short-tandem.ipynb

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   "source": [
    "# Borel Numbers, Zipf's Law, and Short Tandem Repeats"
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   "source": [
    "Some house keeping first, make sure we have statistics and plotting packages available"
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   "source": [
    "using(Gadfly)"
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   "source": [
    "We will start by building a random generator of short tandem repeats"
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       "randomrepeats (generic function with 1 method)"
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   "source": [
    "function randomrepeats(n,m)\n",
    "    d = [string([\"A\", \"C\", \"G\", \"T\"][digits(l,4,m) + 1]...) => 0 for l in 0:(4^m - 1)]\n",
    "    for i = 0:(n - 1)\n",
    "        d[string(rand([\"A\",\"G\",\"C\",\"T\"],m)...)] += 1\n",
    "    end\n",
    "    return d\n",
    "end"
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   "source": [
    "This functions works by:\n",
    "\n",
    "1. Preallocating a dictionary of every possible sequence of length $m$, filled with zeros.\n",
    "2. Creating $n$ random short sequences of lenght $m$.\n",
    "3. Using the $n$ short sequences as a key to increment the dictionary slot."
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   "source": [
    "Lets give this bad boy a whirl!"
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       "Dict{ASCIIString,Int64} with 16 entries:\n",
       "  \"CC\" => 1\n",
       "  \"GC\" => 1\n",
       "  \"GG\" => 0\n",
       "  \"CG\" => 0\n",
       "  \"AT\" => 1\n",
       "  \"CA\" => 0\n",
       "  \"TG\" => 1\n",
       "  \"TA\" => 0\n",
       "  \"GT\" => 0\n",
       "  \"GA\" => 2\n",
       "  \"TT\" => 3\n",
       "  \"AC\" => 0\n",
       "  \"CT\" => 1\n",
       "  \"AA\" => 0\n",
       "  \"AG\" => 0\n",
       "  \"TC\" => 0"
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    "randomrepeats(10,2)"
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   "source": [
    "Lets generate a larger sample and generate a histogram"
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       "                        this.mouseover(fn2, scope);\n",
       "                    } else {\n",
       "                        this.mouseout(fn2, scope);\n",
       "                    }\n",
       "                }\n",
       "                return this;\n",
       "            };\n",
       "        })(events[i]);\n",
       "    }\n",
       "\n",
       "\n",
       "    Element.prototype.mousewheel = function (fn, scope) {\n",
       "        if (Snap.is(fn, \"function\")) {\n",
       "            var el = this;\n",
       "            var fn2 = function (event) {\n",
       "                fn.apply(el, [event]);\n",
       "            };\n",
       "        }\n",
       "\n",
       "        this.node.addEventListener(\n",
       "            /Firefox/i.test(navigator.userAgent) ? \"DOMMouseScroll\" : \"mousewheel\",\n",
       "            fn2);\n",
       "\n",
       "        return this;\n",
       "    };\n",
       "\n",
       "\n",
       "    // Snap's attr function can be too slow for things like panning/zooming.\n",
       "    // This is a function to directly update element attributes without going\n",
       "    // through eve.\n",
       "    Element.prototype.attribute = function(key, val) {\n",
       "        if (val === undefined) {\n",
       "            return this.node.getAttribute(key);\n",
       "        } else {\n",
       "            this.node.setAttribute(key, val);\n",
       "            return this;\n",
       "        }\n",
       "    };\n",
       "\n",
       "    Element.prototype.init_gadfly = function() {\n",
       "        this.mouseenter(Gadfly.plot_mouseover)\n",
       "            .mouseleave(Gadfly.plot_mouseout)\n",
       "            .dblclick(Gadfly.plot_dblclick)\n",
       "            .mousewheel(Gadfly.guide_background_scroll)\n",
       "            .drag(Gadfly.guide_background_drag_onmove,\n",
       "                  Gadfly.guide_background_drag_onstart,\n",
       "                  Gadfly.guide_background_drag_onend);\n",
       "        this.mouseenter(function (event) {\n",
       "            init_pan_zoom(this.plotroot());\n",
       "        });\n",
       "        return this;\n",
       "    };\n",
       "});\n",
       "\n",
       "\n",
       "// When the plot is moused over, emphasize the grid lines.\n",
       "Gadfly.plot_mouseover = function(event) {\n",
       "    var root = this.plotroot();\n",
       "\n",
       "    var keyboard_zoom = function(event) {\n",
       "        if (event.which == 187) { // plus\n",
       "            increase_zoom_by_position(root, 0.1, true);\n",
       "        } else if (event.which == 189) { // minus\n",
       "            increase_zoom_by_position(root, -0.1, true);\n",
       "        }\n",
       "    };\n",
       "    root.data(\"keyboard_zoom\", keyboard_zoom);\n",
       "    window.addEventListener(\"keyup\", keyboard_zoom);\n",
       "\n",
       "    var xgridlines = root.select(\".xgridlines\"),\n",
       "        ygridlines = root.select(\".ygridlines\");\n",
       "\n",
       "    xgridlines.data(\"unfocused_strokedash\",\n",
       "                    xgridlines.attribute(\"stroke-dasharray\").replace(/(\\d)(,|$)/g, \"$1mm$2\"));\n",
       "    ygridlines.data(\"unfocused_strokedash\",\n",
       "                    ygridlines.attribute(\"stroke-dasharray\").replace(/(\\d)(,|$)/g, \"$1mm$2\"));\n",
       "\n",
       "    // emphasize grid lines\n",
       "    var destcolor = root.data(\"focused_xgrid_color\");\n",
       "    xgridlines.attribute(\"stroke-dasharray\", \"none\")\n",
       "              .selectAll(\"path\")\n",
       "              .animate({stroke: destcolor}, 250);\n",
       "\n",
       "    destcolor = root.data(\"focused_ygrid_color\");\n",
       "    ygridlines.attribute(\"stroke-dasharray\", \"none\")\n",
       "              .selectAll(\"path\")\n",
       "              .animate({stroke: destcolor}, 250);\n",
       "\n",
       "    // reveal zoom slider\n",
       "    root.select(\".zoomslider\")\n",
       "        .animate({opacity: 1.0}, 250);\n",
       "};\n",
       "\n",
       "// Reset pan and zoom on double click\n",
       "Gadfly.plot_dblclick = function(event) {\n",
       "  set_plot_pan_zoom(this.plotroot(), 0.0, 0.0, 1.0);\n",
       "};\n",
       "\n",
       "// Unemphasize grid lines on mouse out.\n",
       "Gadfly.plot_mouseout = function(event) {\n",
       "    var root = this.plotroot();\n",
       "\n",
       "    window.removeEventListener(\"keyup\", root.data(\"keyboard_zoom\"));\n",
       "    root.data(\"keyboard_zoom\", undefined);\n",
       "\n",
       "    var xgridlines = root.select(\".xgridlines\"),\n",
       "        ygridlines = root.select(\".ygridlines\");\n",
       "\n",
       "    var destcolor = root.data(\"unfocused_xgrid_color\");\n",
       "\n",
       "    xgridlines.attribute(\"stroke-dasharray\", xgridlines.data(\"unfocused_strokedash\"))\n",
       "              .selectAll(\"path\")\n",
       "              .animate({stroke: destcolor}, 250);\n",
       "\n",
       "    destcolor = root.data(\"unfocused_ygrid_color\");\n",
       "    ygridlines.attribute(\"stroke-dasharray\", ygridlines.data(\"unfocused_strokedash\"))\n",
       "              .selectAll(\"path\")\n",
       "              .animate({stroke: destcolor}, 250);\n",
       "\n",
       "    // hide zoom slider\n",
       "    root.select(\".zoomslider\")\n",
       "        .animate({opacity: 0.0}, 250);\n",
       "};\n",
       "\n",
       "\n",
       "var set_geometry_transform = function(root, tx, ty, scale) {\n",
       "    var xscalable = root.hasClass(\"xscalable\"),\n",
       "        yscalable = root.hasClass(\"yscalable\");\n",
       "\n",
       "    var old_scale = root.data(\"scale\");\n",
       "\n",
       "    var xscale = xscalable ? scale : 1.0,\n",
       "        yscale = yscalable ? scale : 1.0;\n",
       "\n",
       "    tx = xscalable ? tx : 0.0;\n",
       "    ty = yscalable ? ty : 0.0;\n",
       "\n",
       "    var t = new Snap.Matrix().translate(tx, ty).scale(xscale, yscale);\n",
       "\n",
       "    root.selectAll(\".geometry, image\")\n",
       "        .forEach(function (element, i) {\n",
       "            element.transform(t);\n",
       "        });\n",
       "\n",
       "    bounds = root.plotbounds();\n",
       "\n",
       "    if (yscalable) {\n",
       "        var xfixed_t = new Snap.Matrix().translate(0, ty).scale(1.0, yscale);\n",
       "        root.selectAll(\".xfixed\")\n",
       "            .forEach(function (element, i) {\n",
       "                element.transform(xfixed_t);\n",
       "            });\n",
       "\n",
       "        root.select(\".ylabels\")\n",
       "            .transform(xfixed_t)\n",
       "            .selectAll(\"text\")\n",
       "            .forEach(function (element, i) {\n",
       "                if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
       "                    var cx = element.asPX(\"x\"),\n",
       "                        cy = element.asPX(\"y\");\n",
       "                    var st = element.data(\"static_transform\");\n",
       "                    unscale_t = new Snap.Matrix();\n",
       "                    unscale_t.scale(1, 1/scale, cx, cy).add(st);\n",
       "                    element.transform(unscale_t);\n",
       "\n",
       "                    var y = cy * scale + ty;\n",
       "                    element.attr(\"visibility\",\n",
       "                        bounds.y0 <= y && y <= bounds.y1 ? \"visible\" : \"hidden\");\n",
       "                }\n",
       "            });\n",
       "    }\n",
       "\n",
       "    if (xscalable) {\n",
       "        var yfixed_t = new Snap.Matrix().translate(tx, 0).scale(xscale, 1.0);\n",
       "        var xtrans = new Snap.Matrix().translate(tx, 0);\n",
       "        root.selectAll(\".yfixed\")\n",
       "            .forEach(function (element, i) {\n",
       "                element.transform(yfixed_t);\n",
       "            });\n",
       "\n",
       "        root.select(\".xlabels\")\n",
       "            .transform(yfixed_t)\n",
       "            .selectAll(\"text\")\n",
       "            .forEach(function (element, i) {\n",
       "                if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
       "                    var cx = element.asPX(\"x\"),\n",
       "                        cy = element.asPX(\"y\");\n",
       "                    var st = element.data(\"static_transform\");\n",
       "                    unscale_t = new Snap.Matrix();\n",
       "                    unscale_t.scale(1/scale, 1, cx, cy).add(st);\n",
       "\n",
       "                    element.transform(unscale_t);\n",
       "\n",
       "                    var x = cx * scale + tx;\n",
       "                    element.attr(\"visibility\",\n",
       "                        bounds.x0 <= x && x <= bounds.x1 ? \"visible\" : \"hidden\");\n",
       "                    }\n",
       "            });\n",
       "    }\n",
       "\n",
       "    // we must unscale anything that is scale invariance: widths, raiduses, etc.\n",
       "    var size_attribs = [\"font-size\"];\n",
       "    var unscaled_selection = \".geometry, .geometry *\";\n",
       "    if (xscalable) {\n",
       "        size_attribs.push(\"rx\");\n",
       "        unscaled_selection += \", .xgridlines\";\n",
       "    }\n",
       "    if (yscalable) {\n",
       "        size_attribs.push(\"ry\");\n",
       "        unscaled_selection += \", .ygridlines\";\n",
       "    }\n",
       "\n",
       "    root.selectAll(unscaled_selection)\n",
       "        .forEach(function (element, i) {\n",
       "            // circle need special help\n",
       "            if (element.node.nodeName == \"circle\") {\n",
       "                var cx = element.attribute(\"cx\"),\n",
       "                    cy = element.attribute(\"cy\");\n",
       "                unscale_t = new Snap.Matrix().scale(1/xscale, 1/yscale,\n",
       "                                                        cx, cy);\n",
       "                element.transform(unscale_t);\n",
       "                return;\n",
       "            }\n",
       "\n",
       "            for (i in size_attribs) {\n",
       "                var key = size_attribs[i];\n",
       "                var val = parseFloat(element.attribute(key));\n",
       "                if (val !== undefined && val != 0 && !isNaN(val)) {\n",
       "                    element.attribute(key, val * old_scale / scale);\n",
       "                }\n",
       "            }\n",
       "        });\n",
       "};\n",
       "\n",
       "\n",
       "// Find the most appropriate tick scale and update label visibility.\n",
       "var update_tickscale = function(root, scale, axis) {\n",
       "    if (!root.hasClass(axis + \"scalable\")) return;\n",
       "\n",
       "    var tickscales = root.data(axis + \"tickscales\");\n",
       "    var best_tickscale = 1.0;\n",
       "    var best_tickscale_dist = Infinity;\n",
       "    for (tickscale in tickscales) {\n",
       "        var dist = Math.abs(Math.log(tickscale) - Math.log(scale));\n",
       "        if (dist < best_tickscale_dist) {\n",
       "            best_tickscale_dist = dist;\n",
       "            best_tickscale = tickscale;\n",
       "        }\n",
       "    }\n",
       "\n",
       "    if (best_tickscale != root.data(axis + \"tickscale\")) {\n",
       "        root.data(axis + \"tickscale\", best_tickscale);\n",
       "        var mark_inscale_gridlines = function (element, i) {\n",
       "            var inscale = element.attr(\"gadfly:scale\") == best_tickscale;\n",
       "            element.attribute(\"gadfly:inscale\", inscale);\n",
       "            element.attr(\"visibility\", inscale ? \"visible\" : \"hidden\");\n",
       "        };\n",
       "\n",
       "        var mark_inscale_labels = function (element, i) {\n",
       "            var inscale = element.attr(\"gadfly:scale\") == best_tickscale;\n",
       "            element.attribute(\"gadfly:inscale\", inscale);\n",
       "            element.attr(\"visibility\", inscale ? \"visible\" : \"hidden\");\n",
       "        };\n",
       "\n",
       "        root.select(\".\" + axis + \"gridlines\").selectAll(\"path\").forEach(mark_inscale_gridlines);\n",
       "        root.select(\".\" + axis + \"labels\").selectAll(\"text\").forEach(mark_inscale_labels);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "var set_plot_pan_zoom = function(root, tx, ty, scale) {\n",
       "    var old_scale = root.data(\"scale\");\n",
       "    var bounds = root.plotbounds();\n",
       "\n",
       "    var width = bounds.x1 - bounds.x0,\n",
       "        height = bounds.y1 - bounds.y0;\n",
       "\n",
       "    // compute the viewport derived from tx, ty, and scale\n",
       "    var x_min = -width * scale - (scale * width - width),\n",
       "        x_max = width * scale,\n",
       "        y_min = -height * scale - (scale * height - height),\n",
       "        y_max = height * scale;\n",
       "\n",
       "    var x0 = bounds.x0 - scale * bounds.x0,\n",
       "        y0 = bounds.y0 - scale * bounds.y0;\n",
       "\n",
       "    var tx = Math.max(Math.min(tx - x0, x_max), x_min),\n",
       "        ty = Math.max(Math.min(ty - y0, y_max), y_min);\n",
       "\n",
       "    tx += x0;\n",
       "    ty += y0;\n",
       "\n",
       "    // when the scale change, we may need to alter which set of\n",
       "    // ticks is being displayed\n",
       "    if (scale != old_scale) {\n",
       "        update_tickscale(root, scale, \"x\");\n",
       "        update_tickscale(root, scale, \"y\");\n",
       "    }\n",
       "\n",
       "    set_geometry_transform(root, tx, ty, scale);\n",
       "\n",
       "    root.data(\"scale\", scale);\n",
       "    root.data(\"tx\", tx);\n",
       "    root.data(\"ty\", ty);\n",
       "};\n",
       "\n",
       "\n",
       "var scale_centered_translation = function(root, scale) {\n",
       "    var bounds = root.plotbounds();\n",
       "\n",
       "    var width = bounds.x1 - bounds.x0,\n",
       "        height = bounds.y1 - bounds.y0;\n",
       "\n",
       "    var tx0 = root.data(\"tx\"),\n",
       "        ty0 = root.data(\"ty\");\n",
       "\n",
       "    var scale0 = root.data(\"scale\");\n",
       "\n",
       "    // how off from center the current view is\n",
       "    var xoff = tx0 - (bounds.x0 * (1 - scale0) + (width * (1 - scale0)) / 2),\n",
       "        yoff = ty0 - (bounds.y0 * (1 - scale0) + (height * (1 - scale0)) / 2);\n",
       "\n",
       "    // rescale offsets\n",
       "    xoff = xoff * scale / scale0;\n",
       "    yoff = yoff * scale / scale0;\n",
       "\n",
       "    // adjust for the panel position being scaled\n",
       "    var x_edge_adjust = bounds.x0 * (1 - scale),\n",
       "        y_edge_adjust = bounds.y0 * (1 - scale);\n",
       "\n",
       "    return {\n",
       "        x: xoff + x_edge_adjust + (width - width * scale) / 2,\n",
       "        y: yoff + y_edge_adjust + (height - height * scale) / 2\n",
       "    };\n",
       "};\n",
       "\n",
       "\n",
       "// Initialize data for panning zooming if it isn't already.\n",
       "var init_pan_zoom = function(root) {\n",
       "    if (root.data(\"zoompan-ready\")) {\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    // The non-scaling-stroke trick. Rather than try to correct for the\n",
       "    // stroke-width when zooming, we force it to a fixed value.\n",
       "    var px_per_mm = root.node.getCTM().a;\n",
       "\n",
       "    // Drag events report deltas in pixels, which we'd like to convert to\n",
       "    // millimeters.\n",
       "    root.data(\"px_per_mm\", px_per_mm);\n",
       "\n",
       "    root.selectAll(\"path\")\n",
       "        .forEach(function (element, i) {\n",
       "        sw = element.asPX(\"stroke-width\") * px_per_mm;\n",
       "        if (sw > 0) {\n",
       "            element.attribute(\"stroke-width\", sw);\n",
       "            element.attribute(\"vector-effect\", \"non-scaling-stroke\");\n",
       "        }\n",
       "    });\n",
       "\n",
       "    // Store ticks labels original tranformation\n",
       "    root.selectAll(\".xlabels > text, .ylabels > text\")\n",
       "        .forEach(function (element, i) {\n",
       "            var lm = element.transform().localMatrix;\n",
       "            element.data(\"static_transform\",\n",
       "                new Snap.Matrix(lm.a, lm.b, lm.c, lm.d, lm.e, lm.f));\n",
       "        });\n",
       "\n",
       "    var xgridlines = root.select(\".xgridlines\");\n",
       "    var ygridlines = root.select(\".ygridlines\");\n",
       "    var xlabels = root.select(\".xlabels\");\n",
       "    var ylabels = root.select(\".ylabels\");\n",
       "\n",
       "    if (root.data(\"tx\") === undefined) root.data(\"tx\", 0);\n",
       "    if (root.data(\"ty\") === undefined) root.data(\"ty\", 0);\n",
       "    if (root.data(\"scale\") === undefined) root.data(\"scale\", 1.0);\n",
       "    if (root.data(\"xtickscales\") === undefined) {\n",
       "\n",
       "        // index all the tick scales that are listed\n",
       "        var xtickscales = {};\n",
       "        var ytickscales = {};\n",
       "        var add_x_tick_scales = function (element, i) {\n",
       "            xtickscales[element.attribute(\"gadfly:scale\")] = true;\n",
       "        };\n",
       "        var add_y_tick_scales = function (element, i) {\n",
       "            ytickscales[element.attribute(\"gadfly:scale\")] = true;\n",
       "        };\n",
       "\n",
       "        if (xgridlines) xgridlines.selectAll(\"path\").forEach(add_x_tick_scales);\n",
       "        if (ygridlines) ygridlines.selectAll(\"path\").forEach(add_y_tick_scales);\n",
       "        if (xlabels) xlabels.selectAll(\"text\").forEach(add_x_tick_scales);\n",
       "        if (ylabels) ylabels.selectAll(\"text\").forEach(add_y_tick_scales);\n",
       "\n",
       "        root.data(\"xtickscales\", xtickscales);\n",
       "        root.data(\"ytickscales\", ytickscales);\n",
       "        root.data(\"xtickscale\", 1.0);\n",
       "    }\n",
       "\n",
       "    var min_scale = 1.0, max_scale = 1.0;\n",
       "    for (scale in xtickscales) {\n",
       "        min_scale = Math.min(min_scale, scale);\n",
       "        max_scale = Math.max(max_scale, scale);\n",
       "    }\n",
       "    for (scale in ytickscales) {\n",
       "        min_scale = Math.min(min_scale, scale);\n",
       "        max_scale = Math.max(max_scale, scale);\n",
       "    }\n",
       "    root.data(\"min_scale\", min_scale);\n",
       "    root.data(\"max_scale\", max_scale);\n",
       "\n",
       "    // store the original positions of labels\n",
       "    if (xlabels) {\n",
       "        xlabels.selectAll(\"text\")\n",
       "               .forEach(function (element, i) {\n",
       "                   element.data(\"x\", element.asPX(\"x\"));\n",
       "               });\n",
       "    }\n",
       "\n",
       "    if (ylabels) {\n",
       "        ylabels.selectAll(\"text\")\n",
       "               .forEach(function (element, i) {\n",
       "                   element.data(\"y\", element.asPX(\"y\"));\n",
       "               });\n",
       "    }\n",
       "\n",
       "    // mark grid lines and ticks as in or out of scale.\n",
       "    var mark_inscale = function (element, i) {\n",
       "        element.attribute(\"gadfly:inscale\", element.attribute(\"gadfly:scale\") == 1.0);\n",
       "    };\n",
       "\n",
       "    if (xgridlines) xgridlines.selectAll(\"path\").forEach(mark_inscale);\n",
       "    if (ygridlines) ygridlines.selectAll(\"path\").forEach(mark_inscale);\n",
       "    if (xlabels) xlabels.selectAll(\"text\").forEach(mark_inscale);\n",
       "    if (ylabels) ylabels.selectAll(\"text\").forEach(mark_inscale);\n",
       "\n",
       "    // figure out the upper ond lower bounds on panning using the maximum\n",
       "    // and minum grid lines\n",
       "    var bounds = root.plotbounds();\n",
       "    var pan_bounds = {\n",
       "        x0: 0.0,\n",
       "        y0: 0.0,\n",
       "        x1: 0.0,\n",
       "        y1: 0.0\n",
       "    };\n",
       "\n",
       "    if (xgridlines) {\n",
       "        xgridlines\n",
       "            .selectAll(\"path\")\n",
       "            .forEach(function (element, i) {\n",
       "                if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
       "                    var bbox = element.node.getBBox();\n",
       "                    if (bounds.x1 - bbox.x < pan_bounds.x0) {\n",
       "                        pan_bounds.x0 = bounds.x1 - bbox.x;\n",
       "                    }\n",
       "                    if (bounds.x0 - bbox.x > pan_bounds.x1) {\n",
       "                        pan_bounds.x1 = bounds.x0 - bbox.x;\n",
       "                    }\n",
       "                    element.attr(\"visibility\", \"visible\");\n",
       "                }\n",
       "            });\n",
       "    }\n",
       "\n",
       "    if (ygridlines) {\n",
       "        ygridlines\n",
       "            .selectAll(\"path\")\n",
       "            .forEach(function (element, i) {\n",
       "                if (element.attribute(\"gadfly:inscale\") == \"true\") {\n",
       "                    var bbox = element.node.getBBox();\n",
       "                    if (bounds.y1 - bbox.y < pan_bounds.y0) {\n",
       "                        pan_bounds.y0 = bounds.y1 - bbox.y;\n",
       "                    }\n",
       "                    if (bounds.y0 - bbox.y > pan_bounds.y1) {\n",
       "                        pan_bounds.y1 = bounds.y0 - bbox.y;\n",
       "                    }\n",
       "                    element.attr(\"visibility\", \"visible\");\n",
       "                }\n",
       "            });\n",
       "    }\n",
       "\n",
       "    // nudge these values a little\n",
       "    pan_bounds.x0 -= 5;\n",
       "    pan_bounds.x1 += 5;\n",
       "    pan_bounds.y0 -= 5;\n",
       "    pan_bounds.y1 += 5;\n",
       "    root.data(\"pan_bounds\", pan_bounds);\n",
       "\n",
       "    root.data(\"zoompan-ready\", true)\n",
       "};\n",
       "\n",
       "\n",
       "// drag actions, i.e. zooming and panning\n",
       "var pan_action = {\n",
       "    start: function(root, x, y, event) {\n",
       "        root.data(\"dx\", 0);\n",
       "        root.data(\"dy\", 0);\n",
       "        root.data(\"tx0\", root.data(\"tx\"));\n",
       "        root.data(\"ty0\", root.data(\"ty\"));\n",
       "    },\n",
       "    update: function(root, dx, dy, x, y, event) {\n",
       "        var px_per_mm = root.data(\"px_per_mm\");\n",
       "        dx /= px_per_mm;\n",
       "        dy /= px_per_mm;\n",
       "\n",
       "        var tx0 = root.data(\"tx\"),\n",
       "            ty0 = root.data(\"ty\");\n",
       "\n",
       "        var dx0 = root.data(\"dx\"),\n",
       "            dy0 = root.data(\"dy\");\n",
       "\n",
       "        root.data(\"dx\", dx);\n",
       "        root.data(\"dy\", dy);\n",
       "\n",
       "        dx = dx - dx0;\n",
       "        dy = dy - dy0;\n",
       "\n",
       "        var tx = tx0 + dx,\n",
       "            ty = ty0 + dy;\n",
       "\n",
       "        set_plot_pan_zoom(root, tx, ty, root.data(\"scale\"));\n",
       "    },\n",
       "    end: function(root, event) {\n",
       "\n",
       "    },\n",
       "    cancel: function(root) {\n",
       "        set_plot_pan_zoom(root, root.data(\"tx0\"), root.data(\"ty0\"), root.data(\"scale\"));\n",
       "    }\n",
       "};\n",
       "\n",
       "var zoom_box;\n",
       "var zoom_action = {\n",
       "    start: function(root, x, y, event) {\n",
       "        var bounds = root.plotbounds();\n",
       "        var width = bounds.x1 - bounds.x0,\n",
       "            height = bounds.y1 - bounds.y0;\n",
       "        var ratio = width / height;\n",
       "        var xscalable = root.hasClass(\"xscalable\"),\n",
       "            yscalable = root.hasClass(\"yscalable\");\n",
       "        var px_per_mm = root.data(\"px_per_mm\");\n",
       "        x = xscalable ? x / px_per_mm : bounds.x0;\n",
       "        y = yscalable ? y / px_per_mm : bounds.y0;\n",
       "        var w = xscalable ? 0 : width;\n",
       "        var h = yscalable ? 0 : height;\n",
       "        zoom_box = root.rect(x, y, w, h).attr({\n",
       "            \"fill\": \"#000\",\n",
       "            \"opacity\": 0.25\n",
       "        });\n",
       "        zoom_box.data(\"ratio\", ratio);\n",
       "    },\n",
       "    update: function(root, dx, dy, x, y, event) {\n",
       "        var xscalable = root.hasClass(\"xscalable\"),\n",
       "            yscalable = root.hasClass(\"yscalable\");\n",
       "        var px_per_mm = root.data(\"px_per_mm\");\n",
       "        var bounds = root.plotbounds();\n",
       "        if (yscalable) {\n",
       "            y /= px_per_mm;\n",
       "            y = Math.max(bounds.y0, y);\n",
       "            y = Math.min(bounds.y1, y);\n",
       "        } else {\n",
       "            y = bounds.y1;\n",
       "        }\n",
       "        if (xscalable) {\n",
       "            x /= px_per_mm;\n",
       "            x = Math.max(bounds.x0, x);\n",
       "            x = Math.min(bounds.x1, x);\n",
       "        } else {\n",
       "            x = bounds.x1;\n",
       "        }\n",
       "\n",
       "        dx = x - zoom_box.attr(\"x\");\n",
       "        dy = y - zoom_box.attr(\"y\");\n",
       "        if (xscalable && yscalable) {\n",
       "            var ratio = zoom_box.data(\"ratio\");\n",
       "            var width = Math.min(Math.abs(dx), ratio * Math.abs(dy));\n",
       "            var height = Math.min(Math.abs(dy), Math.abs(dx) / ratio);\n",
       "            dx = width * dx / Math.abs(dx);\n",
       "            dy = height * dy / Math.abs(dy);\n",
       "        }\n",
       "        var xoffset = 0,\n",
       "            yoffset = 0;\n",
       "        if (dx < 0) {\n",
       "            xoffset = dx;\n",
       "            dx = -1 * dx;\n",
       "        }\n",
       "        if (dy < 0) {\n",
       "            yoffset = dy;\n",
       "            dy = -1 * dy;\n",
       "        }\n",
       "        if (isNaN(dy)) {\n",
       "            dy = 0.0;\n",
       "        }\n",
       "        if (isNaN(dx)) {\n",
       "            dx = 0.0;\n",
       "        }\n",
       "        zoom_box.transform(\"T\" + xoffset + \",\" + yoffset);\n",
       "        zoom_box.attr(\"width\", dx);\n",
       "        zoom_box.attr(\"height\", dy);\n",
       "    },\n",
       "    end: function(root, event) {\n",
       "        var xscalable = root.hasClass(\"xscalable\"),\n",
       "            yscalable = root.hasClass(\"yscalable\");\n",
       "        var zoom_bounds = zoom_box.getBBox();\n",
       "        if (zoom_bounds.width * zoom_bounds.height <= 0) {\n",
       "            return;\n",
       "        }\n",
       "        var plot_bounds = root.plotbounds();\n",
       "        var zoom_factor = 1.0;\n",
       "        if (yscalable) {\n",
       "            zoom_factor = (plot_bounds.y1 - plot_bounds.y0) / zoom_bounds.height;\n",
       "        } else {\n",
       "            zoom_factor = (plot_bounds.x1 - plot_bounds.x0) / zoom_bounds.width;\n",
       "        }\n",
       "        var tx = (root.data(\"tx\") - zoom_bounds.x) * zoom_factor + plot_bounds.x0,\n",
       "            ty = (root.data(\"ty\") - zoom_bounds.y) * zoom_factor + plot_bounds.y0;\n",
       "        set_plot_pan_zoom(root, tx, ty, root.data(\"scale\") * zoom_factor);\n",
       "        zoom_box.remove();\n",
       "    },\n",
       "    cancel: function(root) {\n",
       "        zoom_box.remove();\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.guide_background_drag_onstart = function(x, y, event) {\n",
       "    var root = this.plotroot();\n",
       "    var scalable = root.hasClass(\"xscalable\") || root.hasClass(\"yscalable\");\n",
       "    var zoomable = !event.altKey && !event.ctrlKey && event.shiftKey && scalable;\n",
       "    var panable = !event.altKey && !event.ctrlKey && !event.shiftKey && scalable;\n",
       "    var drag_action = zoomable ? zoom_action :\n",
       "                      panable  ? pan_action :\n",
       "                                 undefined;\n",
       "    root.data(\"drag_action\", drag_action);\n",
       "    if (drag_action) {\n",
       "        var cancel_drag_action = function(event) {\n",
       "            if (event.which == 27) { // esc key\n",
       "                drag_action.cancel(root);\n",
       "                root.data(\"drag_action\", undefined);\n",
       "            }\n",
       "        };\n",
       "        window.addEventListener(\"keyup\", cancel_drag_action);\n",
       "        root.data(\"cancel_drag_action\", cancel_drag_action);\n",
       "        drag_action.start(root, x, y, event);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.guide_background_drag_onmove = function(dx, dy, x, y, event) {\n",
       "    var root = this.plotroot();\n",
       "    var drag_action = root.data(\"drag_action\");\n",
       "    if (drag_action) {\n",
       "        drag_action.update(root, dx, dy, x, y, event);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.guide_background_drag_onend = function(event) {\n",
       "    var root = this.plotroot();\n",
       "    window.removeEventListener(\"keyup\", root.data(\"cancel_drag_action\"));\n",
       "    root.data(\"cancel_drag_action\", undefined);\n",
       "    var drag_action = root.data(\"drag_action\");\n",
       "    if (drag_action) {\n",
       "        drag_action.end(root, event);\n",
       "    }\n",
       "    root.data(\"drag_action\", undefined);\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.guide_background_scroll = function(event) {\n",
       "    if (event.shiftKey) {\n",
       "        increase_zoom_by_position(this.plotroot(), 0.001 * event.wheelDelta);\n",
       "        event.preventDefault();\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.zoomslider_button_mouseover = function(event) {\n",
       "    this.select(\".button_logo\")\n",
       "         .animate({fill: this.data(\"mouseover_color\")}, 100);\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.zoomslider_button_mouseout = function(event) {\n",
       "     this.select(\".button_logo\")\n",
       "         .animate({fill: this.data(\"mouseout_color\")}, 100);\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.zoomslider_zoomout_click = function(event) {\n",
       "    increase_zoom_by_position(this.plotroot(), -0.1, true);\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.zoomslider_zoomin_click = function(event) {\n",
       "    increase_zoom_by_position(this.plotroot(), 0.1, true);\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.zoomslider_track_click = function(event) {\n",
       "    // TODO\n",
       "};\n",
       "\n",
       "\n",
       "// Map slider position x to scale y using the function y = a*exp(b*x)+c.\n",
       "// The constants a, b, and c are solved using the constraint that the function\n",
       "// should go through the points (0; min_scale), (0.5; 1), and (1; max_scale).\n",
       "var scale_from_slider_position = function(position, min_scale, max_scale) {\n",
       "    var a = (1 - 2 * min_scale + min_scale * min_scale) / (min_scale + max_scale - 2),\n",
       "        b = 2 * Math.log((max_scale - 1) / (1 - min_scale)),\n",
       "        c = (min_scale * max_scale - 1) / (min_scale + max_scale - 2);\n",
       "    return a * Math.exp(b * position) + c;\n",
       "}\n",
       "\n",
       "// inverse of scale_from_slider_position\n",
       "var slider_position_from_scale = function(scale, min_scale, max_scale) {\n",
       "    var a = (1 - 2 * min_scale + min_scale * min_scale) / (min_scale + max_scale - 2),\n",
       "        b = 2 * Math.log((max_scale - 1) / (1 - min_scale)),\n",
       "        c = (min_scale * max_scale - 1) / (min_scale + max_scale - 2);\n",
       "    return 1 / b * Math.log((scale - c) / a);\n",
       "}\n",
       "\n",
       "var increase_zoom_by_position = function(root, delta_position, animate) {\n",
       "    var scale = root.data(\"scale\"),\n",
       "        min_scale = root.data(\"min_scale\"),\n",
       "        max_scale = root.data(\"max_scale\");\n",
       "    var position = slider_position_from_scale(scale, min_scale, max_scale);\n",
       "    position += delta_position;\n",
       "    scale = scale_from_slider_position(position, min_scale, max_scale);\n",
       "    set_zoom(root, scale, animate);\n",
       "}\n",
       "\n",
       "var set_zoom = function(root, scale, animate) {\n",
       "    var min_scale = root.data(\"min_scale\"),\n",
       "        max_scale = root.data(\"max_scale\"),\n",
       "        old_scale = root.data(\"scale\");\n",
       "    var new_scale = Math.max(min_scale, Math.min(scale, max_scale));\n",
       "    if (animate) {\n",
       "        Snap.animate(\n",
       "            old_scale,\n",
       "            new_scale,\n",
       "            function (new_scale) {\n",
       "                update_plot_scale(root, new_scale);\n",
       "            },\n",
       "            200);\n",
       "    } else {\n",
       "        update_plot_scale(root, new_scale);\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "var update_plot_scale = function(root, new_scale) {\n",
       "    var trans = scale_centered_translation(root, new_scale);\n",
       "    set_plot_pan_zoom(root, trans.x, trans.y, new_scale);\n",
       "\n",
       "    root.selectAll(\".zoomslider_thumb\")\n",
       "        .forEach(function (element, i) {\n",
       "            var min_pos = element.data(\"min_pos\"),\n",
       "                max_pos = element.data(\"max_pos\"),\n",
       "                min_scale = root.data(\"min_scale\"),\n",
       "                max_scale = root.data(\"max_scale\");\n",
       "            var xmid = (min_pos + max_pos) / 2;\n",
       "            var xpos = slider_position_from_scale(new_scale, min_scale, max_scale);\n",
       "            element.transform(new Snap.Matrix().translate(\n",
       "                Math.max(min_pos, Math.min(\n",
       "                         max_pos, min_pos + (max_pos - min_pos) * xpos)) - xmid, 0));\n",
       "    });\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.zoomslider_thumb_dragmove = function(dx, dy, x, y, event) {\n",
       "    var root = this.plotroot();\n",
       "    var min_pos = this.data(\"min_pos\"),\n",
       "        max_pos = this.data(\"max_pos\"),\n",
       "        min_scale = root.data(\"min_scale\"),\n",
       "        max_scale = root.data(\"max_scale\"),\n",
       "        old_scale = root.data(\"old_scale\");\n",
       "\n",
       "    var px_per_mm = root.data(\"px_per_mm\");\n",
       "    dx /= px_per_mm;\n",
       "    dy /= px_per_mm;\n",
       "\n",
       "    var xmid = (min_pos + max_pos) / 2;\n",
       "    var xpos = slider_position_from_scale(old_scale, min_scale, max_scale) +\n",
       "                   dx / (max_pos - min_pos);\n",
       "\n",
       "    // compute the new scale\n",
       "    var new_scale = scale_from_slider_position(xpos, min_scale, max_scale);\n",
       "    new_scale = Math.min(max_scale, Math.max(min_scale, new_scale));\n",
       "\n",
       "    update_plot_scale(root, new_scale);\n",
       "    event.stopPropagation();\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.zoomslider_thumb_dragstart = function(x, y, event) {\n",
       "    this.animate({fill: this.data(\"mouseover_color\")}, 100);\n",
       "    var root = this.plotroot();\n",
       "\n",
       "    // keep track of what the scale was when we started dragging\n",
       "    root.data(\"old_scale\", root.data(\"scale\"));\n",
       "    event.stopPropagation();\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.zoomslider_thumb_dragend = function(event) {\n",
       "    this.animate({fill: this.data(\"mouseout_color\")}, 100);\n",
       "    event.stopPropagation();\n",
       "};\n",
       "\n",
       "\n",
       "var toggle_color_class = function(root, color_class, ison) {\n",
       "    var guides = root.selectAll(\".guide.\" + color_class + \",.guide .\" + color_class);\n",
       "    var geoms = root.selectAll(\".geometry.\" + color_class + \",.geometry .\" + color_class);\n",
       "    if (ison) {\n",
       "        guides.animate({opacity: 0.5}, 250);\n",
       "        geoms.animate({opacity: 0.0}, 250);\n",
       "    } else {\n",
       "        guides.animate({opacity: 1.0}, 250);\n",
       "        geoms.animate({opacity: 1.0}, 250);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "Gadfly.colorkey_swatch_click = function(event) {\n",
       "    var root = this.plotroot();\n",
       "    var color_class = this.data(\"color_class\");\n",
       "\n",
       "    if (event.shiftKey) {\n",
       "        root.selectAll(\".colorkey text\")\n",
       "            .forEach(function (element) {\n",
       "                var other_color_class = element.data(\"color_class\");\n",
       "                if (other_color_class != color_class) {\n",
       "                    toggle_color_class(root, other_color_class,\n",
       "                                       element.attr(\"opacity\") == 1.0);\n",
       "                }\n",
       "            });\n",
       "    } else {\n",
       "        toggle_color_class(root, color_class, this.attr(\"opacity\") == 1.0);\n",
       "    }\n",
       "};\n",
       "\n",
       "\n",
       "return Gadfly;\n",
       "\n",
       "}));\n",
       "\n",
       "\n",
       "//@ sourceURL=gadfly.js\n",
       "\n",
       "(function (glob, factory) {\n",
       "    // AMD support\n",
       "      if (typeof require === \"function\" && typeof define === \"function\" && define.amd) {\n",
       "        require([\"Snap.svg\", \"Gadfly\"], function (Snap, Gadfly) {\n",
       "            factory(Snap, Gadfly);\n",
       "        });\n",
       "      } else {\n",
       "          factory(glob.Snap, glob.Gadfly);\n",
       "      }\n",
       "})(window, function (Snap, Gadfly) {\n",
       "    var fig = Snap(\"#img-9dd09a3e\");\n",
       "fig.select(\"#img-9dd09a3e-5\")\n",
       "   .init_gadfly();\n",
       "fig.select(\"#img-9dd09a3e-7\")\n",
       "   .plotroot().data(\"unfocused_ygrid_color\", \"#D0D0E0\")\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-7\")\n",
       "   .plotroot().data(\"focused_ygrid_color\", \"#A0A0A0\")\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-8\")\n",
       "   .plotroot().data(\"unfocused_xgrid_color\", \"#D0D0E0\")\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-8\")\n",
       "   .plotroot().data(\"focused_xgrid_color\", \"#A0A0A0\")\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-13\")\n",
       "   .data(\"mouseover_color\", \"#CD5C5C\")\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-13\")\n",
       "   .data(\"mouseout_color\", \"#6A6A6A\")\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-13\")\n",
       "   .click(Gadfly.zoomslider_zoomin_click)\n",
       ".mouseenter(Gadfly.zoomslider_button_mouseover)\n",
       ".mouseleave(Gadfly.zoomslider_button_mouseout)\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-15\")\n",
       "   .data(\"max_pos\", 120.42)\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-15\")\n",
       "   .data(\"min_pos\", 103.42)\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-15\")\n",
       "   .click(Gadfly.zoomslider_track_click);\n",
       "fig.select(\"#img-9dd09a3e-16\")\n",
       "   .data(\"max_pos\", 120.42)\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-16\")\n",
       "   .data(\"min_pos\", 103.42)\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-16\")\n",
       "   .data(\"mouseover_color\", \"#CD5C5C\")\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-16\")\n",
       "   .data(\"mouseout_color\", \"#6A6A6A\")\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-16\")\n",
       "   .drag(Gadfly.zoomslider_thumb_dragmove,\n",
       "     Gadfly.zoomslider_thumb_dragstart,\n",
       "     Gadfly.zoomslider_thumb_dragend)\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-17\")\n",
       "   .data(\"mouseover_color\", \"#CD5C5C\")\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-17\")\n",
       "   .data(\"mouseout_color\", \"#6A6A6A\")\n",
       ";\n",
       "fig.select(\"#img-9dd09a3e-17\")\n",
       "   .click(Gadfly.zoomslider_zoomout_click)\n",
       ".mouseenter(Gadfly.zoomslider_button_mouseover)\n",
       ".mouseleave(Gadfly.zoomslider_button_mouseout)\n",
       ";\n",
       "    });\n",
       "]]> </script>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "Plot(...)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = randomrepeats(2^14,4)\n",
    "plot(\n",
    "    x = cumsum(sort([v for v in values(h)],rev=true)),\n",
    "    y = sort([v for v in values(h)],rev=true),\n",
    "    Geom.bar,\n",
    "    Guide.xlabel(\"Cumulative samples\"),\n",
    "    Guide.ylabel(\"Samples\")\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "So what is going on here?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Well consider, the probability that the whole sequence of length $n$ is composed of 'A's"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\n",
    "\\left(\\frac{1}{4}\\right)^n\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Conversely, the probability that a sequence of lenght A excludes 'A's is"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\n",
    "\\left(\\frac{3}{4}\\right)^n\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The distribution of, say, 'A's in an sequence of length $n$, is binomial:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\n",
    "    \\mathbb{P}[\\#A=x] = \\binom{n}{x} \\left(\\frac{1}{4}\\right)^x \\left(\\frac{3}{4}\\right)^{n-x} \n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "This is of course true for counting, individually, 'C's, 'G's, and 'T's, as well"
   ]
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Julia 0.5.0",
   "language": "julia",
   "name": "julia-0.5"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "0.5.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}