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42 changes: 21 additions & 21 deletions docs/ex/algaeex.html
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html,body,div,span,applet,object,iframe,h1,h2,h3,h4,h5,h6,p,blockquote,pre,a,abbr,acronym,address,big,cite,code,del,dfn,em,font,img,ins,kbd,q,s,samp,small,strike,strong,sub,sup,tt,var,b,u,i,center,dl,dt,dd,ol,ul,li,fieldset,form,label,legend,table,caption,tbody,tfoot,thead,tr,th,td{margin:0;padding:0;border:0;outline:0;font-size:100%;vertical-align:baseline;background:transparent}body{line-height:1}ol,ul{list-style:none}blockquote,q{quotes:none}blockquote:before,blockquote:after,q:before,q:after{content:'';content:none}:focus{outine:0}ins{text-decoration:none}del{text-decoration:line-through}table{border-collapse:collapse;border-spacing:0}

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</style></head><body><div class="content"><h1></h1><!--introduction--><p><a href="../index.html">MCMC toolbox</a> » <a href="../examples.html">Examples</a> » Algae</p><!--/introduction--><h2>Contents</h2><div><ul><li><a href="#1">Algae example</a></li></ul></div><h2 id="1">Algae example</h2><p>The example uses functions <a href="algaesys.html"><tt>algaesys</tt></a>, <a href="algaefun.html"><tt>algaefun</tt></a> and <a href="algaess.html"><tt>algaess</tt></a>.</p><p>We study the following system</p><p><img vspace="5" hspace="5" src="algaeweb.png" alt=""> </p><p>This is a simplified lake algae dynamics model. We consider phytoplankton <i>A</i>, zooplankton <i>Z</i> and nutrition <i>P</i> (eg. phosphorus) available for <i>A</i> in the water. The system is affected by the water outflow/inflow <i>Q</i>, incoming phosphorus load <i>Pin</i> and temperature <i>T</i>. It is described as a simple predator - pray dynamics between <i>A</i> and <i>Z</i>. The growth of <i>A</i> is limited by the availability of <i>P</i> and it depends on the water temperature <i>T</i>. The inflow/outflow <i>Q</i> affects both <i>A</i> and <i>P</i>, but not <i>Z</i>. We use the following equations:</p><p>
</style></head><body><div class="content"><h1></h1><!--introduction--><p><a href="../index.html">MCMC toolbox</a> » <a href="../examples.html">Examples</a> » Algae</p><!--/introduction--><h2 id="1">Algae example</h2><p>The example uses functions <a href="algaesys.html"><tt>algaesys</tt></a>, <a href="algaefun.html"><tt>algaefun</tt></a> and <a href="algaess.html"><tt>algaess</tt></a>.</p><p>We study the following system</p><p><img vspace="5" hspace="5" src="algaeweb.png" alt=""> </p><p>This is a simplified lake algae dynamics model. We consider phytoplankton <i>A</i>, zooplankton <i>Z</i> and nutrition <i>P</i> (eg. phosphorus) available for <i>A</i> in the water. The system is affected by the water outflow/inflow <i>Q</i>, incoming phosphorus load <i>Pin</i> and temperature <i>T</i>. It is described as a simple predator - pray dynamics between <i>A</i> and <i>Z</i>. The growth of <i>A</i> is limited by the availability of <i>P</i> and it depends on the water temperature <i>T</i>. The inflow/outflow <i>Q</i> affects both <i>A</i> and <i>P</i>, but not <i>Z</i>. We use the following equations:</p><p>
dA/dt = (&mu; - &rho;<sub>a</sub> - Q/V - &alpha;Z) A<br>
dZ/dt = &alpha;ZA-&rho;<sub>z</sub> Z<br>
dP/dt = -Q/V (P-P<sub>in</sub>) +
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</pre><pre class="codeoutput">Using values from the previous run
Sampling these parameters:
name start [min,max] N(mu,s^2)
mumax: 0.443013 [0,Inf] N(0,Inf)
rhoa: 0.0434265 [0,Inf] N(0,Inf)
rhoz: 0.104872 [0,Inf] N(0,Inf)
k: 10.7145 [0,Inf] N(0,Inf)
alpha: 0.0231054 [0,Inf] N(0,Inf)
th: 1.01339 [0,Inf] N(1.14,0.2^2)
A0: 1.48078 [0,Inf] N(0.77,2^2)
Z0: 1.23797 [0,Inf] N(1.3,2^2)
P0: 10.3283 [0,Inf] N(10,2^2)
mumax: 0.398492 [0,Inf] N(0,Inf)
rhoa: 0.0348433 [0,Inf] N(0,Inf)
rhoz: 0.0982238 [0,Inf] N(0,Inf)
k: 9.4219 [0,Inf] N(0,Inf)
alpha: 0.0233986 [0,Inf] N(0,Inf)
th: 1.00471 [0,Inf] N(1.14,0.2^2)
A0: 1.16216 [0,Inf] N(0.77,2^2)
Z0: 2.07551 [0,Inf] N(1.3,2^2)
P0: 8.54463 [0,Inf] N(10,2^2)
</pre><p>Chain plots should reveal that the chain has converged and we can use the results for estimation and predictive inference.</p><pre class="codeinput">figure(2); clf
mcmcplot(chain,[],results,<span class="string">'pairs'</span>);
figure(3); clf
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mean std MC_err tau geweke
---------------------------------------------------------------------
mumax 0.64904 0.13948 0.020242 154.94 0.59507
rhoa 0.053509 0.031097 0.0054147 133.26 0.93348
rhoz 0.10022 0.0050199 0.0004812 37.943 0.99572
k 18.812 7.5254 1.2293 165.15 0.44249
alpha 0.023751 0.0011733 0.00016021 78.551 0.93573
th 1.0035 0.01268 0.0015935 85.238 0.99196
A0 0.93967 0.27383 0.022537 30.847 0.88585
Z0 1.9462 0.48636 0.05209 69.541 0.85093
P0 8.8686 0.87045 0.069375 35.04 0.98544
mumax 7.3133 8.9169 1.8501 612.9 0.034271
rhoa 0.026332 0.014272 0.002472 416.33 0.27418
rhoz 0.097931 0.0057379 0.00069317 81.933 0.9884
k 346.6 434.88 90.632 615.33 0.030661
alpha 0.023557 0.0013095 0.00015524 92.634 0.9452
th 1.0123 0.013533 0.0016359 114.66 0.98836
A0 1.0649 0.30144 0.024486 37.659 0.9064
Z0 1.8118 0.45222 0.050899 66.175 0.82012
P0 8.822 0.85789 0.065738 37.405 0.98279
---------------------------------------------------------------------

</pre><p>In order to use the <tt>mcmcpred</tt> function we need function <tt>modelfun</tt> with input arguments given as <tt>modelfun(xdata,theta)</tt>. We construct this as an anonymous function.</p><pre class="codeinput">modelfun = @(d,th) algaefun(d(:,1),th,th(7:9),d);
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4 changes: 2 additions & 2 deletions docs/ex/algaess.html
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</style></head><body><div class="content"><h1></h1><!--introduction--><p><a href="../index.html">MCMC toolbox</a> » <a href="../examples.html">Examples</a> » Banana</p><!--/introduction--><h2>Contents</h2><div><ul><li><a href="#1">Banana example</a></li></ul></div><h2 id="1">Banana example</h2><p>This techncal example constructs a non Gaussian target distribution by twisting two first dimensions of Gaussian distribution. The Jacobian of the transformation is 1, so it is easy to calculate the right probability regions for the banana and study different adaptive methods.</p><pre class="codeinput">clear <span class="string">model</span> <span class="string">data</span> <span class="string">params</span> <span class="string">options</span>
</style></head><body><div class="content"><h1></h1><!--introduction--><p><a href="../index.html">MCMC toolbox</a> » <a href="../examples.html">Examples</a> » Banana</p><!--/introduction--><h2 id="1">Banana example</h2><p>This techncal example constructs a non Gaussian target distribution by twisting two first dimensions of Gaussian distribution. The Jacobian of the transformation is 1, so it is easy to calculate the right probability regions for the banana and study different adaptive methods.</p><pre class="codeinput">clear <span class="string">model</span> <span class="string">data</span> <span class="string">params</span> <span class="string">options</span>
</pre><p>'banana' sum-of-squares</p><pre class="codeinput">bananafun = @(x,a,b) [a.*x(:,1),x(:,2)./a-b.*((a.*x(:,1)).^2+a^2),x(:,3:end)];
bananainv = @(x,a,b) [x(:,1)./a,x(:,2).*a+a.*b.*(x(:,1).^2+a^2),x(:,3:end)];
bananass = @(x,d) bananainv(x-d.mu,d.a,d.b)*d.lam*bananainv(x-d.mu,d.a,d.b)';
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6 changes: 3 additions & 3 deletions docs/ex/beetleex.html
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</style></head><body><div class="content"><h1></h1><!--introduction--><p><a href="../index.html">MCMC toolbox</a> » <a href="../examples.html">Examples</a> » Beetle</p><!--/introduction--><h2>Contents</h2><div><ul><li><a href="#1">Beetle mortality data</a></li></ul></div><h2 id="1">Beetle mortality data</h2><p>From A. Dobson, <i>An Introduction to Generalized Linear Models</i>, Chapman &amp; Hall/CRC, 2002. Binomial logistic regression example with dose-response data. See <a href="beetless.html">beetless.m</a> for -2log(likelihood) function.</p><pre class="codeinput">clear <span class="string">model</span> <span class="string">data</span> <span class="string">params</span> <span class="string">options</span>
</style></head><body><div class="content"><h1></h1><!--introduction--><p><a href="../index.html">MCMC toolbox</a> » <a href="../examples.html">Examples</a> » Beetle</p><!--/introduction--><h2 id="1">Beetle mortality data</h2><p>From A. Dobson, <i>An Introduction to Generalized Linear Models</i>, Chapman &amp; Hall/CRC, 2002. Binomial logistic regression example with dose-response data. See <a href="beetless.html">beetless.m</a> for -2log(likelihood) function.</p><pre class="codeinput">clear <span class="string">model</span> <span class="string">data</span> <span class="string">params</span> <span class="string">options</span>

data = [
<span class="comment">% dose n y</span>
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