Tutorial_24_Integration.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numerical Integration and Differentiation\n",
    "\n",
    "This notebook illustrates how to perform numerical integration and differentiation. \n",
    "\n",
    "There are several packages for doing this. Here, the focus is on [QuadGK](https://github.com/JuliaMath/QuadGK.jl) and [ForwardDiff](https://github.com/JuliaDiff/ForwardDiff.jl)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load Packages and Extra Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "printyellow (generic function with 1 method)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Printf, QuadGK, ForwardDiff\n",
    "\n",
    "include(\"jlFiles/printmat.jl\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots\n",
    "\n",
    "#pyplot(size=(600,400))\n",
    "gr(size=(480,320))\n",
    "default(fmt = :svg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numerical Integration\n",
    "\n",
    "As a simple illustration, the next cells plot and integrate the $N(0,\\sigma)$ pdf."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Pdf of the N() Distribution \n",
    "\n",
    "Notice: the function takes $\\mu$ and $\\sigma$ (not $\\sigma^2$) as inputs, similar to the Distributions package. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ϕNS (generic function with 3 methods)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function ϕNS(x,μ=0,σ=1)       #pdf of N(μ,σ), defaults to N(0,1)\n",
    "    z   = (x - μ)/σ\n",
    "    pdf = exp(-0.5*z^2)/(sqrt(2*pi)*σ)\n",
    "    return pdf\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x  = -3:0.1:3\n",
    "xb = x[x.<=1.645]\n",
    "\n",
    "p1 = plot( x,ϕNS.(x),\n",
    "           linecolor = :red,\n",
    "           linewidth = 2,\n",
    "           ylims = (0,0.42),\n",
    "           xlims = (-3,3),\n",
    "           legend = nothing,\n",
    "           title = \"pdf of N(0,1)\",\n",
    "           xlabel = \"x\",\n",
    "           ylabel = \"\",\n",
    "           annotation = (1.75,0.25,text(\"the area covers\\nup to x=1.645\",8)) )\n",
    "plot!(xb,ϕNS.(xb),fillcolor=:red,linewidth=2,legend=nothing,fill=(0,:red))\n",
    "display(p1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculating Prob(x<=1.645)\n",
    "\n",
    "The next cell calculates (by numeric integration)\n",
    "\n",
    "$\\int_{-\\infty}^{1.645}\\phi(x,0,\\sigma)dx$,\n",
    "\n",
    "where $\\phi(x,\\mu,\\sigma)$ is the pdf of an $N(\\mu,\\sigma)$ variable.\n",
    "\n",
    "\n",
    "The input to `quadgk` should be a function with only one argument. We do that by creating an anonymous function\n",
    "```\n",
    "x->fn(x,a),\n",
    "```\n",
    "assuming that `a` has a value already."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Pr(x<=1.64) according to N(0,1):     0.950\n",
      "\n",
      "Pr(x<=1.64) according to N(0,2):     0.795\n"
     ]
    }
   ],
   "source": [
    "cdf1, = quadgk(x->ϕNS(x),-Inf,1.645)       #N(0,1)\n",
    "printlnPs(\"\\nPr(x<=1.64) according to N(0,1):\", cdf1)\n",
    "\n",
    "cdf2, = quadgk(x->ϕNS(x,0,2),-Inf,1.645)   #N(0,σ=2)\n",
    "printlnPs(\"\\nPr(x<=1.64) according to N(0,2):\", cdf2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Numerical Derivatives\n",
    "\n",
    "Numerical derivatives can be calculated by a crude finite difference (see `NumDer()` below) or by the much more sophisticated routines in the `ForwardDiff` package.\n",
    "\n",
    "To do this calculation for many `x` values, we loop (either a `for` loop or list comprehension). Dot syntax, `ForwardDiff.derivative.()`, seems to work, but is not documented."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "NumDer (generic function with 1 method)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function NumDer(fun,b0,h)     #crude function for a centered numerical derivative\n",
    "    bminus = b0 .- h\n",
    "    bplus  = b0 .+ h\n",
    "    hh     = bplus - bminus\n",
    "    fplus  = fun(bplus)\n",
    "    fminus = fun(bminus)\n",
    "    D      = (fplus-fminus)/hh\n",
    "    return D\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fn1 (generic function with 1 method)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function fn1(x,a)                 #a simple function, to be differentiated\n",
    "   return (x - 1.1)^2 - a\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The derivative at x=2 is (from two different methods): \n",
      "     1.800     1.800\n",
      "\n"
     ]
    }
   ],
   "source": [
    "x0 = 2\n",
    "\n",
    "dydx_A = NumDer(x->fn1(x,0.5),x0,0.01)           #differentiate fn1(x,0.5) at x = x0\n",
    "dydx_B = ForwardDiff.derivative(x->fn1(x,0.5),x0)\n",
    "\n",
    "println(\"The derivative at x=$x0 is (from two different methods): \")\n",
    "printmat([dydx_A dydx_B])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "now lets plot this\n"
     ]
    }
   ],
   "source": [
    "x = -3:6/99:6          #calculate the derivative at many points\n",
    "\n",
    "dydx_A = [NumDer(x->fn1(x,0.5),x[i],0.01) for i=1:length(x)]  #list comprehension as a quick loop\n",
    "\n",
    "dydx_B = [ForwardDiff.derivative(x->fn1(x,0.5),x[i]) for i=1:length(x)]\n",
    "\n",
    "println(\"now lets plot this\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p1 = plot( [x x x],[fn1.(x,0.5) dydx_A dydx_B],\n",
    "           line = [:solid :dot :dash],\n",
    "           linecolor = [:black :red :blue],\n",
    "           label = [\"fn1()\" \"crude derivative\" \"from ForwardDiff\"],\n",
    "           legend = :top,\n",
    "           title = \"fn1() and its derivative\",\n",
    "           xlabel = \"x\",\n",
    "           ylabel = \"\" )\n",
    "display(p1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Comments on Using the ForwardDiff Package (extra)\n",
    "\n",
    "The `ForwardDiff` package applies an interesting approach to calculate derivatives, using a special number type (\"dual numbers\"). This means that your code must be able to handle such numbers. In most cases, that is not a problem, but you may have to watch out if you create arrays to store (intermediate?) results inside the function. See the examples below"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fnDoesWork (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function fnDoesNotWork(b,a)\n",
    "    z = zeros(length(b))              #will not work with ForwardDiff, since\n",
    "    for i = 1:length(z)               #z cannot store dual numbers\n",
    "        z[i] = b[i]*i\n",
    "    end\n",
    "    return sum(z) + a\n",
    "end\n",
    "\n",
    "function fnDoesWork(b,a)\n",
    "    z = zeros(eltype(b),length(b))   #will work with ForwardDiff, since\n",
    "    for i = 1:length(z)              #when b is a dual number, so is z \n",
    "        z[i] = b[i]*i                #could also start with z = similar(b)\n",
    "    end\n",
    "    return sum(z) + a\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Vector{Float64}:\n",
       " 1.0\n",
       " 2.0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b0 = [1.5,2]\n",
    "\n",
    "#ForwardDiff.gradient(b->fnDoesNotWork(b,1),b0)      #uncomment to get an error\n",
    "ForwardDiff.gradient(b->fnDoesWork(b,1),b0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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