script from the end of this Topic.
\begin{lstlisting}
# Octave script file to compute chance of World Series outcomes.
function w = markov(p,v)
q = 1-p;
......@@ -21030,7 +21028,7 @@ function w = markov(p,v)
v7 = (A**7) * v;
w = v7(11)+v7(16)+v7(20)+v7(23)
endfunction
\end{computercode}
\end{lstlisting}
Using this script, we get that when the American League has a
$p=0.55$ probability of winning each game then their probability
of winning the first-to-win-four series is $0.60829$.
......@@ -21038,7 +21036,7 @@ endfunction
then their probability of winning the series is
$0.71021$.
\partsitem This Octave session
\begin{computercode}
\begin{lstlisting}
octave:1> v0=[1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0];
octave:2> x=(.01:.01:.99)';
octave:3> y=(.01:.01:.99)';
......@@ -21047,7 +21045,7 @@ octave:4> for i=.01:.01:.99
> endfor
octave:5> z=[x, y];
octave:6> gplot z
\end{computercode}
\end{lstlisting}
yields this graph.
By eye we judge that if $p>0.7$ then the team is close to assurred
of the series.
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......@@ -117,6 +117,19 @@
\newcolumntype{.}{D{.}{.}{#2}}\begin{array}{.}}{%
\end{array}}
% Matrix and vector, with numbers centered on decimal point
% Usage: \begin{dmat}{D{.}{.}{1}D{.}{.}{3}} 0 &.123 \\ .2 &.456 \end{dmat}
% (in the D{.}{.}{number} that is the number of decimal places)
\newlength{\dmatcolsep}\setlength{\dmatcolsep}{5pt}
\newenvironment{dmat}[2][\dmatcolsep]{%
\setlength{\arraycolsep}{#1}
\left(\begin{array}{@{}#2@{}}
}{%
\end{array}\right)}
% Usage: \dcolvec[2]{1.23 \\ 4.56} where the optional argument is the number
% of decimal places.
\newcommand{\dcolvec}[2][-1]{\left(\begin{array}{@{}D{.}{.}{#1}@{}} #2 \end{array}\right)}
%=============================================
......@@ -279,7 +292,7 @@
%------------------------- code listings
\usepackage{listings}
\lstset{basicstyle=\ttfamily\small,
\lstset{basicstyle=\ttfamily\scriptsize,
commentstyle=\textit,
keywordstyle=\color{blue}\bfseries,
showstringspaces=false,
......
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