SEM.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Estimation of the mean and its uncertainty"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The mean value, or some other measure of the center of a distribution, is often the first thing one looks for in a data set, or a subsample of a data set.  For example, in a long time series of surface temperature, one might want to know the mean value for each year, together with some idea of how much each mean might vary from its \"true\" value in an ensemble of worlds.  Here the difficulty of the frequentist interpretation is evident--we only have one world.  We can have many realizations of a numerical model of that world, though, so at least via numerical experiments it is possible to generate an ensemble of time series that are different because of some random variations in initial conditions and/or forcing.\n",
    "\n",
    "Let's start exploring these ideas with the simplest possible case, however: random numbers generated from a known distribution, not necessarily a Gaussian.  The experimental approach illustrated here, in which we calculate statistics from a generated ensemble, is complementary to the theoretical approach, in which analytic expressions are calculated for PDFs of interest.  The advantages of the experimental approach are that (1) it doesn't require mathematical sophistication, (2) it can be applied to situations for which the analytic approach would be very difficult or even impossible, (3) it is relatively easy to understand and visualize."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Standard Error of the Mean (SEM)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib notebook\n",
    "\n",
    "# Our standard imports:\n",
    "import numpy as np\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Access to many standard distributions:\n",
    "import scipy.stats as ss"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will work with 2-D arrays of random numbers.  We arbitrarily choose the first dimension to index the member of the ensemble, and the second to index sequential position (e.g. time, perhaps in a uniformly sampled time series) in the ensemble.\n",
    "\n",
    "Start by verifying the usual formula for the Standard Error of the Mean (SEM), which gives the standard deviation of the sample mean of $N$ values from a normal distribution:\n",
    "$$ \\mbox{SEM} = \\frac{\\sigma}{\\sqrt{N}}$$\n",
    "where $\\sigma$ is the standard deviation. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "with 3 points, ybar mean and std are 0.00274, 0.572; SEM is 0.577\n",
      "The mean of the estimated standard deviations is 0.885\n",
      "The median is 0.825\n",
      "The square root of the mean estimated variance is 1.000\n"
     ]
    }
   ],
   "source": [
    "nens = 10000  # number of realizations\n",
    "nt = 3       # length of each time series\n",
    "y = np.random.randn(nens, nt)  # sigma is 1\n",
    "\n",
    "SEM = 1 / np.sqrt(nt)\n",
    "\n",
    "ybar = y.mean(axis=1)\n",
    "ystd = y.std(axis=1, ddof=1)\n",
    " \n",
    "print(\"with %d points, ybar mean and std are %.5f, %.3f; SEM is %.3f\" % \n",
    "      (nt, ybar.mean(), ybar.std(), SEM))\n",
    "print(\"The mean of the estimated standard deviations is %.3f\" % ystd.mean())\n",
    "print(\"The median is %.3f\" % np.median(ystd))\n",
    "print(\"The square root of the mean estimated variance is %.3f\" % \n",
    "      np.sqrt((ystd**2).mean()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the cell above we also calculated the mean of the standard deviations, and the square root of the mean sample variance, $S^2$, using \n",
    "$$S^2 = \\frac{1}{N-1}\\sum_{i=0}^{N-1} (X_i - \\bar{X})^2.$$\n",
    "Notice that in this case we don't know the true standard deviation of the population, so we estimate it from the sample, using the `ddof` (\"deduct degrees of freedom\") argument to reduce the number of degrees of freedom (DOF) from $N$ to $N-1$; we lost one DOF by subtracting the sample mean."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's cycle through some values of nt and see how the SEM varies.  In addition to plotting the experimental and theoretical values, we will plot the *ensemble mean* of the values we get by applying the theoretical formula to *each* member of the ensemble, as if we had a single realization of an experiment, and we wished to estimate the uncertainty in the mean from that experiment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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       "    // Updates the figure title.\n",
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       "            cursor = 'default';\n",
       "            break;\n",
       "        case 2:\n",
       "            cursor = 'crosshair';\n",
       "            break;\n",
       "        case 3:\n",
       "            cursor = 'move';\n",
       "            break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_message = function (fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function (fig, _msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function (fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_history_buttons = function (fig, msg) {\n",
       "    for (var key in msg) {\n",
       "        if (!(key in fig.buttons)) {\n",
       "            continue;\n",
       "        }\n",
       "        fig.buttons[key].disabled = !msg[key];\n",
       "        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {\n",
       "    if (msg['mode'] === 'PAN') {\n",
       "        fig.buttons['Pan'].classList.add('active');\n",
       "        fig.buttons['Zoom'].classList.remove('active');\n",
       "    } else if (msg['mode'] === 'ZOOM') {\n",
       "        fig.buttons['Pan'].classList.remove('active');\n",
       "        fig.buttons['Zoom'].classList.add('active');\n",
       "    } else {\n",
       "        fig.buttons['Pan'].classList.remove('active');\n",
       "        fig.buttons['Zoom'].classList.remove('active');\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function () {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message('ack', {});\n",
       "};\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function (fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = 'image/png';\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src\n",
       "                );\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data\n",
       "            );\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        } else if (\n",
       "            typeof evt.data === 'string' &&\n",
       "            evt.data.slice(0, 21) === 'data:image/png;base64'\n",
       "        ) {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig['handle_' + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\n",
       "                \"No handler for the '\" + msg_type + \"' message type: \",\n",
       "                msg\n",
       "            );\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\n",
       "                    \"Exception inside the 'handler_\" + msg_type + \"' callback:\",\n",
       "                    e,\n",
       "                    e.stack,\n",
       "                    msg\n",
       "                );\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "};\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function (e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e) {\n",
       "        e = window.event;\n",
       "    }\n",
       "    if (e.target) {\n",
       "        targ = e.target;\n",
       "    } else if (e.srcElement) {\n",
       "        targ = e.srcElement;\n",
       "    }\n",
       "    if (targ.nodeType === 3) {\n",
       "        // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "    }\n",
       "\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    var boundingRect = targ.getBoundingClientRect();\n",
       "    var x = e.pageX - (boundingRect.left + document.body.scrollLeft);\n",
       "    var y = e.pageY - (boundingRect.top + document.body.scrollTop);\n",
       "\n",
       "    return { x: x, y: y };\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys(original) {\n",
       "    return Object.keys(original).reduce(function (obj, key) {\n",
       "        if (typeof original[key] !== 'object') {\n",
       "            obj[key] = original[key];\n",
       "        }\n",
       "        return obj;\n",
       "    }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function (event, name) {\n",
       "    var canvas_pos = mpl.findpos(event);\n",
       "\n",
       "    if (name === 'button_press') {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * this.ratio;\n",
       "    var y = canvas_pos.y * this.ratio;\n",
       "\n",
       "    this.send_message(name, {\n",
       "        x: x,\n",
       "        y: y,\n",
       "        button: event.button,\n",
       "        step: event.step,\n",
       "        guiEvent: simpleKeys(event),\n",
       "    });\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function (_event, _name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.key_event = function (event, name) {\n",
       "    // Prevent repeat events\n",
       "    if (name === 'key_press') {\n",
       "        if (event.which === this._key) {\n",
       "            return;\n",
       "        } else {\n",
       "            this._key = event.which;\n",
       "        }\n",
       "    }\n",
       "    if (name === 'key_release') {\n",
       "        this._key = null;\n",
       "    }\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which !== 17) {\n",
       "        value += 'ctrl+';\n",
       "    }\n",
       "    if (event.altKey && event.which !== 18) {\n",
       "        value += 'alt+';\n",
       "    }\n",
       "    if (event.shiftKey && event.which !== 16) {\n",
       "        value += 'shift+';\n",
       "    }\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });\n",
       "    return false;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function (name) {\n",
       "    if (name === 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message('toolbar_button', { name: name });\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "\n",
       "///////////////// REMAINING CONTENT GENERATED BY embed_js.py /////////////////\n",
       "// prettier-ignore\n",
       "var _JSXTOOLS_RESIZE_OBSERVER=function(A){var t,i=new WeakMap,n=new WeakMap,a=new WeakMap,r=new WeakMap,o=new Set;function s(e){if(!(this instanceof s))throw new TypeError(\"Constructor requires 'new' operator\");i.set(this,e)}function h(){throw new TypeError(\"Function is not a constructor\")}function c(e,t,i,n){e=0 in arguments?Number(arguments[0]):0,t=1 in arguments?Number(arguments[1]):0,i=2 in arguments?Number(arguments[2]):0,n=3 in arguments?Number(arguments[3]):0,this.right=(this.x=this.left=e)+(this.width=i),this.bottom=(this.y=this.top=t)+(this.height=n),Object.freeze(this)}function d(){t=requestAnimationFrame(d);var s=new WeakMap,p=new Set;o.forEach((function(t){r.get(t).forEach((function(i){var r=t instanceof window.SVGElement,o=a.get(t),d=r?0:parseFloat(o.paddingTop),f=r?0:parseFloat(o.paddingRight),l=r?0:parseFloat(o.paddingBottom),u=r?0:parseFloat(o.paddingLeft),g=r?0:parseFloat(o.borderTopWidth),m=r?0:parseFloat(o.borderRightWidth),w=r?0:parseFloat(o.borderBottomWidth),b=u+f,F=d+l,v=(r?0:parseFloat(o.borderLeftWidth))+m,W=g+w,y=r?0:t.offsetHeight-W-t.clientHeight,E=r?0:t.offsetWidth-v-t.clientWidth,R=b+v,z=F+W,M=r?t.width:parseFloat(o.width)-R-E,O=r?t.height:parseFloat(o.height)-z-y;if(n.has(t)){var k=n.get(t);if(k[0]===M&&k[1]===O)return}n.set(t,[M,O]);var S=Object.create(h.prototype);S.target=t,S.contentRect=new c(u,d,M,O),s.has(i)||(s.set(i,[]),p.add(i)),s.get(i).push(S)}))})),p.forEach((function(e){i.get(e).call(e,s.get(e),e)}))}return s.prototype.observe=function(i){if(i instanceof window.Element){r.has(i)||(r.set(i,new Set),o.add(i),a.set(i,window.getComputedStyle(i)));var n=r.get(i);n.has(this)||n.add(this),cancelAnimationFrame(t),t=requestAnimationFrame(d)}},s.prototype.unobserve=function(i){if(i instanceof window.Element&&r.has(i)){var n=r.get(i);n.has(this)&&(n.delete(this),n.size||(r.delete(i),o.delete(i))),n.size||r.delete(i),o.size||cancelAnimationFrame(t)}},A.DOMRectReadOnly=c,A.ResizeObserver=s,A.ResizeObserverEntry=h,A}; // eslint-disable-line\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Left button pans, Right button zooms\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
       "\n",
       "mpl.default_extension = \"png\";/* global mpl */\n",
       "\n",
       "var comm_websocket_adapter = function (comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function () {\n",
       "        comm.close();\n",
       "    };\n",
       "    ws.send = function (m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function (msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data']);\n",
       "    });\n",
       "    return ws;\n",
       "};\n",
       "\n",
       "mpl.mpl_figure_comm = function (comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = document.getElementById(id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm);\n",
       "\n",
       "    function ondownload(figure, _format) {\n",
       "        window.open(figure.canvas.toDataURL());\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy, ondownload, element);\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element;\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error('Failed to find cell for figure', id, fig);\n",
       "        return;\n",
       "    }\n",
       "    fig.cell_info[0].output_area.element.on(\n",
       "        'cleared',\n",
       "        { fig: fig },\n",
       "        fig._remove_fig_handler\n",
       "    );\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function (fig, msg) {\n",
       "    var width = fig.canvas.width / fig.ratio;\n",
       "    fig.cell_info[0].output_area.element.off(\n",
       "        'cleared',\n",
       "        fig._remove_fig_handler\n",
       "    );\n",
       "    fig.resizeObserverInstance.unobserve(fig.canvas_div);\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable();\n",
       "    fig.parent_element.innerHTML =\n",
       "        '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "    fig.close_ws(fig, msg);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.close_ws = function (fig, msg) {\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function (_remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width / this.ratio;\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] =\n",
       "        '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function () {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message('ack', {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () {\n",
       "        fig.push_to_output();\n",
       "    }, 1000);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function () {\n",
       "    var fig = this;\n",
       "\n",
       "    var toolbar = document.createElement('div');\n",
       "    toolbar.classList = 'btn-toolbar';\n",
       "    this.root.appendChild(toolbar);\n",
       "\n",
       "    function on_click_closure(name) {\n",
       "        return function (_event) {\n",
       "            return fig.toolbar_button_onclick(name);\n",
       "        };\n",
       "    }\n",
       "\n",
       "    function on_mouseover_closure(tooltip) {\n",
       "        return function (event) {\n",
       "            if (!event.currentTarget.disabled) {\n",
       "                return fig.toolbar_button_onmouseover(tooltip);\n",
       "            }\n",
       "        };\n",
       "    }\n",
       "\n",
       "    fig.buttons = {};\n",
       "    var buttonGroup = document.createElement('div');\n",
       "    buttonGroup.classList = 'btn-group';\n",
       "    var button;\n",
       "    for (var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            /* Instead of a spacer, we start a new button group. */\n",
       "            if (buttonGroup.hasChildNodes()) {\n",
       "                toolbar.appendChild(buttonGroup);\n",
       "            }\n",
       "            buttonGroup = document.createElement('div');\n",
       "            buttonGroup.classList = 'btn-group';\n",
       "            continue;\n",
       "        }\n",
       "\n",
       "        button = fig.buttons[name] = document.createElement('button');\n",
       "        button.classList = 'btn btn-default';\n",
       "        button.href = '#';\n",
       "        button.title = name;\n",
       "        button.innerHTML = '<i class=\"fa ' + image + ' fa-lg\"></i>';\n",
       "        button.addEventListener('click', on_click_closure(method_name));\n",
       "        button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n",
       "        buttonGroup.appendChild(button);\n",
       "    }\n",
       "\n",
       "    if (buttonGroup.hasChildNodes()) {\n",
       "        toolbar.appendChild(buttonGroup);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = document.createElement('span');\n",
       "    status_bar.classList = 'mpl-message pull-right';\n",
       "    toolbar.appendChild(status_bar);\n",
       "    this.message = status_bar;\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = document.createElement('div');\n",
       "    buttongrp.classList = 'btn-group inline pull-right';\n",
       "    button = document.createElement('button');\n",
       "    button.classList = 'btn btn-mini btn-primary';\n",
       "    button.href = '#';\n",
       "    button.title = 'Stop Interaction';\n",
       "    button.innerHTML = '<i class=\"fa fa-power-off icon-remove icon-large\"></i>';\n",
       "    button.addEventListener('click', function (_evt) {\n",
       "        fig.handle_close(fig, {});\n",
       "    });\n",
       "    button.addEventListener(\n",
       "        'mouseover',\n",
       "        on_mouseover_closure('Stop Interaction')\n",
       "    );\n",
       "    buttongrp.appendChild(button);\n",
       "    var titlebar = this.root.querySelector('.ui-dialog-titlebar');\n",
       "    titlebar.insertBefore(buttongrp, titlebar.firstChild);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._remove_fig_handler = function (event) {\n",
       "    var fig = event.data.fig;\n",
       "    if (event.target !== this) {\n",
       "        // Ignore bubbled events from children.\n",
       "        return;\n",
       "    }\n",
       "    fig.close_ws(fig, {});\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function (el) {\n",
       "    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function (el) {\n",
       "    // this is important to make the div 'focusable\n",
       "    el.setAttribute('tabindex', 0);\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    } else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function (event, _name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager) {\n",
       "        manager = IPython.keyboard_manager;\n",
       "    }\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which === 13) {\n",
       "        this.canvas_div.blur();\n",
       "        // select the cell after this one\n",
       "        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
       "        IPython.notebook.select(index + 1);\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_save = function (fig, _msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "};\n",
       "\n",
       "mpl.find_output_cell = function (html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i = 0; i < ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code') {\n",
       "            for (var j = 0; j < cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] === html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "};\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel !== null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target(\n",
       "        'matplotlib',\n",
       "        mpl.mpl_figure_comm\n",
       "    );\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<img src=\"data:image/png;base64,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\" width=\"640\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nens = 10000\n",
    "nts = np.arange(3, 21)\n",
    "n_nts = len(nts)\n",
    "\n",
    "SEMs = 1 / np.sqrt(nts)  # theory, for normal distribution\n",
    "\n",
    "sems_exp = np.zeros(nts.shape)\n",
    "sems_sample = np.zeros(nts.shape)\n",
    "\n",
    "for i, nt in enumerate(nts):\n",
    "    y = np.random.randn(nens, nt)\n",
    "    y_means = y.mean(axis=1)\n",
    "    y_stds = y.std(axis=1, ddof=1)  # note ddof argument!\n",
    "    sems_exp[i] = y_means.std()\n",
    "    sems_sample[i] = y_stds.mean() / np.sqrt(nt)\n",
    "    \n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(nts, sems_exp, 'ro', label='MC std dev of mean')\n",
    "ax.plot(nts, SEMs, 'b-', label='SEM for Normal dist.')\n",
    "ax.plot(nts, sems_sample, 'gx', label='mean sample SEM')\n",
    "#ax.plot(nts, np.median(sems_theory, axis=0), 'bx', label='median SEM')\n",
    "ax.locator_params(axis='x', integer=True)\n",
    "ax.legend(loc='upper right')\n",
    "ax.set_xlabel('Length of sequence');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.close(fig)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is wrong with those green X-marks?  Why do they systematically underestimate the SEM?  They are based on $S$, the estimate of $\\sigma$, and this is biased, as we saw in an earlier cell."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Usually what we want to know is not the SEM but the location of the tails of the sampling distribution of the mean; this can be estimated from the SEM only if the distribution is normal.  The distribution of sample means, with samples taken from a normal population, is not itself Gaussian; rather, it follows the \"Student's t\" distribution.  For small $N$, it has heavier tails than a Gaussian distribution.\n",
    "\n",
    "Let's use the `scipy.stats` module to show the t distribution and compare it to the normal distribution.  (In the following cell you can change `.pdf` to `.cdf` to show the cumulative distribution function instead of the probability distribution function.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 5\n",
    "df = N - 1\n",
    "# Make instances of each distribution:\n",
    "tdist = ss.t(df)\n",
    "ndist = ss.norm()\n",
    "# x-axis array for plotting:\n",
    "x = np.linspace(-5, 5, 1001)\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x, ndist.pdf(x), label='normal')\n",
    "ax.plot(x, tdist.pdf(x), label=r't, $\\nu = %d$' % df)\n",
    "ax.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "plt.close(fig)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can make our own histogram to verify that this distribution is correct.  We will need a large number of samples to resolve the tails and give a reasonably smooth histogram. The t distribution applies to the following random variable:\n",
    "$$\\frac{\\bar{X} - \\mu}{\\frac{S}{  \\sqrt{N}}}$$\n",
    "where $\\mu=0$ is the *true* mean, and $S$ is the standard deviation estimated using the expression given above.  Hence, it is the deviation of the sample mean from the true mean, scaled by the *estimated* SEM.  It is this scaling--the fact that we don't know the *true* $\\sigma$ of the distribution, and therefore the true SEM--that makes the distribution \"t\" rather than \"normal\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "nens = 1000000  # number of realizations\n",
    "nt = 5       # length of each time series\n",
    "y = np.random.randn(nens, nt)\n",
    "ybar = y.mean(axis=1)\n",
    "ystd = y.std(axis=1, ddof=1)\n",
    "normed_ybar = ybar * np.sqrt(nt) / ystd\n",
    "bounds = np.linspace(-5, 5, 51)\n",
    "fig, ax = plt.subplots()\n",
    "ax.hist(normed_ybar, bins=bounds, density=True)\n",
    "x = np.linspace(-5, 5, 501)\n",
    "ax.plot(x, ss.t(nt-1).pdf(x));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "plt.close(fig)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "What does the analogous distribution look like if instead of starting with a Gaussian random variable, you start with a uniform distribution, or with some function of a Gaussian or a uniform distribution?  Try it and find out!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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