1. Workflow example - random walks.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Basic syntax and workflow: Random walk"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An important model in science (biology, chemistry, physics, etc.) is Brownian motion, or random walk, a model of a molecule moving in a medium.\n",
    "\n",
    "The simplest model we can make is a particle jumping at random on the integers. We save the list of consecutive positions in an array.\n",
    "\n",
    "We wrap the code in a function in order to make it more general (e.g. to be able to change the number of steps). It turns out that this also makes it (much) more efficient in Julia:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "walk"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"\"\"\n",
    "    walk(T)\n",
    "\n",
    "Simulate a simple random walk up to time `T`.\n",
    "Returns the trajectory as a `Vector`.\n",
    "\"\"\"\n",
    "function walk(T)\n",
    "\n",
    "    x = 0  # position\n",
    "    trajectory = [x]  # initialise vector\n",
    "\n",
    "    for t in 1:T  # range\n",
    "\n",
    "        if rand() < 0.5\n",
    "            x += 1       # equivalent to `x = x + 1`, i.e. modifies the value of x\n",
    "        else\n",
    "            x -= 1\n",
    "        end\n",
    "        \n",
    "        push!(trajectory, x)  # add the value of `x` to the Vector `trajectory`\n",
    "\n",
    "    end\n",
    "    \n",
    "    return trajectory\n",
    "\n",
    "end"
   ]
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   "execution_count": 2,
   "metadata": {},
   "outputs": [
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     "text": [
      "search: \u001b[1mw\u001b[22m\u001b[1ma\u001b[22m\u001b[1ml\u001b[22m\u001b[1mk\u001b[22mdir sho\u001b[1mw\u001b[22m\u001b[1ma\u001b[22m\u001b[1ml\u001b[22ml ro\u001b[1mw\u001b[22mv\u001b[1ma\u001b[22m\u001b[1ml\u001b[22ms is\u001b[1mw\u001b[22mrit\u001b[1ma\u001b[22mb\u001b[1ml\u001b[22me mime\u001b[1mw\u001b[22mrit\u001b[1ma\u001b[22mb\u001b[1ml\u001b[22me \u001b[1mw\u001b[22m\u001b[1ma\u001b[22mtch_fi\u001b[1ml\u001b[22me\n",
      "\n"
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     "data": {
      "text/markdown": [
       "```\n",
       "walk(T)\n",
       "```\n",
       "\n",
       "Simulate a simple random walk up to time `T`. Returns the trajectory as a `Vector`.\n"
      ],
      "text/plain": [
       "```\n",
       "walk(T)\n",
       "```\n",
       "\n",
       "Simulate a simple random walk up to time `T`. Returns the trajectory as a `Vector`.\n"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "?walk"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We call the function and give the calculated data a name:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "11-element Array{Int64,1}:\n",
       "  0\n",
       "  1\n",
       "  0\n",
       "  1\n",
       "  0\n",
       " -1\n",
       " -2\n",
       " -1\n",
       "  0\n",
       " -1\n",
       " -2"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "traj = walk(10)\n",
    "traj"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use `show(traj)` to use less vertical space:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 1, 0, 1, 0, -1, -2, -1, 0, -1, -2]"
     ]
    }
   ],
   "source": [
    "show(traj)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We wish to plot the trajectory $x(t)$ as a function of $t$. \n",
    "\n",
    "We recommen the `Plots.jl` package, which provides a common interface to several plotting \"backends\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Plots.GRBackend()"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Pkg.add(\"Plots\")    # install the Plots.jl package if you don't already have it -- do once in your life\n",
    "\n",
    "using Plots           # load the package in each session\n",
    "gr()   # choose the GR backend"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
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   "source": [
    "T = 10\n",
    "plot(1:T, walk(T))  # vector of x coordinates, and vector of y coordinates\n",
    "\n",
    "hline!([0], c=:black, ls=:dash, lw=2)  # add dashed horizontal line to plot"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's draw a few walks:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
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     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = 30  # maximum time\n",
    "N = 10  # number of walks\n",
    "\n",
    "plot(legend=false)  # make empty plot with no legend\n",
    "\n",
    "for i in 1:N\n",
    "    plot!(1:T, walk(T))   # ! means \"add to plot\"\n",
    "end\n",
    "\n",
    "hline!([0], c=:black, ls=:dash, lw=2)  # add"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we want to explore the space of parameters `T` (maximum time of the walk) and `N` (number of walks), it would be nice to be able to manipulate them interactively.\n",
    "\n",
    "This may be done using `Interact.jl` (patched a few hours ago!).\n",
    "\n",
    "For `Interact` to work correctly, you need version 1.5.1 of `IJulia.jl`. Check this using"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " - IJulia                        1.5.1\n"
     ]
    }
   ],
   "source": [
    "Pkg.status(\"IJulia\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you don't have a recent enough version, do a "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mUpdating METADATA...\n",
      "\u001b[39m"
     ]
    },
    {
     "ename": "LoadError",
     "evalue": "\u001b[91mInterruptException:\u001b[39m",
     "output_type": "error",
     "traceback": [
      "\u001b[91mInterruptException:\u001b[39m",
      "",
      "Stacktrace:",
      " [1] \u001b[1mStringVector\u001b[22m\u001b[22m at \u001b[1m./iobuffer.jl:30\u001b[22m\u001b[22m [inlined]",
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      " [3] \u001b[1mreadstring\u001b[22m\u001b[22m at \u001b[1m./io.jl:540\u001b[22m\u001b[22m [inlined]",
      " [4] \u001b[1mopen\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::Base.#readstring, ::String\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m./iostream.jl:152\u001b[22m\u001b[22m",
      " [5] \u001b[1mreadchomp\u001b[22m\u001b[22m at \u001b[1m./io.jl:491\u001b[22m\u001b[22m [inlined]",
      " [6] \u001b[1mavailable\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::Array{String,1}\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m./pkg/read.jl:24\u001b[22m\u001b[22m",
      " [7] \u001b[1mupdate\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::String, ::Set{String}\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m./pkg/entry.jl:383\u001b[22m\u001b[22m",
      " [8] \u001b[1m(::Base.Pkg.Dir.##4#7{Array{Any,1},Base.Pkg.Entry.#update,Tuple{String,Set{String}}})\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m./pkg/dir.jl:36\u001b[22m\u001b[22m",
      " [9] \u001b[1mcd\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::Base.Pkg.Dir.##4#7{Array{Any,1},Base.Pkg.Entry.#update,Tuple{String,Set{String}}}, ::String\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m./file.jl:70\u001b[22m\u001b[22m",
      " [10] \u001b[1m#cd#1\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m::Array{Any,1}, ::Function, ::Function, ::String, ::Vararg{Any,N} where N\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m./pkg/dir.jl:36\u001b[22m\u001b[22m",
      " [11] \u001b[1mupdate\u001b[22m\u001b[22m\u001b[1m(\u001b[22m\u001b[22m\u001b[1m)\u001b[22m\u001b[22m at \u001b[1m./pkg/pkg.jl:228\u001b[22m\u001b[22m"
     ]
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   ],
   "source": [
    "Pkg.update()"
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  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
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       "\t    prompt_area: true,\n",
       "\t    events: IPython.events,\n",
       "\t    keyboard_manager: IPython.keyboard_manager\n",
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       "\n",
       "\tswitch (type) {\n",
       "\tcase \"image/png\":\n",
       "            var _src = 'data:' + type + ';base64,' + val;\n",
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       "\n",
       "\n",
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       "\t    $([IPython.events]).on(\n",
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       "                // console.log(md.comm_id);\n",
       "                toinsert.addClass(\"signal-\" + md.comm_id);\n",
       "                toinsert.data(\"type\", type);\n",
       "                // Signal back indicating the mimetype required\n",
       "                var comm_manager = IPython.notebook.kernel.comm_manager;\n",
       "                var comm = comm_manager.comms[md.comm_id];\n",
       "                comm.then(function (c) {\n",
       "                    c.send({action: \"subscribe_mime\",\n",
       "                       mime: type});\n",
       "                    toinsert.bind(\"destroyed\", function() {\n",
       "                        c.send({action: \"unsubscribe_mime\",\n",
       "                               mime: type});\n",
       "                    });\n",
       "                })\n",
       "\t\t    }\n",
       "\t    });\n",
       "\t}\n",
       "\n",
       "\ttry {\n",
       "\t    // try to initialize right away. otherwise, wait on the status_started event.\n",
       "\t    initComm(undefined, IPython.notebook);\n",
       "\t} catch (e) {\n",
       "\t    $([IPython.events]).on('kernel_created.Kernel kernel_created.Session', initComm);\n",
       "\t}\n",
       "    });\n",
       "})(IPython, jQuery, _, MathJax);\n",
       "</script>\n",
       "    <script>\n",
       "        window.interactLoadedFlag = true\n",
       "       $(\"#interact-js-shim\").bind(\"destroyed\", function () {\n",
       "           if (window.interactLoadedFlag) {\n",
       "               console.warn(\"JavaScript required by Interact will be removed if you remove this cell or run using Interact more than once.\")\n",
       "           }\n",
       "       })\n",
       "       $([IPython.events]).on(\"kernel_starting.Kernel kernel_restarting.Kernel\", function () { window.interactLoadedFlag = false })\n",
       "   </script>\n",
       "</div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Pkg.add(\"Interact\")\n",
    "\n",
    "using Interact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [],
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       "Interact.Options{:SelectionSlider,Int64}(1: \"input\" = 50 Int64 , \"T\", 50, \"50\", Interact.OptionDict(DataStructures.OrderedDict(\"1\"=>1,\"2\"=>2,\"3\"=>3,\"4\"=>4,\"5\"=>5,\"6\"=>6,\"7\"=>7,\"8\"=>8,\"9\"=>9,\"10\"=>10…), Dict(68=>\"68\",2=>\"2\",89=>\"89\",11=>\"11\",39=>\"39\",46=>\"46\",85=>\"85\",25=>\"25\",55=>\"55\",42=>\"42\"…)), Any[], Any[], true, \"horizontal\")"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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       "Interact.Options{:SelectionSlider,Int64}(3: \"input-2\" = 500 Int64 , \"N\", 500, \"500\", Interact.OptionDict(DataStructures.OrderedDict(\"1\"=>1,\"2\"=>2,\"3\"=>3,\"4\"=>4,\"5\"=>5,\"6\"=>6,\"7\"=>7,\"8\"=>8,\"9\"=>9,\"10\"=>10…), Dict(306=>\"306\",29=>\"29\",74=>\"74\",905=>\"905\",176=>\"176\",892=>\"892\",285=>\"285\",318=>\"318\",873=>\"873\",975=>\"975\"…)), Any[], Any[], true, \"horizontal\")"
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     },
     "metadata": {},
     "output_type": "display_data"
    },
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       "  210.724,202.614 216.252,199.008 221.78,195.402 227.307,191.795 232.835,195.402 238.362,199.008 243.89,202.614 249.417,199.008 254.945,195.402 260.472,199.008 \n",
       "  266,202.614 271.528,206.22 277.055,202.614 282.583,206.22 288.11,202.614 293.638,199.008 299.165,202.614 304.693,199.008 310.22,195.402 315.748,199.008 \n",
       "  \n",
       "  \"/>\n",
       "<polyline clip-path=\"url(#clip02)\" style=\"stroke:#000000; stroke-width:2; stroke-opacity:1; fill:none\" stroke-dasharray=\"8, 5\" points=\"\n",
       "  39.3701,188.189 592.126,188.189 \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 11,
     "metadata": {
      "comm_id": "8d042030-ba16-4529-8160-b47087233db7",
      "reactive": true
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "max_T = 100\n",
    "max_N = 1000\n",
    "\n",
    "@manipulate for T in 1:max_T, N in 1:max_N\n",
    "\n",
    "    p = plot(legend=false, xlim=(0, max_T), ylim=(-5sqrt(max_T), 5sqrt(max_T)))  # make empty plot with no legend; fix axes limits\n",
    "\n",
    "    for i in 1:N\n",
    "        plot!(1:T, walk(T))   # ! means \"add to plot\"\n",
    "    end\n",
    "\n",
    "    hline!([0], c=:black, ls=:dash, lw=2)  # add\n",
    "    \n",
    "    p\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Data analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we have a feel for the behaviour of the system, let's collect some data and do some data analysis.\n",
    "Let's calculate the variance of the positions of `N` walks after time `T`:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A simple implementation could be as follows. First let's rewrite the `walk` function to only return the last value:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "walk (generic function with 1 method)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function walk(T)\n",
    "\n",
    "    x = 0  \n",
    "\n",
    "    for t in 1:T  \n",
    "\n",
    "        if rand() < 0.5\n",
    "            x += 1 \n",
    "        else\n",
    "            x -= 1\n",
    "        end\n",
    "        \n",
    "    end\n",
    "    \n",
    "    return x\n",
    "\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "walks (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function walks(N, T)\n",
    "    \n",
    "    sumsq = 0.0\n",
    "    \n",
    "    for i in 1:N\n",
    "        w = walk(T)\n",
    "        sumsq += w^2\n",
    "    end \n",
    "    \n",
    "    sumsq /= N\n",
    "    \n",
    "    return sumsq\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  2.466867 seconds (5.11 k allocations: 277.438 KiB)\n",
      "  2.469756 seconds (5 allocations: 176 bytes)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "9934.24212"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time walks(100000, 10000)\n",
    "@time walks(100000, 10000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Note the memory allocation in the first run. This is due to the function being compiled. \n",
    "The first run should always be discarded for benchmarking purposes.  \n",
    "\n",
    "For microbenchmarks, prefer the `BenchmarkTools.jl` package:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mRecompiling stale cache file /Users/dpsanders/.julia/lib/v0.6/FileIO.ji for module FileIO.\n",
      "\u001b[39m\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mRecompiling stale cache file /Users/dpsanders/.julia/lib/v0.6/JLD.ji for module JLD.\n",
      "\u001b[39m"
     ]
    }
   ],
   "source": [
    "using BenchmarkTools"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: \n",
       "  memory estimate:  0 bytes\n",
       "  allocs estimate:  0\n",
       "  --------------\n",
       "  minimum time:     290.541 ns (0.00% GC)\n",
       "  median time:      301.728 ns (0.00% GC)\n",
       "  mean time:        320.571 ns (0.00% GC)\n",
       "  maximum time:     1.889 μs (0.00% GC)\n",
       "  --------------\n",
       "  samples:          10000\n",
       "  evals/sample:     320"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@benchmark walks(10, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Vectorized (Matlab-ish) style:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can easily try out different simulation methods for our `walk` function.\n",
    "To do so, we write a generic `walks` function that **takes a function as argument**.\n",
    "\n",
    "We also use **array comprehensions** or **generator expressions**:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Bool,1}:\n",
       "  true\n",
       "  true\n",
       "  true\n",
       "  true\n",
       " false\n",
       "  true\n",
       " false\n",
       "  true\n",
       " false\n",
       "  true"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[rand() < 0.5 for i in 1:10]  # array comprehension"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Base.Generator{UnitRange{Int64},##4#5}(#4, 1:10)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(rand() < 0.5 for i in 1:10)  # generator expression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "walks (generic function with 2 methods)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function walk2(T) \n",
    "    right_steps = sum(rand() < 0.5 for i in 1:T)   # generator\n",
    "    left_steps = T - right_steps\n",
    "    \n",
    "    return right_steps - left_steps\n",
    "end\n",
    "  \n",
    "walks(f, N, T) = var(f(T) for i in 1:N)  # generator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  2.487515 seconds (42.02 k allocations: 2.219 MiB)\n"
     ]
    },
    {
     "data": {
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       "10006.59314735185"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = 10^5\n",
    "T = 10^4\n",
    "\n",
    "@time walks(walk, N, T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  7.049938 seconds (357.77 k allocations: 10.699 MiB)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "9908.27690706957"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time walks(walk2, N, T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "walk3 (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function walk3(T) \n",
    "    right_steps = sum(rand(Bool, T))   # generator\n",
    "    left_steps = T - right_steps\n",
    "    \n",
    "    return right_steps - left_steps\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  2.620494 seconds (148.49 k allocations: 973.026 MiB, 3.83% gc time)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "9978.530287427275"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time walks(walk3, 100000, 10000) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "walk4 (generic function with 1 method)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function walk4(T) \n",
    "    right_steps = sum(bitrand(T))\n",
    "    left_steps = T - right_steps\n",
    "    \n",
    "    return right_steps - left_steps\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.234214 seconds (668.14 k allocations: 154.680 MiB, 4.62% gc time)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "10086.881314092727"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time walks(walk4, 100000, 10000)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course, it's *possible* to do these things in C++, but it's certainly much *harder*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Look at the data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is easy to collect data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "walk_data (generic function with 1 method)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "walk_data(N, T) = [walk(T) for i in 1:N]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "\t    if (o.handler) {\n",
       "\t\to.handler.apply(this, arguments)\n",
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       "    var OutputArea = IPython.version >= \"4.0.0\" ? require(\"notebook/js/outputarea\").OutputArea : IPython.OutputArea;\n",
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       "\t    prompt_area: true,\n",
       "\t    events: IPython.events,\n",
       "\t    keyboard_manager: IPython.keyboard_manager\n",
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       "\n",
       "\n",
       "    $(document).ready(function() {\n",
       "\tfunction initComm(evt, data) {\n",
       "\t    var comm_manager = data.kernel.comm_manager;\n",
       "        //_.extend(comm_manager.targets, require(\"widgets/js/widget\"))\n",
       "\t    comm_manager.register_target(\"Signal\", function (comm) {\n",
       "            comm.on_msg(function (msg) {\n",
       "                var val = msg.content.data.value;\n",
       "                $(\".signal-\" + comm.comm_id).each(function() {\n",
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       "            });\n",
       "\t    });\n",
       "\n",
       "\t    // coordingate with Comm and redraw Signals\n",
       "\t    // XXX: Test using Reactive here to improve performance\n",
       "\t    $([IPython.events]).on(\n",
       "\t\t'output_appended.OutputArea', function (event, type, value, md, toinsert) {\n",
       "\t\t    if (md && md.reactive) {\n",
       "                // console.log(md.comm_id);\n",
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       "\t    $([IPython.events]).on('kernel_created.Kernel kernel_created.Session', initComm);\n",
       "\t}\n",
       "    });\n",
       "})(IPython, jQuery, _, MathJax);\n",
       "</script>\n",
       "    <script>\n",
       "        window.interactLoadedFlag = true\n",
       "       $(\"#interact-js-shim\").bind(\"destroyed\", function () {\n",
       "           if (window.interactLoadedFlag) {\n",
       "               console.warn(\"JavaScript required by Interact will be removed if you remove this cell or run using Interact more than once.\")\n",
       "           }\n",
       "       })\n",
       "       $([IPython.events]).on(\"kernel_starting.Kernel kernel_restarting.Kernel\", function () { window.interactLoadedFlag = false })\n",
       "   </script>\n",
       "</div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "using Plots; gr()\n",
    "using Interact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
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      ]
     },
     "execution_count": 25,
     "metadata": {
      "comm_id": "a0356c09-7ae3-48ae-9e32-2c6e695885c6",
      "reactive": true
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = walk_data(10000, 10000)\n",
    "\n",
    "@manipulate for bins in 10:1000\n",
    "    histogram(data, bins=bins, normed=true, ylim=(0,0.006), xlim=(-300,300))\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's compare this to the known analytical result: a normal distribution with variance equal to the number of steps:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use Unicode variable names and operators:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100.0"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "μ = 0        # type \\mu<TAB>\n",
    "σ = √10000   # type \\sqrt<TAB>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will use the probability density function (PDF) from the `Distributions.jl` package:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Pkg.add(\"Distributions\")\n",
    "using Distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
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       "</g>\n",
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      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "histogram(data, bins=100, normed=true, label=\"data\")\n",
    "plot!(-250:250, x->pdf(Normal(0, σ), x), label=\"exact\", lw=3)  # probability \n",
    "xaxis!(\"distance\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To save the figure, we use `savefig`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "savefig(\"test.pdf\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can open the file. `;` sends the command to the command line (shell); on Mac, `open` opens the file:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    ";open test.pdf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Summary:\n",
    "\n",
    "- Julia allows us to calculate **fast**\n",
    "\n",
    "\n",
    "- **And** interactively visualize our data\n",
    "\n",
    "\n",
    "- All in one place"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Save the data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose we wish to save our data to a file on disk. The simplest way is using `writedlm` (\"write delimited\"):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "writedlm(\"random_walk.dat\", data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's read the same data back in:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10000×1 Array{Float64,2}:\n",
       "  -20.0\n",
       "   56.0\n",
       "  -24.0\n",
       "  136.0\n",
       "  -26.0\n",
       "   46.0\n",
       " -100.0\n",
       "  -60.0\n",
       "   14.0\n",
       "  160.0\n",
       " -108.0\n",
       " -118.0\n",
       "  -68.0\n",
       "    ⋮  \n",
       "  -76.0\n",
       "  110.0\n",
       "   42.0\n",
       "  102.0\n",
       "  -20.0\n",
       "   64.0\n",
       "   38.0\n",
       "    6.0\n",
       "   -6.0\n",
       "  -24.0\n",
       "  124.0\n",
       "    4.0"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data_new = readdlm(\"random_walk.dat\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the format has changed. Although there are some options to `readdlm`, this is not a good solution to the problem."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The good solution is to use the `JLD.jl` (\"JuLia Data\") package (which uses the standard HDF5 format underneath). We will also save another variable for illustrative purposes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: using JLD.translate in module Main conflicts with an existing identifier.\n"
     ]
    }
   ],
   "source": [
    "# Pkg.add(\"JLD\")\n",
    "\n",
    "using JLD"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "17"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i = 17"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "save(\"random_walk.jld\", \"final_pos\", data, \"arbitrary\", i)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Dict{String,Any} with 2 entries:\n",
       "  \"arbitrary\" => 17\n",
       "  \"final_pos\" => [-20, 56, -24, 136, -26, 46, -100, -60, 14, 160  …  42, 102, -…"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "all = load(\"random_walk.jld\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The data is read in as a **dictionary**, associating to each variable name its value. We can extract these as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "17"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "j = all[\"arbitrary\"]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Alternatively, we can *read only part of the data file*:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10000-element Array{Int64,1}:\n",
       "  -20\n",
       "   56\n",
       "  -24\n",
       "  136\n",
       "  -26\n",
       "   46\n",
       " -100\n",
       "  -60\n",
       "   14\n",
       "  160\n",
       " -108\n",
       " -118\n",
       "  -68\n",
       "    ⋮\n",
       "  -76\n",
       "  110\n",
       "   42\n",
       "  102\n",
       "  -20\n",
       "   64\n",
       "   38\n",
       "    6\n",
       "   -6\n",
       "  -24\n",
       "  124\n",
       "    4"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "final_pos = load(\"random_walk.jld\", \"final_pos\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the type of the data has now been correctly preserved. This is the point of using `JLD`. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Workflow: Extract code into a Julia script"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since we have developed a quantity of code in this notebook, it is a good moment to extract the code into a Julia script file, with termination `.jl`, that we edit, e.g. with Juno.\n",
    "\n",
    "We can then read in the file as if we typed it with"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "include(\"random_walk.jl\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Later on, we could make it into a `module` (separate namespace) and then into a Julia pacakge."
   ]
  }
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