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Course: SIMUW 08, Fall 2009School: Washington
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6 Day WS 1 -- Markov Chains system:sage <font color='purple'><h1>Markov Chains</h1></font> A <b><i>Markov chain</i></b> is a random process $X_0, X_1, X_2, \ldots$ with the property that $X_{i+1}$ depends only on $X_i$. <br><br> NOTE: A "random process"...
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6 Day WS 1 -- Markov Chains
system:sage
<font color='purple'><h1>Markov Chains</h1></font>
A <b><i>Markov chain</i></b> is a random process $X_0, X_1, X_2, \ldots$ with the property that $X_{i+1}$ depends only on $X_i$.
<br><br>
NOTE: A "random process" as an procedure that generates a sequence of random objects. They do not
have to be numbers.
<br><br>
<br><br>
<b>Example:</b> Tossing a coin repeatedly is a Markov chain; in fact $X_{i+1}$ doesn't even depend on $X_i$!
This is basically the standard Brownian motion model of the stock market.
<br><br>
<b>Example:</b> Generate a sequence $X_i$ as follows.
Start with $X_i=-1$.
<ol>
<li> If $X_i = -1$, toss a single coin. If it comes up heads then $X_{i+1} = 1$
and if it comes up tails then $X_{i+1} = -1$.
<li> On the other hand, if $X_i = +1$, then toss two coins. If they both come
up heads, then $X_{i+1} = 1$; otherwise $X_{i+1} = -1$.
</ol>
Let's simulate this:
{{{id=0|
@interact
def f(length=(2..200)):
X = [-1]
for i in range(length-1):
if X[i] == -1:
if random() <= 0.5: # tails
z = -1
else:
z = +1
else:
if random() >= 0.5 and random() >= 0.5:
z = 1
else:
z = -1
X.append(z)
print X
T = finance.TimeSeries(X)
if len(set(X)) > 1:
show(T.plot(points=True,pointsize=50,rgbcolor="orange") + T.plot(), frame=True)
T.plot_histogram(2).show(frame=True)
///
<html><div padding=6 id='div-interact-0'> <table width=800px height=400px bgcolor='#c5c5c5'
cellpadding=15><tr><td bgcolor='#f9f9f9' valign=top align=left><table><tr><td align=right><font color="black">length </font></td><td><table><tr><td>
<div id='slider-length-0' class='ui-slider-3' style='padding:0px;margin:0px;'><span class='ui-slider-handle'></span></div>
</td><td><font color='black' id='slider-length-0-lbl'></font></td></tr></table><script>(function(){ var values = ["2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50","51","52","53","54","55","56","57","58","59","60","61","62","63","64","65","66","67","68","69","70","71","72","73","74","75","76","77","78","79","80","81","82","83","84","85","86","87","88","89","90","91","92","93","94","95","96","97","98","99","100","101","102","103","104","105","106","107","108","109","110","111","112","113","114","115","116","117","118","119","120","121","122","123","124","125","126","127","128","129","130","131","132","133","134","135","136","137","138","139","140","141","142","143","144","145","146","147","148","149","150","151","152","153","154","155","156","157","158","159","160","161","162","163","164","165","166","167","168","169","170","171","172","173","174","175","176","177","178","179","180","181","182","183","184","185","186","187","188","189","190","191","192","193","194","195","196","197","198","199","200"]; setTimeout(function() {
$('#slider-length-0').slider({
stepping: 1, minValue: 0, maxValue: 198, startValue: 0,
change: function () { var position = Math.ceil($('#slider-length-0').slider('value')); if(values!=null) $('#slider-length-0-lbl').text(values[position]); interact(0, "sage.server.notebook.interact.update(0, \"length\", 1, sage.server.notebook.interact.standard_b64decode(\""+encode64(position)+"\"), globals())"); },
slide: function() { if(values!=null) $('#slider-length-0-lbl').text(values[Math.ceil($('#slider-length-0').slider('value'))]); }
});
if(values != null) $('#slider-length-0-lbl').text(values[Math.ceil($('#slider-length-0').slider('value'))]);
}, 1); })();</script></td></tr>
</table><div id='cell-interact-0'><?__SAGE__START>
<table border=0 bgcolor='#white' width=100% height=100%>
<tr><td bgcolor=white align=left valign=top><pre><?__SAGE__TEXT></pre></td></tr>
<tr><td align=left valign=top><?__SAGE__HTML></td></tr>
</table><?__SAGE__END></div></td>
</tr></table></div>
</html>
CPU time: 0.00 s, Wall time: 0.00 s
}}}
{{{id=2|
///
}}}
<br><br>
<b>Exercise:</b> Let $X_i$ be the sequence above.
<ol>
<li> What is the probability that a given $X_i$ is $-1$ (for arbitrary $i$)?
<li> What is the probability that a given $X_i$ is $+1$ (for arbitrary $i$)?
</ol>
{{{id=4|
///
}}}
{{{id=5|
///
}}}
<br><br>
<b>Exercise:</b> example, For another way to make
a Markov chain is to draw a directed graph, and label the edge from $v$ to $w$ with a number
between $0$ and $1$, so the sum of the outgoing edge labels is $1$. The result is a Markov
chain that hops from one vertex to another at random.
<ol>
<li> Make up your own example of such a Markov chain with $5$ vertices. Draw it on a piece of paper.
<li> Simulate 10 steps in your Markov chain. The sage command below computes
random numbers between 0 and 1, which you may find useful.
<li> You can also think of the above Markov chain process as being described by
a $5\times 5$ matrix $A=(a_{ij})$ whose rows sum to $1$ and such that $a_{ij}$
is the probability of going from vertex $i$ to vertex $j$. Write down this matrix
for your example.
</ol>
{{{id=7|
@interact
def _(numbers=(10,(1..200))):
for i in range(numbers): print random(),
///
<html><div padding=6 id='div-interact-7'> <table width=800px height=400px bgcolor='#c5c5c5'
cellpadding=15><tr><td bgcolor='#f9f9f9' valign=top align=left><table><tr><td align=right><font color="black">numbers </font></td><td><table><tr><td>
<div id='slider-numbers-7' class='ui-slider-3' style='padding:0px;margin:0px;'><span class='ui-slider-handle'></span></div>
</td><td><font color='black' id='slider-numbers-7-lbl'></font></td></tr></table><script>(function(){ var values = ["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50","51","52","53","54","55","56","57","58","59","60","61","62","63","64","65","66","67","68","69","70","71","72","73","74","75","76","77","78","79","80","81","82","83","84","85","86","87","88","89","90","91","92","93","94","95","96","97","98","99","100","101","102","103","104","105","106","107","108","109","110","111","112","113","114","115","116","117","118","119","120","121","122","123","124","125","126","127","128","129","130","131","132","133","134","135","136","137","138","139","140","141","142","143","144","145","146","147","148","149","150","151","152","153","154","155","156","157","158","159","160","161","162","163","164","165","166","167","168","169","170...
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