# Frequentist definition

A nonparametric procedure is a statistical procedure that has a certain desirable properties that hold under relatively mild assumptions regarding the underlying populations from which the data are obtained.

Hollander, Myles, Douglas A. Wolfe, and Eric Chicken. Nonparametric statistical methods. John Wiley and Sons, 2013.</sub>

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# Frequentist definition

The term nonparametric is imprecise. The related term distribution-free has a more precise meaning...The distribution-free property enables one to obtain the distribution of the statistic under the null hypothesis without specifying the underlying distribution of the data

Hollander, Myles, Douglas A. Wolfe, and Eric Chicken. Nonparametric statistical methods. John Wiley and Sons, 2013.</sub>

# BNP definition

Inferring the distribution of the data when is completely unknown.

Bayesian parametric

Bayesian nonparametric

# Distribution on distributions?

$$x_1,\dots,x_n\sim P ~~~\textit{ and }~~~ P \sim \mu$$

## Nonparametric Prior

Probability Space: Let $\mathbb{X}$ be a standard Borel space with Borel $\sigma$-field $\mathcal{B}_\mathbb{X}$ and $\mathbb{P}$ be the space of probability measures on $(\mathbb{X},\mathcal{B}_{\mathbb{X}})$.

Nonparametric Prior: Let $\mathbb{M}$ be the class of all probability measures on $(\mathbb{P},\mathcal{B}_\mathbb{P})$. We call the elements $\mu \in \mathbb{M}$ nonparametric priors.

## Dirichlet Process

#### developed by Ferguson

$$x_1,\dots,x_n\sim P ~~~\textit{ and }~~~ P \sim \mu$$

We are interested on a particular $\mu$:

$$x_1,\dots,x_n\sim P ~~~\textit{ and }~~~ P \sim Dp(s,\alpha)$$

## Dirichlet Process

#### developed by Ferguson

An element $\mu \in \mathbb{M}$ is called a Dirichlet process $\mathcal{D}p(s,\alpha)$ on $\mathbb{X}$, with

• prior strength $s\in \mathbb{R}^+$
• prior probability measure $\alpha$ on $\mathbb{X}$

if for every finite measurable partition $B_1,\dots,B_m$ of $\mathbb{X}$,

the vector $(P(B_1),\dots,P(B_m))$

has a Dirichlet distribution with parameters $(s\alpha(B_1),\dots,s\alpha(B_m))$

## Prior Base measure

It means that $\alpha(\mathbb{X})=s\in \mathbb{R}$ is finite and, thus, we can normalize $\alpha(\cdot)$!

Normalization: $$\alpha(\cdot)=s ~\alpha(\cdot) ~~~ \text{ with ~~~\alpha(\mathbb{X})=1}$$
Note that $\alpha(\cdot)$ is now a probability measure.

We equivalently say :$~~~P\sim \mathcal{D}(\alpha)~~~$ or $~~~P \sim \mathcal{Dp}(s,\alpha)$

## Example 1/3

• $\mathbb{X}=\mathbb{R}$
• prior strength measure: $s=4$
• prior probability measure: $\alpha=0.5\delta_{\{-1\}}+0.5\delta_{\{1\}}$



In :

using Compose
myplot=plot(x=rand(1)*0.0,y=rand(1)*0.0,Guide.annotation(compose(context(),line([(-1, 0), (-1, 0.5)]),line([(1, 0), (1, 0.5)]),stroke(colorant"blue"),linewidth(0.4))),Stat.xticks(ticks=[-2.0,-1.5,-1.0,-0.5,0.0, 0.5, 1.0,1.5,2]),Stat.yticks(ticks=[0.0, 0.5, 1.0]),xintercept=[0.0], Geom.line,Guide.ylabel("Probability"),Theme(major_label_font_size=15pt,minor_label_font_size=13pt))
draw(PNG("plots/real.png", 6inch, 3inch), myplot)




WARNING: The following aesthetics are mapped, but not used by any geometry:
xintercept



## Dirichlet Process

We say that $P\sim \mathcal{D}p(s,\alpha)$ if

• for every finite measurable partition $B_1,\dots,B_m$ of $\mathbb{X}$,
• the vector $(P(B_1),\dots,P(B_m))$ has a Dirichlet distribution with parameters $(s\alpha(B_1),\dots,s\alpha(B_m))$

## Example 2/3

$$P \sim Dp(s,\alpha)$$

if

• Partition: $B_1=(-\infty,0]$, $B_2=(0,\infty)$

• then

$$\begin{array}{rcl} (P(B_1),P(B_2))&\sim& Dir(s\alpha(B_1),s\alpha(B_2))\\ &=&Beta(s\alpha(B_1),s\alpha(B_2)) \end{array}$$

## Example 3/3

Since $s=4$ and $\alpha=0.5\delta_{\{-1\}}+0.5\delta_{\{1\}}$ then

$$\begin{array}{rcl} (P(B_1),P(B_2))&\sim& Dir(s\alpha(B_1),s\alpha(B_2))\\ &=& Dir(2,2)=Beta(2,2) \end{array}$$


In :

xv=(0:0.01:1)
yv=6*xv.^(1).*(1-xv).^(1)

densplot=plot(x=xv,y=yv, Geom.line,Guide.xlabel("P(B<sub>1</sub>)"),Guide.ylabel("Density"),Theme(panel_fill=colorant"black", default_color=colorant"orange",major_label_font_size=15pt,minor_label_font_size=13pt))
draw(PNG("plots/densplot.png", 6inch, 3inch), densplot)



## Posterior Dirichlet Process

One of the most remarkable properties of the DP priors is that the posterior of $P$ is again a DP.

• let $x_1,\dots,x_n \sim P$ i.i.d.
• assume that $P \sim Dp(s,\alpha)$

Then the posterior distribution of $P$ is
$$P|x_1,\dots,x_n \sim Dp\left(s+n, \frac{s}{s+n} \alpha+ \frac{1}{s+n} \sum\limits_{i=1}^n \delta_{x_i}\right),$$

where $\delta_{x_i}$ is an atomic probability measure centered at $x_i$.

## Example cont.

• $\mathbb{X}=\mathbb{R}$
• prior strength: $\alpha(\mathbb{R})=s=4$
• prior probability measure: $\alpha=0.5\delta_{\{-1\}}+0.5\delta_{\{1\}}$

## Example cont.

• Observations: $\{-1.5,-0.5, 0.25, 0.75, 1.25, 1.75\}$
• posterior strength: $s_n=s+n=4+6$
• posterior probability measure: $\alpha_n=\frac{s}{s+n} \alpha+ \frac{1}{s+n} \sum\limits_{i=1}^n \delta_{x_i}$



In :

using Compose
myplot=plot(x=rand(1)*0.0,y=rand(1)*0.0,Guide.annotation(compose(context(),line([(-1, 0), (-1, 0.4)]),line([(1, 0), (1, 0.4)]),stroke(colorant"blue"),linewidth(0.4))),Guide.annotation(compose(context(),line([(-1.5, 0), (-1.5, 0.1)]),line([(-0.5, 0), (-0.5, 0.1)]),line([(0.25, 0), (0.25, 0.1)]),line([(0.75, 0), (0.75, 0.1)]),line([(1.25, 0), (1.25, 0.1)]),line([(1.75, 0), (1.75, 0.1)]),stroke(colorant"orange"),linewidth(0.4))),Stat.xticks(ticks=[-2.0,-1.5,-1.0,-0.5,0.0, 0.5, 1.0,1.5,2]),Stat.yticks(ticks=[0.0, 0.5, 1.0]),xintercept=[0.0], Geom.line,Guide.ylabel("Probability"),Theme(major_label_font_size=15pt,minor_label_font_size=13pt))
draw(PNG("plots/realpost.png", 6inch, 3inch), myplot)




WARNING: The following aesthetics are mapped, but not used by any geometry:
xintercept


• Partition: $B_1=(-\infty,0]$, $B_2=(0,\infty)$
$$\begin{array}{rcl} (P(B_1),P(B_2))&\sim& Dir(s_n\alpha_n(B_1),s_n\alpha_n(B_2))\\ &=&Dir(4,6)=Beta(4,6) \end{array}$$


In :

xv=(0:0.01:1)
yv=504*xv.^(3).*(1-xv).^(5)

densplot=plot(x=xv,y=yv, Geom.line,Guide.xlabel("P(B<sub>1</sub>)"),Guide.ylabel("Density"),Theme(panel_fill=colorant"black", default_color=colorant"orange",major_label_font_size=15pt,minor_label_font_size=13pt))
draw(PNG("plots/densplotpost.png", 6inch, 3inch), densplot)



## Another example

• Prior measure $s=1$, $\alpha = \delta_{-1.5}$
• Observations $\{-1,-0.5, 0.25, 0.75, 1.25, 1.75\}$
• posterior strength: $s_n=s+n=1+6=7$
• posterior probability measure: $\alpha_n=\frac{s}{s+n} \alpha+ \frac{1}{s+n} \sum\limits_{i=1}^n \delta_{x_i}$
$$\begin{array}{rcl} (P(B_1),P(B_2))&\sim&Beta(3,4) \end{array}$$


In :

using Compose
myplot=plot(x=rand(1)*0.0,y=rand(1)*0.0,Guide.annotation(compose(context(),line([(-1.5, 0), (-1.5, 1/7)]),stroke(colorant"blue"),linewidth(0.4))),Guide.annotation(compose(context(),line([(-1, 0), (-1, 1/7)]),line([(-0.5, 0), (-0.5, 1/7)]),line([(0.25, 0), (0.25, 1/7)]),line([(0.75, 0), (0.75, 1/7)]),line([(1.25, 0), (1.25, 1/7)]),line([(1.75, 0), (1.75, 1/7)]),stroke(colorant"orange"),linewidth(0.4))),Stat.xticks(ticks=[-2.0,-1.5,-1.0,-0.5,0.0, 0.5, 1.0,1.5,2]),Stat.yticks(ticks=[0.0, 0.5, 1.0]),xintercept=[0.0], Geom.line,Guide.ylabel("Probability"),Theme(major_label_font_size=15pt,minor_label_font_size=13pt))
draw(PNG("plots/realpost1.png", 6inch, 3inch), myplot)




WARNING: The following aesthetics are mapped, but not used by any geometry:
xintercept


• Partition: $B_1=(-\infty,-0.75]$, $B_2=(-0.75,0.5]$ $B_3=[0.5,\infty)$
$$\begin{array}{rcl} (P(B_1),P(B_2),P(B_3))&\sim& Dir(s_n\alpha_n(B_1),s_n\alpha_n(B_2),s_n\alpha_n(B_3))\\ &=&Dir(2,2,3) \end{array}$$

## Finest Partition

• $\alpha_n=\frac{s}{s+n} \alpha+ \frac{1}{s+n} \sum\limits_{i=1}^n \delta_{x_i}$, $~~~~s_n=s+n$

• $B_1=(-\infty,-1.5]$, $~B_2=(-1.5,-1]$, $~B_3=(-1,-0.5]$,.....

## Finest Partition

We can actually only consider

• $B_1=(.,-1.5]$, $~B_2=(.,-1]$, $~B_3=(.,-0.5]$,...

$$\begin{array}{rcl} (P(B_1),\dots,P(B_{8}))&\sim& Dir(s_n\alpha_n(B_1),\dots,s_n\alpha_n(B_8))\\ &=& Dir(s,1,1,1,1,1,1) \end{array}$$

• Note that the support of $P$ is atomic, hence

$$P=w_0 \delta_{\{-1.5\}}+ w_1 \delta_{-1}+ w_2 \delta_{-0.5}+\dots+ w_7 \delta_{1.75},$$
with $(w_0,w_1,\dots,w_n)\sim Dir(s,1,\dots,1)$

So we have a Hierarchical model:

## General property

$$P=w_0 P_0+ \sum_{i=1}^n w_i \delta_{X_i},$$

where

• $(w_0,w_1,\dots,w_n)\sim Dir(s,1,\dots,1)$
• $\sum_{i=0}^n w_i = 1$
• $P_0 \sim Dp(s,\alpha)$

# Bayesian NonParametric Statistics

#### It reduces to ask the right question to the posterior

$$P=w_0 P_0+ \sum_{i=1}^n w_i \delta_{x_i},$$

with

• $(w_0,w_1,\dots,w_n)\sim Dir(s,1,\dots,1)$
• $\sum_{i=0}^n w_i = 1$,
• $P_0 \sim Dp(s,\alpha)$.

## Sign Test

Goal: is the median of some distribution $P$ greater than zero?

• Known $P$: just check whether $P(0,\infty)>P(-\infty,0]~~$ equiv. $~~P(0,\infty)>0.5$

• Unknown $P$: use observations from $P$ to estimate $P$ and query the posterior (Bayesian way)

Our estimate of $P$ is $$P=w_0 P_0+ \sum_{i=1}^n w_i \delta_{x_i},$$ with

• $(w_0,w_1,\dots,w_n)\sim Dir(s,1,\dots,1)$
• $P_0 \sim Dp(s,\alpha)$.

## Sign Test

$$P(\infty,0] ~~~ P(0,\infty)$$

This is equal to

$$\begin{array}{rclcl} P(-\infty,0]&=&w_0 P_0(-\infty,0]&+& \sum_{i=1}^n w_i I_{(-\infty,0]}(x_i)\\ P(0,\infty)&=&w_0 P_0(0,\infty)&+& \sum_{i=1}^n w_i I_{(0,\infty)}(x_i) \end{array}$$

## Bayesian Sign Test

The Bayesian sign test reduces to compute

$$\mathcal{P}\Big(P(0,\infty)>P(-\infty,0]\Big) =\mathcal{P}\Big(P(0,\infty)>0.5\Big)$$

where $\mathcal{P}$ is computed w.r.t. $(w_0,w_1,\dots,w_n)\sim Dir(s,1,\dots,1)$ and $P_0 \sim Dp(s,\alpha)$

## Example cont.

$$\begin{array}{rcl} P(-\infty,0]&=&w_0 P_0(-\infty,0]+ \sum_{i=1}^n w_i I_{(-\infty,0]}(x_i)\\ &=&w_0 + w_1 +w_2\\ P(0,\infty)&=& w_0 P_0(0,\infty)+ \sum_{i=1}^n w_i I_{(0,\infty)}(x_i)\\ &=& w_3 +w_4+w_5+w_6 \end{array}$$

The goal is to compute

$$\mathcal{P}(P(0,\infty)>0.5) =\mathcal{P}(w_3 +w_4+w_5+w_6 >0.5)$$

where $\mathcal{P}$ is computed w.r.t. $(w_0,w_1,\dots,w_n)\sim Dir(s,1,\dots,1)$



In :

include("plots/Bsigntest1.jl")

#Bayesian Sign Test
observ=[-1;-0.5;0.25;0.75;1.25;1.75];
data=Bsigntest(observ,zeros(length(observ),1),0,[1,0]);
#Plot
df0 = DataFrame(x=data[2,:][:])
p0=plot(df0, x=:x, Geom.histogram,Theme(major_label_font_size=13pt,minor_label_font_size=12pt,key_label_font_size=11pt),Guide.xlabel("P(0,&#8734;)"))
set_default_plot_size(15cm, 8cm)
display(p0)
prob=length(find(z->z> 0.5,data[2,:][:]))/length(data[2,:][:])
println("Probability that P(0,\u221e)>0.5 is $prob")   P(0,∞) -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 -1.00 -0.95 -0.90 -0.85 -0.80 -0.75 -0.70 -0.65 -0.60 -0.55 -0.50 -0.45 -0.40 -0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 -1 0 1 2 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500 3000 3500 -1500 -1450 -1400 -1350 -1300 -1250 -1200 -1150 -1100 -1050 -1000 -950 -900 -850 -800 -750 -700 -650 -600 -550 -500 -450 -400 -350 -300 -250 -200 -150 -100 -50 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950 2000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500 2550 2600 2650 2700 2750 2800 2850 2900 2950 3000 -2000 0 2000 4000 -1500 -1400 -1300 -1200 -1100 -1000 -900 -800 -700 -600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 (function(N){var k=/[\.\/]/,L=/\s*,\s*/,C=function(a,d){return a-d},a,v,y={n:{}},M=function(){for(var a=0,d=this.length;a<d;a++)if("undefined"!=typeof this[a])return this[a]},A=function(){for(var a=this.length;--a;)if("undefined"!=typeof this[a])return this[a]},w=function(k,d){k=String(k);var f=v,n=Array.prototype.slice.call(arguments,2),u=w.listeners(k),p=0,b,q=[],e={},l=[],r=a;l.firstDefined=M;l.lastDefined=A;a=k;for(var s=v=0,x=u.length;s<x;s++)"zIndex"in u[s]&&(q.push(u[s].zIndex),0>u[s].zIndex&& (e[u[s].zIndex]=u[s]));for(q.sort(C);0>q[p];)if(b=e[q[p++] ],l.push(b.apply(d,n)),v)return v=f,l;for(s=0;s<x;s++)if(b=u[s],"zIndex"in b)if(b.zIndex==q[p]){l.push(b.apply(d,n));if(v)break;do if(p++,(b=e[q[p] ])&&l.push(b.apply(d,n)),v)break;while(b)}else e[b.zIndex]=b;else if(l.push(b.apply(d,n)),v)break;v=f;a=r;return l};w._events=y;w.listeners=function(a){a=a.split(k);var d=y,f,n,u,p,b,q,e,l=[d],r=[];u=0;for(p=a.length;u<p;u++){e=[];b=0;for(q=l.length;b<q;b++)for(d=l[b].n,f=[d[a[u] ],d["*"] ],n=2;n--;)if(d= f[n])e.push(d),r=r.concat(d.f||[]);l=e}return r};w.on=function(a,d){a=String(a);if("function"!=typeof d)return function(){};for(var f=a.split(L),n=0,u=f.length;n<u;n++)(function(a){a=a.split(k);for(var b=y,f,e=0,l=a.length;e<l;e++)b=b.n,b=b.hasOwnProperty(a[e])&&b[a[e] ]||(b[a[e] ]={n:{}});b.f=b.f||[];e=0;for(l=b.f.length;e<l;e++)if(b.f[e]==d){f=!0;break}!f&&b.f.push(d)})(f[n]);return function(a){+a==+a&&(d.zIndex=+a)}};w.f=function(a){var d=[].slice.call(arguments,1);return function(){w.apply(null, [a,null].concat(d).concat([].slice.call(arguments,0)))}};w.stop=function(){v=1};w.nt=function(k){return k?(new RegExp("(?:\\.|\\/|^)"+k+"(?:\\.|\\/|$)")).test(a):a};w.nts=function(){return a.split(k)};w.off=w.unbind=function(a,d){if(a){var f=a.split(L);if(1<f.length)for(var n=0,u=f.length;n<u;n++)w.off(f[n],d);else{for(var f=a.split(k),p,b,q,e,l=[y],n=0,u=f.length;n<u;n++)for(e=0;e<l.length;e+=q.length-2){q=[e,1];p=l[e].n;if("*"!=f[n])p[f[n] ]&&q.push(p[f[n] ]);else for(b in p)p.hasOwnProperty(b)&&
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"DOMMouseScroll" : "mousewheel", fn2); return this; }; // Snap's attr function can be too slow for things like panning/zooming. // This is a function to directly update element attributes without going // through eve. Element.prototype.attribute = function(key, val) { if (val === undefined) { return this.node.getAttribute(key); } else { this.node.setAttribute(key, val); return this; } }; Element.prototype.init_gadfly = function() { this.mouseenter(Gadfly.plot_mouseover) .mouseleave(Gadfly.plot_mouseout) .dblclick(Gadfly.plot_dblclick) .mousewheel(Gadfly.guide_background_scroll) .drag(Gadfly.guide_background_drag_onmove, Gadfly.guide_background_drag_onstart, Gadfly.guide_background_drag_onend); this.mouseenter(function (event) { init_pan_zoom(this.plotroot()); }); return this; }; }); // When the plot is moused over, emphasize the grid lines. Gadfly.plot_mouseover = function(event) { var root = this.plotroot(); var keyboard_zoom = function(event) { if (event.which == 187) { // plus increase_zoom_by_position(root, 0.1, true); } else if (event.which == 189) { // minus increase_zoom_by_position(root, -0.1, true); } }; root.data("keyboard_zoom", keyboard_zoom); window.addEventListener("keyup", keyboard_zoom); var xgridlines = root.select(".xgridlines"), ygridlines = root.select(".ygridlines"); xgridlines.data("unfocused_strokedash", xgridlines.attribute("stroke-dasharray").replace(/(\d)(,|$)/g, "$1mm$2"));
ygridlines.data("unfocused_strokedash",
ygridlines.attribute("stroke-dasharray").replace(/(\d)(,|$)/g, "$1mm$2")); // emphasize grid lines var destcolor = root.data("focused_xgrid_color"); xgridlines.attribute("stroke-dasharray", "none") .selectAll("path") .animate({stroke: destcolor}, 250); destcolor = root.data("focused_ygrid_color"); ygridlines.attribute("stroke-dasharray", "none") .selectAll("path") .animate({stroke: destcolor}, 250); // reveal zoom slider root.select(".zoomslider") .animate({opacity: 1.0}, 250); }; // Reset pan and zoom on double click Gadfly.plot_dblclick = function(event) { set_plot_pan_zoom(this.plotroot(), 0.0, 0.0, 1.0); }; // Unemphasize grid lines on mouse out. Gadfly.plot_mouseout = function(event) { var root = this.plotroot(); window.removeEventListener("keyup", root.data("keyboard_zoom")); root.data("keyboard_zoom", undefined); var xgridlines = root.select(".xgridlines"), ygridlines = root.select(".ygridlines"); var destcolor = root.data("unfocused_xgrid_color"); xgridlines.attribute("stroke-dasharray", xgridlines.data("unfocused_strokedash")) .selectAll("path") .animate({stroke: destcolor}, 250); destcolor = root.data("unfocused_ygrid_color"); ygridlines.attribute("stroke-dasharray", ygridlines.data("unfocused_strokedash")) .selectAll("path") .animate({stroke: destcolor}, 250); // hide zoom slider root.select(".zoomslider") .animate({opacity: 0.0}, 250); }; var set_geometry_transform = function(root, tx, ty, scale) { var xscalable = root.hasClass("xscalable"), yscalable = root.hasClass("yscalable"); var old_scale = root.data("scale"); var xscale = xscalable ? scale : 1.0, yscale = yscalable ? scale : 1.0; tx = xscalable ? tx : 0.0; ty = yscalable ? ty : 0.0; var t = new Snap.Matrix().translate(tx, ty).scale(xscale, yscale); root.selectAll(".geometry, image") .forEach(function (element, i) { element.transform(t); }); bounds = root.plotbounds(); if (yscalable) { var xfixed_t = new Snap.Matrix().translate(0, ty).scale(1.0, yscale); root.selectAll(".xfixed") .forEach(function (element, i) { element.transform(xfixed_t); }); root.select(".ylabels") .transform(xfixed_t) .selectAll("text") .forEach(function (element, i) { if (element.attribute("gadfly:inscale") == "true") { var cx = element.asPX("x"), cy = element.asPX("y"); var st = element.data("static_transform"); unscale_t = new Snap.Matrix(); unscale_t.scale(1, 1/scale, cx, cy).add(st); element.transform(unscale_t); var y = cy * scale + ty; element.attr("visibility", bounds.y0 <= y && y <= bounds.y1 ? "visible" : "hidden"); } }); } if (xscalable) { var yfixed_t = new Snap.Matrix().translate(tx, 0).scale(xscale, 1.0); var xtrans = new Snap.Matrix().translate(tx, 0); root.selectAll(".yfixed") .forEach(function (element, i) { element.transform(yfixed_t); }); root.select(".xlabels") .transform(yfixed_t) .selectAll("text") .forEach(function (element, i) { if (element.attribute("gadfly:inscale") == "true") { var cx = element.asPX("x"), cy = element.asPX("y"); var st = element.data("static_transform"); unscale_t = new Snap.Matrix(); unscale_t.scale(1/scale, 1, cx, cy).add(st); element.transform(unscale_t); var x = cx * scale + tx; element.attr("visibility", bounds.x0 <= x && x <= bounds.x1 ? "visible" : "hidden"); } }); } // we must unscale anything that is scale invariance: widths, raiduses, etc. var size_attribs = ["font-size"]; var unscaled_selection = ".geometry, .geometry *"; if (xscalable) { size_attribs.push("rx"); unscaled_selection += ", .xgridlines"; } if (yscalable) { size_attribs.push("ry"); unscaled_selection += ", .ygridlines"; } root.selectAll(unscaled_selection) .forEach(function (element, i) { // circle need special help if (element.node.nodeName == "circle") { var cx = element.attribute("cx"), cy = element.attribute("cy"); unscale_t = new Snap.Matrix().scale(1/xscale, 1/yscale, cx, cy); element.transform(unscale_t); return; } for (i in size_attribs) { var key = size_attribs[i]; var val = parseFloat(element.attribute(key)); if (val !== undefined && val != 0 && !isNaN(val)) { element.attribute(key, val * old_scale / scale); } } }); }; // Find the most appropriate tick scale and update label visibility. var update_tickscale = function(root, scale, axis) { if (!root.hasClass(axis + "scalable")) return; var tickscales = root.data(axis + "tickscales"); var best_tickscale = 1.0; var best_tickscale_dist = Infinity; for (tickscale in tickscales) { var dist = Math.abs(Math.log(tickscale) - Math.log(scale)); if (dist < best_tickscale_dist) { best_tickscale_dist = dist; best_tickscale = tickscale; } } if (best_tickscale != root.data(axis + "tickscale")) { root.data(axis + "tickscale", best_tickscale); var mark_inscale_gridlines = function (element, i) { var inscale = element.attr("gadfly:scale") == best_tickscale; element.attribute("gadfly:inscale", inscale); element.attr("visibility", inscale ? "visible" : "hidden"); }; var mark_inscale_labels = function (element, i) { var inscale = element.attr("gadfly:scale") == best_tickscale; element.attribute("gadfly:inscale", inscale); element.attr("visibility", inscale ? "visible" : "hidden"); }; root.select("." + axis + "gridlines").selectAll("path").forEach(mark_inscale_gridlines); root.select("." + axis + "labels").selectAll("text").forEach(mark_inscale_labels); } }; var set_plot_pan_zoom = function(root, tx, ty, scale) { var old_scale = root.data("scale"); var bounds = root.plotbounds(); var width = bounds.x1 - bounds.x0, height = bounds.y1 - bounds.y0; // compute the viewport derived from tx, ty, and scale var x_min = -width * scale - (scale * width - width), x_max = width * scale, y_min = -height * scale - (scale * height - height), y_max = height * scale; var x0 = bounds.x0 - scale * bounds.x0, y0 = bounds.y0 - scale * bounds.y0; var tx = Math.max(Math.min(tx - x0, x_max), x_min), ty = Math.max(Math.min(ty - y0, y_max), y_min); tx += x0; ty += y0; // when the scale change, we may need to alter which set of // ticks is being displayed if (scale != old_scale) { update_tickscale(root, scale, "x"); update_tickscale(root, scale, "y"); } set_geometry_transform(root, tx, ty, scale); root.data("scale", scale); root.data("tx", tx); root.data("ty", ty); }; var scale_centered_translation = function(root, scale) { var bounds = root.plotbounds(); var width = bounds.x1 - bounds.x0, height = bounds.y1 - bounds.y0; var tx0 = root.data("tx"), ty0 = root.data("ty"); var scale0 = root.data("scale"); // how off from center the current view is var xoff = tx0 - (bounds.x0 * (1 - scale0) + (width * (1 - scale0)) / 2), yoff = ty0 - (bounds.y0 * (1 - scale0) + (height * (1 - scale0)) / 2); // rescale offsets xoff = xoff * scale / scale0; yoff = yoff * scale / scale0; // adjust for the panel position being scaled var x_edge_adjust = bounds.x0 * (1 - scale), y_edge_adjust = bounds.y0 * (1 - scale); return { x: xoff + x_edge_adjust + (width - width * scale) / 2, y: yoff + y_edge_adjust + (height - height * scale) / 2 }; }; // Initialize data for panning zooming if it isn't already. var init_pan_zoom = function(root) { if (root.data("zoompan-ready")) { return; } // The non-scaling-stroke trick. Rather than try to correct for the // stroke-width when zooming, we force it to a fixed value. var px_per_mm = root.node.getCTM().a; // Drag events report deltas in pixels, which we'd like to convert to // millimeters. root.data("px_per_mm", px_per_mm); root.selectAll("path") .forEach(function (element, i) { sw = element.asPX("stroke-width") * px_per_mm; if (sw > 0) { element.attribute("stroke-width", sw); element.attribute("vector-effect", "non-scaling-stroke"); } }); // Store ticks labels original tranformation root.selectAll(".xlabels > text, .ylabels > text") .forEach(function (element, i) { var lm = element.transform().localMatrix; element.data("static_transform", new Snap.Matrix(lm.a, lm.b, lm.c, lm.d, lm.e, lm.f)); }); var xgridlines = root.select(".xgridlines"); var ygridlines = root.select(".ygridlines"); var xlabels = root.select(".xlabels"); var ylabels = root.select(".ylabels"); if (root.data("tx") === undefined) root.data("tx", 0); if (root.data("ty") === undefined) root.data("ty", 0); if (root.data("scale") === undefined) root.data("scale", 1.0); if (root.data("xtickscales") === undefined) { // index all the tick scales that are listed var xtickscales = {}; var ytickscales = {}; var add_x_tick_scales = function (element, i) { xtickscales[element.attribute("gadfly:scale")] = true; }; var add_y_tick_scales = function (element, i) { ytickscales[element.attribute("gadfly:scale")] = true; }; if (xgridlines) xgridlines.selectAll("path").forEach(add_x_tick_scales); if (ygridlines) ygridlines.selectAll("path").forEach(add_y_tick_scales); if (xlabels) xlabels.selectAll("text").forEach(add_x_tick_scales); if (ylabels) ylabels.selectAll("text").forEach(add_y_tick_scales); root.data("xtickscales", xtickscales); root.data("ytickscales", ytickscales); root.data("xtickscale", 1.0); } var min_scale = 1.0, max_scale = 1.0; for (scale in xtickscales) { min_scale = Math.min(min_scale, scale); max_scale = Math.max(max_scale, scale); } for (scale in ytickscales) { min_scale = Math.min(min_scale, scale); max_scale = Math.max(max_scale, scale); } root.data("min_scale", min_scale); root.data("max_scale", max_scale); // store the original positions of labels if (xlabels) { xlabels.selectAll("text") .forEach(function (element, i) { element.data("x", element.asPX("x")); }); } if (ylabels) { ylabels.selectAll("text") .forEach(function (element, i) { element.data("y", element.asPX("y")); }); } // mark grid lines and ticks as in or out of scale. var mark_inscale = function (element, i) { element.attribute("gadfly:inscale", element.attribute("gadfly:scale") == 1.0); }; if (xgridlines) xgridlines.selectAll("path").forEach(mark_inscale); if (ygridlines) ygridlines.selectAll("path").forEach(mark_inscale); if (xlabels) xlabels.selectAll("text").forEach(mark_inscale); if (ylabels) ylabels.selectAll("text").forEach(mark_inscale); // figure out the upper ond lower bounds on panning using the maximum // and minum grid lines var bounds = root.plotbounds(); var pan_bounds = { x0: 0.0, y0: 0.0, x1: 0.0, y1: 0.0 }; if (xgridlines) { xgridlines .selectAll("path") .forEach(function (element, i) { if (element.attribute("gadfly:inscale") == "true") { var bbox = element.node.getBBox(); if (bounds.x1 - bbox.x < pan_bounds.x0) { pan_bounds.x0 = bounds.x1 - bbox.x; } if (bounds.x0 - bbox.x > pan_bounds.x1) { pan_bounds.x1 = bounds.x0 - bbox.x; } element.attr("visibility", "visible"); } }); } if (ygridlines) { ygridlines .selectAll("path") .forEach(function (element, i) { if (element.attribute("gadfly:inscale") == "true") { var bbox = element.node.getBBox(); if (bounds.y1 - bbox.y < pan_bounds.y0) { pan_bounds.y0 = bounds.y1 - bbox.y; } if (bounds.y0 - bbox.y > pan_bounds.y1) { pan_bounds.y1 = bounds.y0 - bbox.y; } element.attr("visibility", "visible"); } }); } // nudge these values a little pan_bounds.x0 -= 5; pan_bounds.x1 += 5; pan_bounds.y0 -= 5; pan_bounds.y1 += 5; root.data("pan_bounds", pan_bounds); root.data("zoompan-ready", true) }; // drag actions, i.e. zooming and panning var pan_action = { start: function(root, x, y, event) { root.data("dx", 0); root.data("dy", 0); root.data("tx0", root.data("tx")); root.data("ty0", root.data("ty")); }, update: function(root, dx, dy, x, y, event) { var px_per_mm = root.data("px_per_mm"); dx /= px_per_mm; dy /= px_per_mm; var tx0 = root.data("tx"), ty0 = root.data("ty"); var dx0 = root.data("dx"), dy0 = root.data("dy"); root.data("dx", dx); root.data("dy", dy); dx = dx - dx0; dy = dy - dy0; var tx = tx0 + dx, ty = ty0 + dy; set_plot_pan_zoom(root, tx, ty, root.data("scale")); }, end: function(root, event) { }, cancel: function(root) { set_plot_pan_zoom(root, root.data("tx0"), root.data("ty0"), root.data("scale")); } }; var zoom_box; var zoom_action = { start: function(root, x, y, event) { var bounds = root.plotbounds(); var width = bounds.x1 - bounds.x0, height = bounds.y1 - bounds.y0; var ratio = width / height; var xscalable = root.hasClass("xscalable"), yscalable = root.hasClass("yscalable"); var px_per_mm = root.data("px_per_mm"); x = xscalable ? x / px_per_mm : bounds.x0; y = yscalable ? y / px_per_mm : bounds.y0; var w = xscalable ? 0 : width; var h = yscalable ? 0 : height; zoom_box = root.rect(x, y, w, h).attr({ "fill": "#000", "opacity": 0.25 }); zoom_box.data("ratio", ratio); }, update: function(root, dx, dy, x, y, event) { var xscalable = root.hasClass("xscalable"), yscalable = root.hasClass("yscalable"); var px_per_mm = root.data("px_per_mm"); var bounds = root.plotbounds(); if (yscalable) { y /= px_per_mm; y = Math.max(bounds.y0, y); y = Math.min(bounds.y1, y); } else { y = bounds.y1; } if (xscalable) { x /= px_per_mm; x = Math.max(bounds.x0, x); x = Math.min(bounds.x1, x); } else { x = bounds.x1; } dx = x - zoom_box.attr("x"); dy = y - zoom_box.attr("y"); if (xscalable && yscalable) { var ratio = zoom_box.data("ratio"); var width = Math.min(Math.abs(dx), ratio * Math.abs(dy)); var height = Math.min(Math.abs(dy), Math.abs(dx) / ratio); dx = width * dx / Math.abs(dx); dy = height * dy / Math.abs(dy); } var xoffset = 0, yoffset = 0; if (dx < 0) { xoffset = dx; dx = -1 * dx; } if (dy < 0) { yoffset = dy; dy = -1 * dy; } if (isNaN(dy)) { dy = 0.0; } if (isNaN(dx)) { dx = 0.0; } zoom_box.transform("T" + xoffset + "," + yoffset); zoom_box.attr("width", dx); zoom_box.attr("height", dy); }, end: function(root, event) { var xscalable = root.hasClass("xscalable"), yscalable = root.hasClass("yscalable"); var zoom_bounds = zoom_box.getBBox(); if (zoom_bounds.width * zoom_bounds.height <= 0) { return; } var plot_bounds = root.plotbounds(); var zoom_factor = 1.0; if (yscalable) { zoom_factor = (plot_bounds.y1 - plot_bounds.y0) / zoom_bounds.height; } else { zoom_factor = (plot_bounds.x1 - plot_bounds.x0) / zoom_bounds.width; } var tx = (root.data("tx") - zoom_bounds.x) * zoom_factor + plot_bounds.x0, ty = (root.data("ty") - zoom_bounds.y) * zoom_factor + plot_bounds.y0; set_plot_pan_zoom(root, tx, ty, root.data("scale") * zoom_factor); zoom_box.remove(); }, cancel: function(root) { zoom_box.remove(); } }; Gadfly.guide_background_drag_onstart = function(x, y, event) { var root = this.plotroot(); var scalable = root.hasClass("xscalable") || root.hasClass("yscalable"); var zoomable = !event.altKey && !event.ctrlKey && event.shiftKey && scalable; var panable = !event.altKey && !event.ctrlKey && !event.shiftKey && scalable; var drag_action = zoomable ? zoom_action : panable ? pan_action : undefined; root.data("drag_action", drag_action); if (drag_action) { var cancel_drag_action = function(event) { if (event.which == 27) { // esc key drag_action.cancel(root); root.data("drag_action", undefined); } }; window.addEventListener("keyup", cancel_drag_action); root.data("cancel_drag_action", cancel_drag_action); drag_action.start(root, x, y, event); } }; Gadfly.guide_background_drag_onmove = function(dx, dy, x, y, event) { var root = this.plotroot(); var drag_action = root.data("drag_action"); if (drag_action) { drag_action.update(root, dx, dy, x, y, event); } }; Gadfly.guide_background_drag_onend = function(event) { var root = this.plotroot(); window.removeEventListener("keyup", root.data("cancel_drag_action")); root.data("cancel_drag_action", undefined); var drag_action = root.data("drag_action"); if (drag_action) { drag_action.end(root, event); } root.data("drag_action", undefined); }; Gadfly.guide_background_scroll = function(event) { if (event.shiftKey) { increase_zoom_by_position(this.plotroot(), 0.001 * event.wheelDelta); event.preventDefault(); } }; Gadfly.zoomslider_button_mouseover = function(event) { this.select(".button_logo") .animate({fill: this.data("mouseover_color")}, 100); }; Gadfly.zoomslider_button_mouseout = function(event) { this.select(".button_logo") .animate({fill: this.data("mouseout_color")}, 100); }; Gadfly.zoomslider_zoomout_click = function(event) { increase_zoom_by_position(this.plotroot(), -0.1, true); }; Gadfly.zoomslider_zoomin_click = function(event) { increase_zoom_by_position(this.plotroot(), 0.1, true); }; Gadfly.zoomslider_track_click = function(event) { // TODO }; // Map slider position x to scale y using the function y = a*exp(b*x)+c. // The constants a, b, and c are solved using the constraint that the function // should go through the points (0; min_scale), (0.5; 1), and (1; max_scale). var scale_from_slider_position = function(position, min_scale, max_scale) { var a = (1 - 2 * min_scale + min_scale * min_scale) / (min_scale + max_scale - 2), b = 2 * Math.log((max_scale - 1) / (1 - min_scale)), c = (min_scale * max_scale - 1) / (min_scale + max_scale - 2); return a * Math.exp(b * position) + c; } // inverse of scale_from_slider_position var slider_position_from_scale = function(scale, min_scale, max_scale) { var a = (1 - 2 * min_scale + min_scale * min_scale) / (min_scale + max_scale - 2), b = 2 * Math.log((max_scale - 1) / (1 - min_scale)), c = (min_scale * max_scale - 1) / (min_scale + max_scale - 2); return 1 / b * Math.log((scale - c) / a); } var increase_zoom_by_position = function(root, delta_position, animate) { var scale = root.data("scale"), min_scale = root.data("min_scale"), max_scale = root.data("max_scale"); var position = slider_position_from_scale(scale, min_scale, max_scale); position += delta_position; scale = scale_from_slider_position(position, min_scale, max_scale); set_zoom(root, scale, animate); } var set_zoom = function(root, scale, animate) { var min_scale = root.data("min_scale"), max_scale = root.data("max_scale"), old_scale = root.data("scale"); var new_scale = Math.max(min_scale, Math.min(scale, max_scale)); if (animate) { Snap.animate( old_scale, new_scale, function (new_scale) { update_plot_scale(root, new_scale); }, 200); } else { update_plot_scale(root, new_scale); } } var update_plot_scale = function(root, new_scale) { var trans = scale_centered_translation(root, new_scale); set_plot_pan_zoom(root, trans.x, trans.y, new_scale); root.selectAll(".zoomslider_thumb") .forEach(function (element, i) { var min_pos = element.data("min_pos"), max_pos = element.data("max_pos"), min_scale = root.data("min_scale"), max_scale = root.data("max_scale"); var xmid = (min_pos + max_pos) / 2; var xpos = slider_position_from_scale(new_scale, min_scale, max_scale); element.transform(new Snap.Matrix().translate( Math.max(min_pos, Math.min( max_pos, min_pos + (max_pos - min_pos) * xpos)) - xmid, 0)); }); }; Gadfly.zoomslider_thumb_dragmove = function(dx, dy, x, y, event) { var root = this.plotroot(); var min_pos = this.data("min_pos"), max_pos = this.data("max_pos"), min_scale = root.data("min_scale"), max_scale = root.data("max_scale"), old_scale = root.data("old_scale"); var px_per_mm = root.data("px_per_mm"); dx /= px_per_mm; dy /= px_per_mm; var xmid = (min_pos + max_pos) / 2; var xpos = slider_position_from_scale(old_scale, min_scale, max_scale) + dx / (max_pos - min_pos); // compute the new scale var new_scale = scale_from_slider_position(xpos, min_scale, max_scale); new_scale = Math.min(max_scale, Math.max(min_scale, new_scale)); update_plot_scale(root, new_scale); event.stopPropagation(); }; Gadfly.zoomslider_thumb_dragstart = function(x, y, event) { this.animate({fill: this.data("mouseover_color")}, 100); var root = this.plotroot(); // keep track of what the scale was when we started dragging root.data("old_scale", root.data("scale")); event.stopPropagation(); }; Gadfly.zoomslider_thumb_dragend = function(event) { this.animate({fill: this.data("mouseout_color")}, 100); event.stopPropagation(); }; var toggle_color_class = function(root, color_class, ison) { var guides = root.selectAll(".guide." + color_class + ",.guide ." + color_class); var geoms = root.selectAll(".geometry." + color_class + ",.geometry ." + color_class); if (ison) { guides.animate({opacity: 0.5}, 250); geoms.animate({opacity: 0.0}, 250); } else { guides.animate({opacity: 1.0}, 250); geoms.animate({opacity: 1.0}, 250); } }; Gadfly.colorkey_swatch_click = function(event) { var root = this.plotroot(); var color_class = this.data("color_class"); if (event.shiftKey) { root.selectAll(".colorkey text") .forEach(function (element) { var other_color_class = element.data("color_class"); if (other_color_class != color_class) { toggle_color_class(root, other_color_class, element.attr("opacity") == 1.0); } }); } else { toggle_color_class(root, color_class, this.attr("opacity") == 1.0); } }; return Gadfly; })); //@ sourceURL=gadfly.js (function (glob, factory) { // AMD support if (typeof require === "function" && typeof define === "function" && define.amd) { require(["Snap.svg", "Gadfly"], function (Snap, Gadfly) { factory(Snap, Gadfly); }); } else { factory(glob.Snap, glob.Gadfly); } })(window, function (Snap, Gadfly) { var fig = Snap("#img-0b522373"); fig.select("#img-0b522373-5") .init_gadfly(); fig.select("#img-0b522373-7") .plotroot().data("unfocused_ygrid_color", "#D0D0E0") ; fig.select("#img-0b522373-7") .plotroot().data("focused_ygrid_color", "#A0A0A0") ; fig.select("#img-0b522373-8") .plotroot().data("unfocused_xgrid_color", "#D0D0E0") ; fig.select("#img-0b522373-8") .plotroot().data("focused_xgrid_color", "#A0A0A0") ; fig.select("#img-0b522373-13") .data("mouseover_color", "#CD5C5C") ; fig.select("#img-0b522373-13") .data("mouseout_color", "#6A6A6A") ; fig.select("#img-0b522373-13") .click(Gadfly.zoomslider_zoomin_click) .mouseenter(Gadfly.zoomslider_button_mouseover) .mouseleave(Gadfly.zoomslider_button_mouseout) ; fig.select("#img-0b522373-15") .data("max_pos", 129) ; fig.select("#img-0b522373-15") .data("min_pos", 112) ; fig.select("#img-0b522373-15") .click(Gadfly.zoomslider_track_click); fig.select("#img-0b522373-16") .data("max_pos", 129) ; fig.select("#img-0b522373-16") .data("min_pos", 112) ; fig.select("#img-0b522373-16") .data("mouseover_color", "#CD5C5C") ; fig.select("#img-0b522373-16") .data("mouseout_color", "#6A6A6A") ; fig.select("#img-0b522373-16") .drag(Gadfly.zoomslider_thumb_dragmove, Gadfly.zoomslider_thumb_dragstart, Gadfly.zoomslider_thumb_dragend) ; fig.select("#img-0b522373-17") .data("mouseover_color", "#CD5C5C") ; fig.select("#img-0b522373-17") .data("mouseout_color", "#6A6A6A") ; fig.select("#img-0b522373-17") .click(Gadfly.zoomslider_zoomout_click) .mouseenter(Gadfly.zoomslider_button_mouseover) .mouseleave(Gadfly.zoomslider_button_mouseout) ; }); Probability that P(0,∞)>0.5 is 0.65675  ## Bayesian Answer The median is greater than zero with posterior probability$0.66$. • No Null hypothesis • No$p$-value ## Sensitivity to the prior (IDP) $$\begin{array}{rcl} P(-\infty,0]&=&w_0 P_0(-\infty,0]+ \sum_{i=1}^n w_i I_{(-\infty,0]}(x_i)\\ &=& w_1 +w_2\\ P(0,\infty)&=& w_0 P_0(0,\infty)+ \sum_{i=1}^n w_i I_{(0,\infty)}(x_i)\\ &=& w_0 +w_3 +w_4+w_5+w_6 \end{array}$$ The goal is to compute $$\mathcal{P}(P(0,\infty)>0.5) =\mathcal{P}(w_0 +w_3 +w_4+w_5+w_6 >0.5)$$ where$\mathcal{P}$is computed w.r.t.$(w_0,w_1,\dots,w_n)\sim Dir(s,1,\dots,1)$ In : using Distributions, Gadfly, Compose, DataFrames include("plots/Bsigntest1.jl") #Bayesian Sign Test observ=[-1;-0.5;0.25;0.75;1.25;1.75]; data=Bsigntest(observ,zeros(length(observ),1),0,[0,1]); #Plot df0 = DataFrame(x=data[2,:][:]) p0=plot(df0, x=:x, Geom.histogram,Theme(major_label_font_size=13pt,minor_label_font_size=12pt,key_label_font_size=11pt),Guide.xlabel("P(0,&#8734;)")) set_default_plot_size(15cm, 8cm) display(p0) prob=length(find(z->z> 0.5,data[2,:][:]))/length(data[2,:][:]) println("Probability that P(0,\u221e)>0.5 is$prob")




P(0,∞)

-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
-1.00
-0.95
-0.90
-0.85
-0.80
-0.75
-0.70
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.80
1.85
1.90
1.95
2.00
-1
0
1
2
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0

-2500
-2000
-1500
-1000
-500
0
500
1000
1500
2000
2500
3000
3500
4000
4500
-2000
-1900
-1800
-1700
-1600
-1500
-1400
-1300
-1200
-1100
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
2800
2900
3000
3100
3200
3300
3400
3500
3600
3700
3800
3900
4000
-2000
0
2000
4000
-2000
-1800
-1600
-1400
-1200
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
3200
3400
3600
3800
4000

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to:u(v),f:n(p)}}function z(a){return a}function d(a){return function(b){return+b.toFixed(3)+a}}function f(b){return a.rgb(b,b,b)}function n(a){var b=0,d,f,k,n,h,p,q=[];d=0;for(f=a.length;d<f;d++){h="[";p=['"'+a[d]+'"'];k=1;for(n=a[d].length;k<n;k++)p[k]="val["+b++ +"]";h+=p+"]";q[d]=h}return Function("val","return Snap.path.toString.call(["+q+"])")}function u(a){for(var b=[],d=0,f=a.length;d<f;d++)for(var k=1,n=a[d].length;k<n;k++)b.push(a[d][k]);return b}var p={},b=/[a-z]+$/i,q=String; p.stroke=p.fill="colour";v.prototype.equal=function(a,b){return k("snap.util.equal",this,a,b).firstDefined()};k.on("snap.util.equal",function(e,k){var r,s;r=q(this.attr(e)||"");var x=this;if(r==+r&&k==+k)return{from:+r,to:+k,f:z};if("colour"==p[e])return r=a.color(r),s=a.color(k),{from:[r.r,r.g,r.b,r.opacity],to:[s.r,s.g,s.b,s.opacity],f:f};if("transform"==e||"gradientTransform"==e||"patternTransform"==e)return k instanceof a.Matrix&&(k=k.toTransformString()),a._.rgTransform.test(k)||(k=a._.svgTransform2string(k)), w(r,k,function(){return x.getBBox(1)});if("d"==e||"path"==e)return r=a.path.toCubic(r,k),{from:u(r),to:u(r),f:n(r)};if("points"==e)return r=q(r).split(a._.separator),s=q(k).split(a._.separator),{from:r,to:s,f:function(a){return a}};aUnit=r.match(b);s=q(k).match(b);return aUnit&&aUnit==s?{from:parseFloat(r),to:parseFloat(k),f:d(aUnit)}:{from:this.asPX(e),to:this.asPX(e,k),f:z}})});C.plugin(function(a,v,y,C){var A=v.prototype,w="createTouch"in C.doc;v="click dblclick mousedown mousemove mouseout mouseover mouseup touchstart touchmove touchend touchcancel".split(" "); 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function(){k.stop();var d=z(this.node,"filter");if(d)return(d=w(d).match(A))&&a.select(d)});k.on("snap.util.attr.filter",function(d){if(d instanceof v&&"filter"==d.type){k.stop();var f=d.node.id;f||(z(d.node,{id:d.id}),f=d.id);z(this.node,{filter:a.url(f)})}d&&"none"!=d||(k.stop(),this.node.removeAttribute("filter"))});a.filter.blur=function(d,f){null==d&&(d=2);return a.format('<feGaussianBlur stdDeviation="{def}"/>',{def:null==f?d:[d,f]})};a.filter.blur.toString=function(){return this()};a.filter.shadow=
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(function (glob, factory) {
// AMD support
if (typeof define === "function" && define.amd) {
// Define as an anonymous module
return factory(Snap);
});
} else {
// Browser globals (glob is window)
// Snap adds itself to window
}
}(this, function (Snap) {

// Get an x/y coordinate value in pixels
var xPX = function(fig, x) {
var client_box = fig.node.getBoundingClientRect();
return x * fig.node.viewBox.baseVal.width / client_box.width;
};

var yPX = function(fig, y) {
var client_box = fig.node.getBoundingClientRect();
return y * fig.node.viewBox.baseVal.height / client_box.height;
};

Snap.plugin(function (Snap, Element, Paper, global) {
// Traverse upwards from a snap element to find and return the first
// note with the "plotroot" class.
Element.prototype.plotroot = function () {
var element = this;
while (!element.hasClass("plotroot") && element.parent() != null) {
element = element.parent();
}
return element;
};

Element.prototype.svgroot = function () {
var element = this;
while (element.node.nodeName != "svg" && element.parent() != null) {
element = element.parent();
}
return element;
};

Element.prototype.plotbounds = function () {
var root = this.plotroot()
var bbox = root.select(".guide.background").node.getBBox();
return {
x0: bbox.x,
x1: bbox.x + bbox.width,
y0: bbox.y,
y1: bbox.y + bbox.height
};
};

Element.prototype.plotcenter = function () {
var root = this.plotroot()
var bbox = root.select(".guide.background").node.getBBox();
return {
x: bbox.x + bbox.width / 2,
y: bbox.y + bbox.height / 2
};
};

// Emulate IE style mouseenter/mouseleave events, since Microsoft always
// does everything right.
// See: http://www.dynamic-tools.net/toolbox/isMouseLeaveOrEnter/
var events = ["mouseenter", "mouseleave"];

for (i in events) {
(function (event_name) {
var event_name = events[i];
Element.prototype[event_name] = function (fn, scope) {
if (Snap.is(fn, "function")) {
var fn2 = function (event) {
if (event.type != "mouseover" && event.type != "mouseout") {
return;
}

var reltg = event.relatedTarget ? event.relatedTarget :
event.type == "mouseout" ? event.toElement : event.fromElement;
while (reltg && reltg != this.node) reltg = reltg.parentNode;

if (reltg != this.node) {
return fn.apply(this, event);
}
};

if (event_name == "mouseenter") {
this.mouseover(fn2, scope);
} else {
this.mouseout(fn2, scope);
}
}
return this;
};
})(events[i]);
}

Element.prototype.mousewheel = function (fn, scope) {
if (Snap.is(fn, "function")) {
var el = this;
var fn2 = function (event) {
fn.apply(el, [event]);
};
}

/Firefox/i.test(navigator.userAgent) ? "DOMMouseScroll" : "mousewheel",
fn2);

return this;
};

// Snap's attr function can be too slow for things like panning/zooming.
// This is a function to directly update element attributes without going
// through eve.
Element.prototype.attribute = function(key, val) {
if (val === undefined) {
return this.node.getAttribute(key);
} else {
this.node.setAttribute(key, val);
return this;
}
};

this.mouseenter(function (event) {
init_pan_zoom(this.plotroot());
});
return this;
};
});

// When the plot is moused over, emphasize the grid lines.
var root = this.plotroot();

var keyboard_zoom = function(event) {
if (event.which == 187) { // plus
increase_zoom_by_position(root, 0.1, true);
} else if (event.which == 189) { // minus
increase_zoom_by_position(root, -0.1, true);
}
};
root.data("keyboard_zoom", keyboard_zoom);

var xgridlines = root.select(".xgridlines"),
ygridlines = root.select(".ygridlines");

xgridlines.data("unfocused_strokedash",
xgridlines.attribute("stroke-dasharray").replace(/(\d)(,|$)/g, "$1mm$2")); ygridlines.data("unfocused_strokedash", ygridlines.attribute("stroke-dasharray").replace(/(\d)(,|$)/g, "$1mm$2"));

// emphasize grid lines
var destcolor = root.data("focused_xgrid_color");
xgridlines.attribute("stroke-dasharray", "none")
.selectAll("path")
.animate({stroke: destcolor}, 250);

destcolor = root.data("focused_ygrid_color");
ygridlines.attribute("stroke-dasharray", "none")
.selectAll("path")
.animate({stroke: destcolor}, 250);

// reveal zoom slider
root.select(".zoomslider")
.animate({opacity: 1.0}, 250);
};

// Reset pan and zoom on double click
set_plot_pan_zoom(this.plotroot(), 0.0, 0.0, 1.0);
};

// Unemphasize grid lines on mouse out.
var root = this.plotroot();

window.removeEventListener("keyup", root.data("keyboard_zoom"));
root.data("keyboard_zoom", undefined);

var xgridlines = root.select(".xgridlines"),
ygridlines = root.select(".ygridlines");

var destcolor = root.data("unfocused_xgrid_color");

xgridlines.attribute("stroke-dasharray", xgridlines.data("unfocused_strokedash"))
.selectAll("path")
.animate({stroke: destcolor}, 250);

destcolor = root.data("unfocused_ygrid_color");
ygridlines.attribute("stroke-dasharray", ygridlines.data("unfocused_strokedash"))
.selectAll("path")
.animate({stroke: destcolor}, 250);

// hide zoom slider
root.select(".zoomslider")
.animate({opacity: 0.0}, 250);
};

var set_geometry_transform = function(root, tx, ty, scale) {
var xscalable = root.hasClass("xscalable"),
yscalable = root.hasClass("yscalable");

var old_scale = root.data("scale");

var xscale = xscalable ? scale : 1.0,
yscale = yscalable ? scale : 1.0;

tx = xscalable ? tx : 0.0;
ty = yscalable ? ty : 0.0;

var t = new Snap.Matrix().translate(tx, ty).scale(xscale, yscale);

root.selectAll(".geometry, image")
.forEach(function (element, i) {
element.transform(t);
});

bounds = root.plotbounds();

if (yscalable) {
var xfixed_t = new Snap.Matrix().translate(0, ty).scale(1.0, yscale);
root.selectAll(".xfixed")
.forEach(function (element, i) {
element.transform(xfixed_t);
});

root.select(".ylabels")
.transform(xfixed_t)
.selectAll("text")
.forEach(function (element, i) {
var cx = element.asPX("x"),
cy = element.asPX("y");
var st = element.data("static_transform");
unscale_t = new Snap.Matrix();
element.transform(unscale_t);

var y = cy * scale + ty;
element.attr("visibility",
bounds.y0 <= y && y <= bounds.y1 ? "visible" : "hidden");
}
});
}

if (xscalable) {
var yfixed_t = new Snap.Matrix().translate(tx, 0).scale(xscale, 1.0);
var xtrans = new Snap.Matrix().translate(tx, 0);
root.selectAll(".yfixed")
.forEach(function (element, i) {
element.transform(yfixed_t);
});

root.select(".xlabels")
.transform(yfixed_t)
.selectAll("text")
.forEach(function (element, i) {
var cx = element.asPX("x"),
cy = element.asPX("y");
var st = element.data("static_transform");
unscale_t = new Snap.Matrix();

element.transform(unscale_t);

var x = cx * scale + tx;
element.attr("visibility",
bounds.x0 <= x && x <= bounds.x1 ? "visible" : "hidden");
}
});
}

// we must unscale anything that is scale invariance: widths, raiduses, etc.
var size_attribs = ["font-size"];
var unscaled_selection = ".geometry, .geometry *";
if (xscalable) {
size_attribs.push("rx");
unscaled_selection += ", .xgridlines";
}
if (yscalable) {
size_attribs.push("ry");
unscaled_selection += ", .ygridlines";
}

root.selectAll(unscaled_selection)
.forEach(function (element, i) {
// circle need special help
if (element.node.nodeName == "circle") {
var cx = element.attribute("cx"),
cy = element.attribute("cy");
unscale_t = new Snap.Matrix().scale(1/xscale, 1/yscale,
cx, cy);
element.transform(unscale_t);
return;
}

for (i in size_attribs) {
var key = size_attribs[i];
var val = parseFloat(element.attribute(key));
if (val !== undefined && val != 0 && !isNaN(val)) {
element.attribute(key, val * old_scale / scale);
}
}
});
};

// Find the most appropriate tick scale and update label visibility.
var update_tickscale = function(root, scale, axis) {
if (!root.hasClass(axis + "scalable")) return;

var tickscales = root.data(axis + "tickscales");
var best_tickscale = 1.0;
var best_tickscale_dist = Infinity;
for (tickscale in tickscales) {
var dist = Math.abs(Math.log(tickscale) - Math.log(scale));
if (dist < best_tickscale_dist) {
best_tickscale_dist = dist;
best_tickscale = tickscale;
}
}

if (best_tickscale != root.data(axis + "tickscale")) {
root.data(axis + "tickscale", best_tickscale);
var mark_inscale_gridlines = function (element, i) {
var inscale = element.attr("gadfly:scale") == best_tickscale;
element.attr("visibility", inscale ? "visible" : "hidden");
};

var mark_inscale_labels = function (element, i) {
var inscale = element.attr("gadfly:scale") == best_tickscale;
element.attr("visibility", inscale ? "visible" : "hidden");
};

root.select("." + axis + "gridlines").selectAll("path").forEach(mark_inscale_gridlines);
root.select("." + axis + "labels").selectAll("text").forEach(mark_inscale_labels);
}
};

var set_plot_pan_zoom = function(root, tx, ty, scale) {
var old_scale = root.data("scale");
var bounds = root.plotbounds();

var width = bounds.x1 - bounds.x0,
height = bounds.y1 - bounds.y0;

// compute the viewport derived from tx, ty, and scale
var x_min = -width * scale - (scale * width - width),
x_max = width * scale,
y_min = -height * scale - (scale * height - height),
y_max = height * scale;

var x0 = bounds.x0 - scale * bounds.x0,
y0 = bounds.y0 - scale * bounds.y0;

var tx = Math.max(Math.min(tx - x0, x_max), x_min),
ty = Math.max(Math.min(ty - y0, y_max), y_min);

tx += x0;
ty += y0;

// when the scale change, we may need to alter which set of
// ticks is being displayed
if (scale != old_scale) {
update_tickscale(root, scale, "x");
update_tickscale(root, scale, "y");
}

set_geometry_transform(root, tx, ty, scale);

root.data("scale", scale);
root.data("tx", tx);
root.data("ty", ty);
};

var scale_centered_translation = function(root, scale) {
var bounds = root.plotbounds();

var width = bounds.x1 - bounds.x0,
height = bounds.y1 - bounds.y0;

var tx0 = root.data("tx"),
ty0 = root.data("ty");

var scale0 = root.data("scale");

// how off from center the current view is
var xoff = tx0 - (bounds.x0 * (1 - scale0) + (width * (1 - scale0)) / 2),
yoff = ty0 - (bounds.y0 * (1 - scale0) + (height * (1 - scale0)) / 2);

// rescale offsets
xoff = xoff * scale / scale0;
yoff = yoff * scale / scale0;

// adjust for the panel position being scaled
var x_edge_adjust = bounds.x0 * (1 - scale),
y_edge_adjust = bounds.y0 * (1 - scale);

return {
x: xoff + x_edge_adjust + (width - width * scale) / 2,
y: yoff + y_edge_adjust + (height - height * scale) / 2
};
};

// Initialize data for panning zooming if it isn't already.
var init_pan_zoom = function(root) {
return;
}

// The non-scaling-stroke trick. Rather than try to correct for the
// stroke-width when zooming, we force it to a fixed value.
var px_per_mm = root.node.getCTM().a;

// Drag events report deltas in pixels, which we'd like to convert to
// millimeters.
root.data("px_per_mm", px_per_mm);

root.selectAll("path")
.forEach(function (element, i) {
sw = element.asPX("stroke-width") * px_per_mm;
if (sw > 0) {
element.attribute("stroke-width", sw);
element.attribute("vector-effect", "non-scaling-stroke");
}
});

// Store ticks labels original tranformation
root.selectAll(".xlabels > text, .ylabels > text")
.forEach(function (element, i) {
var lm = element.transform().localMatrix;
element.data("static_transform",
new Snap.Matrix(lm.a, lm.b, lm.c, lm.d, lm.e, lm.f));
});

var xgridlines = root.select(".xgridlines");
var ygridlines = root.select(".ygridlines");
var xlabels = root.select(".xlabels");
var ylabels = root.select(".ylabels");

if (root.data("tx") === undefined) root.data("tx", 0);
if (root.data("ty") === undefined) root.data("ty", 0);
if (root.data("scale") === undefined) root.data("scale", 1.0);
if (root.data("xtickscales") === undefined) {

// index all the tick scales that are listed
var xtickscales = {};
var ytickscales = {};
var add_x_tick_scales = function (element, i) {
};
var add_y_tick_scales = function (element, i) {
};

root.data("xtickscales", xtickscales);
root.data("ytickscales", ytickscales);
root.data("xtickscale", 1.0);
}

var min_scale = 1.0, max_scale = 1.0;
for (scale in xtickscales) {
min_scale = Math.min(min_scale, scale);
max_scale = Math.max(max_scale, scale);
}
for (scale in ytickscales) {
min_scale = Math.min(min_scale, scale);
max_scale = Math.max(max_scale, scale);
}
root.data("min_scale", min_scale);
root.data("max_scale", max_scale);

// store the original positions of labels
if (xlabels) {
xlabels.selectAll("text")
.forEach(function (element, i) {
element.data("x", element.asPX("x"));
});
}

if (ylabels) {
ylabels.selectAll("text")
.forEach(function (element, i) {
element.data("y", element.asPX("y"));
});
}

// mark grid lines and ticks as in or out of scale.
var mark_inscale = function (element, i) {
};

if (xgridlines) xgridlines.selectAll("path").forEach(mark_inscale);
if (ygridlines) ygridlines.selectAll("path").forEach(mark_inscale);
if (xlabels) xlabels.selectAll("text").forEach(mark_inscale);
if (ylabels) ylabels.selectAll("text").forEach(mark_inscale);

// figure out the upper ond lower bounds on panning using the maximum
// and minum grid lines
var bounds = root.plotbounds();
var pan_bounds = {
x0: 0.0,
y0: 0.0,
x1: 0.0,
y1: 0.0
};

if (xgridlines) {
xgridlines
.selectAll("path")
.forEach(function (element, i) {
var bbox = element.node.getBBox();
if (bounds.x1 - bbox.x < pan_bounds.x0) {
pan_bounds.x0 = bounds.x1 - bbox.x;
}
if (bounds.x0 - bbox.x > pan_bounds.x1) {
pan_bounds.x1 = bounds.x0 - bbox.x;
}
element.attr("visibility", "visible");
}
});
}

if (ygridlines) {
ygridlines
.selectAll("path")
.forEach(function (element, i) {
var bbox = element.node.getBBox();
if (bounds.y1 - bbox.y < pan_bounds.y0) {
pan_bounds.y0 = bounds.y1 - bbox.y;
}
if (bounds.y0 - bbox.y > pan_bounds.y1) {
pan_bounds.y1 = bounds.y0 - bbox.y;
}
element.attr("visibility", "visible");
}
});
}

// nudge these values a little
pan_bounds.x0 -= 5;
pan_bounds.x1 += 5;
pan_bounds.y0 -= 5;
pan_bounds.y1 += 5;
root.data("pan_bounds", pan_bounds);

};

// drag actions, i.e. zooming and panning
var pan_action = {
start: function(root, x, y, event) {
root.data("dx", 0);
root.data("dy", 0);
root.data("tx0", root.data("tx"));
root.data("ty0", root.data("ty"));
},
update: function(root, dx, dy, x, y, event) {
var px_per_mm = root.data("px_per_mm");
dx /= px_per_mm;
dy /= px_per_mm;

var tx0 = root.data("tx"),
ty0 = root.data("ty");

var dx0 = root.data("dx"),
dy0 = root.data("dy");

root.data("dx", dx);
root.data("dy", dy);

dx = dx - dx0;
dy = dy - dy0;

var tx = tx0 + dx,
ty = ty0 + dy;

set_plot_pan_zoom(root, tx, ty, root.data("scale"));
},
end: function(root, event) {

},
cancel: function(root) {
set_plot_pan_zoom(root, root.data("tx0"), root.data("ty0"), root.data("scale"));
}
};

var zoom_box;
var zoom_action = {
start: function(root, x, y, event) {
var bounds = root.plotbounds();
var width = bounds.x1 - bounds.x0,
height = bounds.y1 - bounds.y0;
var ratio = width / height;
var xscalable = root.hasClass("xscalable"),
yscalable = root.hasClass("yscalable");
var px_per_mm = root.data("px_per_mm");
x = xscalable ? x / px_per_mm : bounds.x0;
y = yscalable ? y / px_per_mm : bounds.y0;
var w = xscalable ? 0 : width;
var h = yscalable ? 0 : height;
zoom_box = root.rect(x, y, w, h).attr({
"fill": "#000",
"opacity": 0.25
});
zoom_box.data("ratio", ratio);
},
update: function(root, dx, dy, x, y, event) {
var xscalable = root.hasClass("xscalable"),
yscalable = root.hasClass("yscalable");
var px_per_mm = root.data("px_per_mm");
var bounds = root.plotbounds();
if (yscalable) {
y /= px_per_mm;
y = Math.max(bounds.y0, y);
y = Math.min(bounds.y1, y);
} else {
y = bounds.y1;
}
if (xscalable) {
x /= px_per_mm;
x = Math.max(bounds.x0, x);
x = Math.min(bounds.x1, x);
} else {
x = bounds.x1;
}

dx = x - zoom_box.attr("x");
dy = y - zoom_box.attr("y");
if (xscalable && yscalable) {
var ratio = zoom_box.data("ratio");
var width = Math.min(Math.abs(dx), ratio * Math.abs(dy));
var height = Math.min(Math.abs(dy), Math.abs(dx) / ratio);
dx = width * dx / Math.abs(dx);
dy = height * dy / Math.abs(dy);
}
var xoffset = 0,
yoffset = 0;
if (dx < 0) {
xoffset = dx;
dx = -1 * dx;
}
if (dy < 0) {
yoffset = dy;
dy = -1 * dy;
}
if (isNaN(dy)) {
dy = 0.0;
}
if (isNaN(dx)) {
dx = 0.0;
}
zoom_box.transform("T" + xoffset + "," + yoffset);
zoom_box.attr("width", dx);
zoom_box.attr("height", dy);
},
end: function(root, event) {
var xscalable = root.hasClass("xscalable"),
yscalable = root.hasClass("yscalable");
var zoom_bounds = zoom_box.getBBox();
if (zoom_bounds.width * zoom_bounds.height <= 0) {
return;
}
var plot_bounds = root.plotbounds();
var zoom_factor = 1.0;
if (yscalable) {
zoom_factor = (plot_bounds.y1 - plot_bounds.y0) / zoom_bounds.height;
} else {
zoom_factor = (plot_bounds.x1 - plot_bounds.x0) / zoom_bounds.width;
}
var tx = (root.data("tx") - zoom_bounds.x) * zoom_factor + plot_bounds.x0,
ty = (root.data("ty") - zoom_bounds.y) * zoom_factor + plot_bounds.y0;
set_plot_pan_zoom(root, tx, ty, root.data("scale") * zoom_factor);
zoom_box.remove();
},
cancel: function(root) {
zoom_box.remove();
}
};

Gadfly.guide_background_drag_onstart = function(x, y, event) {
var root = this.plotroot();
var scalable = root.hasClass("xscalable") || root.hasClass("yscalable");
var zoomable = !event.altKey && !event.ctrlKey && event.shiftKey && scalable;
var panable = !event.altKey && !event.ctrlKey && !event.shiftKey && scalable;
var drag_action = zoomable ? zoom_action :
panable  ? pan_action :
undefined;
root.data("drag_action", drag_action);
if (drag_action) {
var cancel_drag_action = function(event) {
if (event.which == 27) { // esc key
drag_action.cancel(root);
root.data("drag_action", undefined);
}
};
root.data("cancel_drag_action", cancel_drag_action);
drag_action.start(root, x, y, event);
}
};

Gadfly.guide_background_drag_onmove = function(dx, dy, x, y, event) {
var root = this.plotroot();
var drag_action = root.data("drag_action");
if (drag_action) {
drag_action.update(root, dx, dy, x, y, event);
}
};

var root = this.plotroot();
window.removeEventListener("keyup", root.data("cancel_drag_action"));
root.data("cancel_drag_action", undefined);
var drag_action = root.data("drag_action");
if (drag_action) {
drag_action.end(root, event);
}
root.data("drag_action", undefined);
};

if (event.shiftKey) {
increase_zoom_by_position(this.plotroot(), 0.001 * event.wheelDelta);
event.preventDefault();
}
};

this.select(".button_logo")
.animate({fill: this.data("mouseover_color")}, 100);
};

this.select(".button_logo")
.animate({fill: this.data("mouseout_color")}, 100);
};

increase_zoom_by_position(this.plotroot(), -0.1, true);
};

increase_zoom_by_position(this.plotroot(), 0.1, true);
};

// TODO
};

// Map slider position x to scale y using the function y = a*exp(b*x)+c.
// The constants a, b, and c are solved using the constraint that the function
// should go through the points (0; min_scale), (0.5; 1), and (1; max_scale).
var scale_from_slider_position = function(position, min_scale, max_scale) {
var a = (1 - 2 * min_scale + min_scale * min_scale) / (min_scale + max_scale - 2),
b = 2 * Math.log((max_scale - 1) / (1 - min_scale)),
c = (min_scale * max_scale - 1) / (min_scale + max_scale - 2);
return a * Math.exp(b * position) + c;
}

// inverse of scale_from_slider_position
var slider_position_from_scale = function(scale, min_scale, max_scale) {
var a = (1 - 2 * min_scale + min_scale * min_scale) / (min_scale + max_scale - 2),
b = 2 * Math.log((max_scale - 1) / (1 - min_scale)),
c = (min_scale * max_scale - 1) / (min_scale + max_scale - 2);
return 1 / b * Math.log((scale - c) / a);
}

var increase_zoom_by_position = function(root, delta_position, animate) {
var scale = root.data("scale"),
min_scale = root.data("min_scale"),
max_scale = root.data("max_scale");
var position = slider_position_from_scale(scale, min_scale, max_scale);
position += delta_position;
scale = scale_from_slider_position(position, min_scale, max_scale);
set_zoom(root, scale, animate);
}

var set_zoom = function(root, scale, animate) {
var min_scale = root.data("min_scale"),
max_scale = root.data("max_scale"),
old_scale = root.data("scale");
var new_scale = Math.max(min_scale, Math.min(scale, max_scale));
if (animate) {
Snap.animate(
old_scale,
new_scale,
function (new_scale) {
update_plot_scale(root, new_scale);
},
200);
} else {
update_plot_scale(root, new_scale);
}
}

var update_plot_scale = function(root, new_scale) {
var trans = scale_centered_translation(root, new_scale);
set_plot_pan_zoom(root, trans.x, trans.y, new_scale);

root.selectAll(".zoomslider_thumb")
.forEach(function (element, i) {
var min_pos = element.data("min_pos"),
max_pos = element.data("max_pos"),
min_scale = root.data("min_scale"),
max_scale = root.data("max_scale");
var xmid = (min_pos + max_pos) / 2;
var xpos = slider_position_from_scale(new_scale, min_scale, max_scale);
element.transform(new Snap.Matrix().translate(
Math.max(min_pos, Math.min(
max_pos, min_pos + (max_pos - min_pos) * xpos)) - xmid, 0));
});
};

Gadfly.zoomslider_thumb_dragmove = function(dx, dy, x, y, event) {
var root = this.plotroot();
var min_pos = this.data("min_pos"),
max_pos = this.data("max_pos"),
min_scale = root.data("min_scale"),
max_scale = root.data("max_scale"),
old_scale = root.data("old_scale");

var px_per_mm = root.data("px_per_mm");
dx /= px_per_mm;
dy /= px_per_mm;

var xmid = (min_pos + max_pos) / 2;
var xpos = slider_position_from_scale(old_scale, min_scale, max_scale) +
dx / (max_pos - min_pos);

// compute the new scale
var new_scale = scale_from_slider_position(xpos, min_scale, max_scale);
new_scale = Math.min(max_scale, Math.max(min_scale, new_scale));

update_plot_scale(root, new_scale);
event.stopPropagation();
};

Gadfly.zoomslider_thumb_dragstart = function(x, y, event) {
this.animate({fill: this.data("mouseover_color")}, 100);
var root = this.plotroot();

// keep track of what the scale was when we started dragging
root.data("old_scale", root.data("scale"));
event.stopPropagation();
};

this.animate({fill: this.data("mouseout_color")}, 100);
event.stopPropagation();
};

var toggle_color_class = function(root, color_class, ison) {
var guides = root.selectAll(".guide." + color_class + ",.guide ." + color_class);
var geoms = root.selectAll(".geometry." + color_class + ",.geometry ." + color_class);
if (ison) {
guides.animate({opacity: 0.5}, 250);
geoms.animate({opacity: 0.0}, 250);
} else {
guides.animate({opacity: 1.0}, 250);
geoms.animate({opacity: 1.0}, 250);
}
};

var root = this.plotroot();
var color_class = this.data("color_class");

if (event.shiftKey) {
root.selectAll(".colorkey text")
.forEach(function (element) {
var other_color_class = element.data("color_class");
if (other_color_class != color_class) {
toggle_color_class(root, other_color_class,
element.attr("opacity") == 1.0);
}
});
} else {
toggle_color_class(root, color_class, this.attr("opacity") == 1.0);
}
};

}));

(function (glob, factory) {
// AMD support
if (typeof require === "function" && typeof define === "function" && define.amd) {
});
} else {
}
var fig = Snap("#img-46920db4");
fig.select("#img-46920db4-5")
fig.select("#img-46920db4-7")
.plotroot().data("unfocused_ygrid_color", "#D0D0E0")
;
fig.select("#img-46920db4-7")
.plotroot().data("focused_ygrid_color", "#A0A0A0")
;
fig.select("#img-46920db4-8")
.plotroot().data("unfocused_xgrid_color", "#D0D0E0")
;
fig.select("#img-46920db4-8")
.plotroot().data("focused_xgrid_color", "#A0A0A0")
;
fig.select("#img-46920db4-13")
.data("mouseover_color", "#CD5C5C")
;
fig.select("#img-46920db4-13")
.data("mouseout_color", "#6A6A6A")
;
fig.select("#img-46920db4-13")
;
fig.select("#img-46920db4-15")
.data("max_pos", 129)
;
fig.select("#img-46920db4-15")
.data("min_pos", 112)
;
fig.select("#img-46920db4-15")
fig.select("#img-46920db4-16")
.data("max_pos", 129)
;
fig.select("#img-46920db4-16")
.data("min_pos", 112)
;
fig.select("#img-46920db4-16")
.data("mouseover_color", "#CD5C5C")
;
fig.select("#img-46920db4-16")
.data("mouseout_color", "#6A6A6A")
;
fig.select("#img-46920db4-16")
;
fig.select("#img-46920db4-17")
.data("mouseover_color", "#CD5C5C")
;
fig.select("#img-46920db4-17")
.data("mouseout_color", "#6A6A6A")
;
fig.select("#img-46920db4-17")
;
});

Probability that P(0,∞)>0.5 is 0.88956