{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Plotting smooth curves\n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
\n", " \n", " Loading BokehJS ...\n", "
" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/javascript": [ "\n", "(function(root) {\n", " function now() {\n", " return new Date();\n", " }\n", "\n", " var force = true;\n", "\n", " if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n", " root._bokeh_onload_callbacks = [];\n", " root._bokeh_is_loading = undefined;\n", " }\n", "\n", " var JS_MIME_TYPE = 'application/javascript';\n", " var HTML_MIME_TYPE = 'text/html';\n", " var EXEC_MIME_TYPE = 'application/vnd.bokehjs_exec.v0+json';\n", " var CLASS_NAME = 'output_bokeh rendered_html';\n", "\n", " /**\n", " * Render data to the DOM node\n", " */\n", " function render(props, node) {\n", " var script = document.createElement(\"script\");\n", " node.appendChild(script);\n", " }\n", "\n", " /**\n", " * Handle when an output is cleared or removed\n", " */\n", " function handleClearOutput(event, handle) {\n", " var cell = handle.cell;\n", "\n", " var id = cell.output_area._bokeh_element_id;\n", " var server_id = cell.output_area._bokeh_server_id;\n", " // Clean up Bokeh references\n", " if (id != null && id in Bokeh.index) {\n", " Bokeh.index[id].model.document.clear();\n", " delete Bokeh.index[id];\n", " }\n", "\n", " if (server_id !== undefined) {\n", " // Clean up Bokeh references\n", " var cmd = \"from bokeh.io.state import curstate; print(curstate().uuid_to_server['\" + server_id + \"'].get_sessions()[0].document.roots[0]._id)\";\n", " cell.notebook.kernel.execute(cmd, {\n", " iopub: {\n", " output: function(msg) {\n", " var id = msg.content.text.trim();\n", " if (id in Bokeh.index) {\n", " Bokeh.index[id].model.document.clear();\n", " delete Bokeh.index[id];\n", " }\n", " }\n", " }\n", " });\n", " // Destroy server and session\n", " var cmd = \"import bokeh.io.notebook as ion; ion.destroy_server('\" + server_id + \"')\";\n", " cell.notebook.kernel.execute(cmd);\n", " }\n", " }\n", "\n", " /**\n", " * Handle when a new output is added\n", " */\n", " function handleAddOutput(event, handle) {\n", " var output_area = handle.output_area;\n", " var output = handle.output;\n", "\n", " // limit handleAddOutput to display_data with EXEC_MIME_TYPE content only\n", " if ((output.output_type != \"display_data\") || (!output.data.hasOwnProperty(EXEC_MIME_TYPE))) {\n", " return\n", " }\n", "\n", " var toinsert = output_area.element.find(\".\" + CLASS_NAME.split(' ')[0]);\n", "\n", " if (output.metadata[EXEC_MIME_TYPE][\"id\"] !== undefined) {\n", " toinsert[toinsert.length - 1].firstChild.textContent = output.data[JS_MIME_TYPE];\n", " // store reference to embed id on output_area\n", " output_area._bokeh_element_id = output.metadata[EXEC_MIME_TYPE][\"id\"];\n", " }\n", " if (output.metadata[EXEC_MIME_TYPE][\"server_id\"] !== undefined) {\n", " var bk_div = document.createElement(\"div\");\n", " bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n", " var script_attrs = bk_div.children[0].attributes;\n", " for (var i = 0; i < script_attrs.length; i++) {\n", " toinsert[toinsert.length - 1].firstChild.setAttribute(script_attrs[i].name, script_attrs[i].value);\n", " }\n", " // store reference to server id on output_area\n", " output_area._bokeh_server_id = output.metadata[EXEC_MIME_TYPE][\"server_id\"];\n", " }\n", " }\n", "\n", " function register_renderer(events, OutputArea) {\n", "\n", " function append_mime(data, metadata, element) {\n", " // create a DOM node to render to\n", " var toinsert = this.create_output_subarea(\n", " metadata,\n", " CLASS_NAME,\n", " EXEC_MIME_TYPE\n", " );\n", " this.keyboard_manager.register_events(toinsert);\n", " // Render to node\n", " var props = {data: data, metadata: metadata[EXEC_MIME_TYPE]};\n", " render(props, toinsert[toinsert.length - 1]);\n", " element.append(toinsert);\n", " return toinsert\n", " }\n", "\n", " /* Handle when an output is cleared or removed */\n", " events.on('clear_output.CodeCell', handleClearOutput);\n", " events.on('delete.Cell', handleClearOutput);\n", "\n", " /* Handle when a new output is added */\n", " events.on('output_added.OutputArea', handleAddOutput);\n", "\n", " /**\n", " * Register the mime type and append_mime function with output_area\n", " */\n", " OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n", " /* Is output safe? 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\\n\"+\n", " \"

\\n\"+\n", " \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n", " \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n", " \"

\\n\"+\n", " \"\\n\"+\n", " \"\\n\"+\n", " \"from bokeh.resources import INLINE\\n\"+\n", " \"output_notebook(resources=INLINE)\\n\"+\n", " \"\\n\"+\n", " \"
\"}};\n", "\n", " function display_loaded() {\n", " var el = document.getElementById(\"1001\");\n", " if (el != null) {\n", " el.textContent = \"BokehJS is loading...\";\n", " }\n", " if (root.Bokeh !== undefined) {\n", " if (el != null) {\n", " el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n", " }\n", " } else if (Date.now() < root._bokeh_timeout) {\n", " setTimeout(display_loaded, 100)\n", " }\n", " }\n", "\n", "\n", " function run_callbacks() {\n", " try {\n", " root._bokeh_onload_callbacks.forEach(function(callback) {\n", " if (callback != null)\n", " callback();\n", " });\n", " } finally {\n", " delete root._bokeh_onload_callbacks\n", " }\n", " console.debug(\"Bokeh: all callbacks have finished\");\n", " }\n", "\n", " function load_libs(css_urls, js_urls, callback) {\n", " if (css_urls == null) css_urls = [];\n", " if (js_urls == null) js_urls = [];\n", "\n", " root._bokeh_onload_callbacks.push(callback);\n", " if (root._bokeh_is_loading > 0) {\n", " console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n", " return null;\n", " }\n", " if (js_urls == null || js_urls.length === 0) {\n", " run_callbacks();\n", " return null;\n", " }\n", " console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n", " root._bokeh_is_loading = css_urls.length + js_urls.length;\n", "\n", " function on_load() {\n", " root._bokeh_is_loading--;\n", " if (root._bokeh_is_loading === 0) {\n", " console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n", " run_callbacks()\n", " }\n", " }\n", "\n", " function on_error() {\n", " console.error(\"failed to load \" + url);\n", " }\n", "\n", " for (var i = 0; i < css_urls.length; i++) {\n", " var url = css_urls[i];\n", " const element = document.createElement(\"link\");\n", " element.onload = on_load;\n", " element.onerror = on_error;\n", " element.rel = \"stylesheet\";\n", " element.type = \"text/css\";\n", " element.href = url;\n", " console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n", " document.body.appendChild(element);\n", " }\n", "\n", " for (var i = 0; i < js_urls.length; i++) {\n", " var url = js_urls[i];\n", " var element = document.createElement('script');\n", " element.onload = on_load;\n", " element.onerror = on_error;\n", " element.async = false;\n", " element.src = url;\n", " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", " document.head.appendChild(element);\n", " }\n", " };var element = document.getElementById(\"1001\");\n", " if (element == null) {\n", " console.error(\"Bokeh: ERROR: autoload.js configured with elementid '1001' but no matching script tag was found. \")\n", " return false;\n", " }\n", "\n", " function inject_raw_css(css) {\n", " const element = document.createElement(\"style\");\n", " element.appendChild(document.createTextNode(css));\n", " document.body.appendChild(element);\n", " }\n", "\n", " var js_urls = [\"https://cdn.pydata.org/bokeh/release/bokeh-1.3.4.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-widgets-1.3.4.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-tables-1.3.4.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-gl-1.3.4.min.js\"];\n", " var css_urls = [];\n", "\n", " var inline_js = [\n", " function(Bokeh) {\n", " Bokeh.set_log_level(\"info\");\n", " },\n", " \n", " function(Bokeh) {\n", " \n", " },\n", " function(Bokeh) {} // ensure no trailing comma for IE\n", " ];\n", "\n", " function run_inline_js() {\n", " \n", " if ((root.Bokeh !== undefined) || (force === true)) {\n", " for (var i = 0; i < inline_js.length; i++) {\n", " inline_js[i].call(root, root.Bokeh);\n", " }if (force === true) {\n", " display_loaded();\n", " }} else if (Date.now() < root._bokeh_timeout) {\n", " setTimeout(run_inline_js, 100);\n", " } else if (!root._bokeh_failed_load) {\n", " console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n", " root._bokeh_failed_load = true;\n", " } else if (force !== true) {\n", " var cell = $(document.getElementById(\"1001\")).parents('.cell').data().cell;\n", " cell.output_area.append_execute_result(NB_LOAD_WARNING)\n", " }\n", "\n", " }\n", "\n", " if (root._bokeh_is_loading === 0) {\n", " console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n", " run_inline_js();\n", " } else {\n", " load_libs(css_urls, js_urls, function() {\n", " console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n", " run_inline_js();\n", " });\n", " }\n", "}(window));" ], "application/vnd.bokehjs_load.v0+json": "\n(function(root) {\n function now() {\n return new Date();\n }\n\n var force = true;\n\n if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n root._bokeh_onload_callbacks = [];\n root._bokeh_is_loading = undefined;\n }\n\n \n\n \n if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n root._bokeh_timeout = Date.now() + 5000;\n root._bokeh_failed_load = false;\n }\n\n var NB_LOAD_WARNING = {'data': {'text/html':\n \"
\\n\"+\n \"

\\n\"+\n \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n \"

\\n\"+\n \"\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"\\n\"+\n \"
\"}};\n\n function display_loaded() {\n var el = document.getElementById(\"1001\");\n if (el != null) {\n el.textContent = \"BokehJS is loading...\";\n }\n if (root.Bokeh !== undefined) {\n if (el != null) {\n el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n }\n } else if (Date.now() < root._bokeh_timeout) {\n setTimeout(display_loaded, 100)\n }\n }\n\n\n function run_callbacks() {\n try {\n root._bokeh_onload_callbacks.forEach(function(callback) {\n if (callback != null)\n callback();\n });\n } finally {\n delete root._bokeh_onload_callbacks\n }\n console.debug(\"Bokeh: all callbacks have finished\");\n }\n\n function load_libs(css_urls, js_urls, callback) {\n if (css_urls == null) css_urls = [];\n if (js_urls == null) js_urls = [];\n\n root._bokeh_onload_callbacks.push(callback);\n if (root._bokeh_is_loading > 0) {\n console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n return null;\n }\n if (js_urls == null || js_urls.length === 0) {\n run_callbacks();\n return null;\n }\n console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n function on_load() {\n root._bokeh_is_loading--;\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n run_callbacks()\n }\n }\n\n function on_error() {\n console.error(\"failed to load \" + url);\n }\n\n for (var i = 0; i < css_urls.length; i++) {\n var url = css_urls[i];\n const element = document.createElement(\"link\");\n element.onload = on_load;\n element.onerror = on_error;\n element.rel = \"stylesheet\";\n element.type = \"text/css\";\n element.href = url;\n console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n document.body.appendChild(element);\n }\n\n for (var i = 0; i < js_urls.length; i++) {\n var url = js_urls[i];\n var element = document.createElement('script');\n element.onload = on_load;\n element.onerror = on_error;\n element.async = false;\n element.src = url;\n console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n document.head.appendChild(element);\n }\n };var element = document.getElementById(\"1001\");\n if (element == null) {\n console.error(\"Bokeh: ERROR: autoload.js configured with elementid '1001' but no matching script tag was found. \")\n return false;\n }\n\n function inject_raw_css(css) {\n const element = document.createElement(\"style\");\n element.appendChild(document.createTextNode(css));\n document.body.appendChild(element);\n }\n\n var js_urls = [\"https://cdn.pydata.org/bokeh/release/bokeh-1.3.4.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-widgets-1.3.4.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-tables-1.3.4.min.js\", \"https://cdn.pydata.org/bokeh/release/bokeh-gl-1.3.4.min.js\"];\n var css_urls = [];\n\n var inline_js = [\n function(Bokeh) {\n Bokeh.set_log_level(\"info\");\n },\n \n function(Bokeh) {\n \n },\n function(Bokeh) {} // ensure no trailing comma for IE\n ];\n\n function run_inline_js() {\n \n if ((root.Bokeh !== undefined) || (force === true)) {\n for (var i = 0; i < inline_js.length; i++) {\n inline_js[i].call(root, root.Bokeh);\n }if (force === true) {\n display_loaded();\n }} else if (Date.now() < root._bokeh_timeout) {\n setTimeout(run_inline_js, 100);\n } else if (!root._bokeh_failed_load) {\n console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n root._bokeh_failed_load = true;\n } else if (force !== true) {\n var cell = $(document.getElementById(\"1001\")).parents('.cell').data().cell;\n cell.output_area.append_execute_result(NB_LOAD_WARNING)\n }\n\n }\n\n if (root._bokeh_is_loading === 0) {\n console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n run_inline_js();\n } else {\n load_libs(css_urls, js_urls, function() {\n console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n run_inline_js();\n });\n }\n}(window));" }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import scipy.special\n", "\n", "import bokeh.plotting\n", "import bokeh.io\n", "bokeh.io.output_notebook()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "\n", "Sometimes you want to plot smooth functions, as opposed to measured data like we have done so far. To do this, you can use Numpy and/or Scipy to generate arrays of values of smooth functions.\n", "\n", "We will plot the [Airy disk](https://en.wikipedia.org/wiki/Airy_disk), which we encounter in biology when doing microscopy as the diffraction pattern of light passing through a pinhole. Here is a picture of the diffraction pattern from a laser (with the main peak overexposed).\n", "\n", "
\n", "\n", "![Airy disk](airy_disk.png)\n", " \n", "
\n", "\n", "The equation for the radial light intensity of an Airy disk is\n", "\n", "\\begin{align}\n", "\\frac{I(x)}{I_0} = 4 \\left(\\frac{J_1(x)}{x}\\right)^2,\n", "\\end{align}\n", "\n", "where $I_0$ is the maximum intensity (the intensity at the center of the image) and $x$ is the radial distance from the center. Here, $J_1(x)$ is the first order Bessel function of the first kind. Yeesh. How do we plot *that*?\n", "\n", "Fortunately, SciPy has lots of special functions available. Specifically, `scipy.special.j1()` computes exactly what we are after! We pass in a NumPy array that has the values of $x$ we want to plot and then compute the $y$-values using the expression for the normalized intensity.\n", "\n", "To plot a smooth curve, we use the `np.linspace()` function with lots of points. We then connect the points with straight lines, which to the eye look like a smooth curve. Let's try it. We'll use 400 points, which I find is a good rule of thumb for not-too-quickly-oscillating functions." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# The x-values we want\n", "x = np.linspace(-15, 15, 400)\n", "\n", "# The normalized intensity\n", "norm_I = 4 * (scipy.special.j1(x) / x)**2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now that we have the values we want to plot, we could construct a Pandas `DataFrame` to pass in as the `source` to `p.line()`. We do not need to take this extra step, though. If we instead leave `source` unspecified, and pass in NumPy arrays for `x` and `y`, Bokeh will directly use those in constructing the plot." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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x_axis_label='x',\n", " y_axis_label='I(x)/I₀',\n", ")\n", "\n", "p.line(\n", " x=x,\n", " y=norm_I,\n", " line_join='bevel',\n", " line_width=2,\n", ")\n", "\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We could also plot dots (which doesn't make sense here, but we'll show it just to see who the line joining works to make a plot of a smooth function)." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "\n", "\n", "
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"\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "There is one detail I swept under the rug here. What happens if we compute the function for $x = 0$?" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/Users/Justin/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:1: RuntimeWarning: invalid value encountered in double_scalars\n", " \"\"\"Entry point for launching an IPython kernel.\n" ] }, { "data": { "text/plain": [ "nan" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "4 * (scipy.special.j1(0) / 0)**2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We get a `RuntimeWarning` because we divided by zero. We know that\n", "\n", "\\begin{align}\n", "\\lim_{x\\to 0} \\frac{J_1(x)}{x} = \\frac{1}{2},\n", "\\end{align}\n", "\n", "so we could write a new function that checks if $x = 0$ and returns the appropriate limit for $x = 0$. In the `x` array I constructed for the plot, we hopped over zero, so it was never evaluated. If we were being careful, we could write our own Airy function that deals with this." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "def airy_disk(x):\n", " \"\"\"Compute the Airy disk.\"\"\"\n", " # Set up output array\n", " res = np.empty_like(x)\n", " \n", " # Where x is very close to zero (use np.isclose)\n", " near_zero = np.isclose(x, 0)\n", " \n", " # Compute values where x is close to zero\n", " res[near_zero] = 1.0\n", "\n", " # Everywhere else\n", " x_vals = x[~near_zero]\n", " res[~near_zero] = 4 * (scipy.special.j1(x_vals) / x_vals)**2\n", " \n", " return res" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Computing environment" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "CPython 3.7.4\n", "IPython 7.1.1\n", "\n", "numpy 1.17.2\n", "scipy 1.3.1\n", "bokeh 1.3.4\n", "jupyterlab 1.1.4\n" ] } ], "source": [ "%load_ext watermark\n", "%watermark -v -p numpy,scipy,bokeh,jupyterlab" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.4" } }, "nbformat": 4, "nbformat_minor": 4 }