{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Visualizing distributions\n", "\n", "[Data set download](https://s3.amazonaws.com/bebi103.caltech.edu/data/gfmt_sleep.csv)\n", "\n", "
" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "nbsphinx": "hidden", "tags": [] }, "outputs": [], "source": [ "#| code-fold: true\n", "\n", "# Colab setup ------------------\n", "import os, sys, subprocess\n", "if \"google.colab\" in sys.modules:\n", " cmd = \"pip install --upgrade polars iqplot watermark\"\n", " process = subprocess.Popen(cmd.split(), stdout=subprocess.PIPE, stderr=subprocess.PIPE)\n", " stdout, stderr = process.communicate()\n", " data_path = \"https://s3.amazonaws.com/bebi103.caltech.edu/data/\"\n", "else:\n", " data_path = \"../data/\"" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/html": [ " \n", "
\n", " \n", " Loading BokehJS ...\n", "
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\\n\"+\n \"\\n\"+\n \"\\n\"+\n \"from bokeh.resources import INLINE\\n\"+\n \"output_notebook(resources=INLINE)\\n\"+\n \"\\n\"+\n \"
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\n", "\n", "You can think of an experiment as **sampling** out of a probability distribution. To make this more concrete, imagine measuring the lengths of eggs laid by a given hen. Each egg will be of a different length than others, but all of the eggs will be about six centimeters long. The probability of getting an egg more than eight centimeters long is very small, whereas the probability of getting an egg between 5.5 and 6.5 centimeters is high. The **generative probability distribution**, the distribution from which experimental data are sampled, then, has high probability mass around six centimeters and low probability mass away from that. We cannot know the generative distribution. We can approximate it with generative models (which we will do in the latter part of the course).\n", "\n", "Here, we will investigate how to make plots to investigate properties about the (unknown) generative distribution of a set of repeated measurements. We will use the [iqplot](https://iqplot.github.io/) to make the plots, but will withhold dwelling on its syntax until the next notebook of this lesson. Instead, we will make a given plot and then discuss how it helps us visualize the underlying generative probability distribution of a data set.\n", "\n", "We will continue using the facial recognition data set. We will make the usual adjustments by adding an `'insomnia'` column and also a `'sleeper'` column that more meaningfully indicates where the subject is an insomniac or a normal sleeper (with the words \"normal\" or \"insomniac\" instead of `True` and `False`)." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ "shape: (5, 17)\n", "┌────────────────────┬────────┬─────┬─────────────┬───┬──────┬─────┬──────────┬───────────┐\n", "│ participant number ┆ gender ┆ age ┆ correct hit ┆ … ┆ psqi ┆ ess ┆ insomnia ┆ sleeper │\n", "│ --- ┆ --- ┆ --- ┆ percentage ┆ ┆ --- ┆ --- ┆ --- ┆ --- │\n", "│ i64 ┆ str ┆ i64 ┆ --- ┆ ┆ i64 ┆ i64 ┆ bool ┆ str │\n", "│ ┆ ┆ ┆ i64 ┆ ┆ ┆ ┆ ┆ │\n", "╞════════════════════╪════════╪═════╪═════════════╪═══╪══════╪═════╪══════════╪═══════════╡\n", "│ 8 ┆ f ┆ 39 ┆ 65 ┆ … ┆ 13 ┆ 2 ┆ true ┆ insomniac │\n", "│ 16 ┆ m ┆ 42 ┆ 90 ┆ … ┆ 11 ┆ 7 ┆ true ┆ insomniac │\n", "│ 18 ┆ f ┆ 31 ┆ 90 ┆ … ┆ 9 ┆ 3 ┆ true ┆ insomniac │\n", "│ 22 ┆ f ┆ 35 ┆ 100 ┆ … ┆ 8 ┆ 20 ┆ true ┆ insomniac │\n", "│ 27 ┆ f ┆ 74 ┆ 60 ┆ … ┆ 9 ┆ 12 ┆ true ┆ insomniac │\n", "└────────────────────┴────────┴─────┴─────────────┴───┴──────┴─────┴──────────┴───────────┘" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fname = os.path.join(data_path, \"gfmt_sleep.csv\")\n", "df = (\n", " pl.read_csv(fname, null_values=\"*\")\n", " .with_columns(\n", " insomnia := (pl.col('sci') <= 16).alias('insomnia'),\n", " insomnia\n", " .replace_strict({True: 'insomniac', False: 'normal'}, return_dtype=pl.String)\n", " .alias('sleeper')\n", " )\n", ")\n", "\n", "# Take a look as a reminder\n", "df.head()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Box plots\n", "\n", "We saw in the previous part of the lesson that bar graphs throw out most of the information present in a data set. They only give an approximation of the mean of the generative distribution and nothing else. We can instead report more information.\n", "\n", "A box-and-whisker plot, also just called a box plot is a better option than a bar graph. Indeed, it was invented by John Tukey himself. Instead of condensing your measurements into one value (or two, if you include an error bar) like in a bar graph, you condense them into at least five. It is easier to describe a box plot if you have one to look at." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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That means that 75 percent of the measurements were less than the top of the box. The bottom of the box is the 25th percentile. The line in the middle of the box is the 50th percentile, also called the **median**. Half of the measured quantities were less than the median, and half were above. The total height of the box encompasses the measurements between the 25th and 75th percentile, and is called the **interquartile region**, or **IQR**. The top whisker extends to the minimum of these two quantities: the largest measured data point and the 75th percentile plus 1.5 times the IQR. Similarly, the bottom whisker extends to the maximum of the smallest measured data point and the 25th percentile minus 1.5 times the IQR. Any data points not falling between the whiskers are then plotted individually, and are typically termed **outliers**. Note that \"outlier\" is just a name; it does not imply anything special we should consider in those data points.\n", "\n", "So, box-and-whisker plots give much more information than a bar plot. They give a reasonable summary of how data are distributed by providing quantiles.\n", "\n", "Not to draw our focus away from visualizing how a data set is distributed, going forward in this notebook, we will not split the data set by gender and sleep preference, but will instead look at the entire data set. Here is a box plot for that." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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Conversely, when plotting x-y data in a scatter plot, you plot all of your data points. Shouldn't the same be true for plots with categorical variables? You went through all the work to get the data; you should show them all!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Strip plots\n", "\n", "One convenient way to plot all of your data is a **strip plot**. In a strip plot, every point is plotted." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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We suspect there is more probability density around 85% or so, with a heavy tail heading toward lower values, ending at 40. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Spike plots\n", "\n", "The swarm plot above essentially provides a count of the number of times a given measurement was observed. Because of the nature of the GFMT, the percent correct values are discrete, coming in 2.5% increments. When we have discrete values like that, a **spike plot** serves to provide the counts of each measurement value." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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A histogram is constructed by dividing the measurement values into bins and then counting how many measurements fall into each bin. The bins are then displayed graphically.\n", "\n", "A problem with histograms is that they require a choice of binning. Different choices of bins can lead to qualitatively different appearances of the plot. One choice of number of bins is the [Freedman-Diaconis rule](https://en.wikipedia.org/wiki/Freedman–Diaconis_rule), which serves to minimize the integral of the squared difference between the (unknown) underlying probability density function and the histogram. This is the default of iqplot." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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As an example I will generate Normally distributed data and plot the histogram." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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"This looks similar to the standard Normal curve we are used to seeing and is a useful comparison to a **probability density function** (PDF). However, Histograms suffer from **binning bias**. By binning the data, you are not plotting all of them. In general, if you can **plot all of your data**, you should. For that reason, I prefer not to use histograms for studying how data are distributed, but rather prefer to use ECDFs, which enable plotting of all data.\n", "\n", "The ECDF evaluated at x for a set of measurements is defined as\n", "\n", "\\begin{align}\n", "\\text{ECDF}(x) = \\text{fraction of measurements } \\le x.\n", "\\end{align}\n", "\n", "While the histogram is an attempt to visualize a probability density function (PDF) of a distribution, the ECDF visualizes the **cumulative density function** (CDF). The CDF, $F(x)$, and PDF, $f(x)$, both completely define a univariate distribution and are related by\n", "\n", "\\begin{align}\n", "f(x) = \\frac{\\mathrm{d}F}{\\mathrm{d}x}.\n", "\\end{align}\n", "\n", "The definition of the ECDF is all that you need for interpretation. For a given value on the x-axis, the value of the ECDF is the fraction of observations that are less than or equal to that value. Once you get used to looking at CDFs, they will become as familiar as PDFs. A peak in a PDF corresponds to an inflection point in a CDF.\n", "\n", "To make this more clear, let us look at plot of a PDF and ECDF for familiar distributions, the Gaussian and Binomial.\n", "\n", "
\n", "\n", "![PDFs and CDFs](pdf_cdf.png)\n", "\n", "
\n", "\n", "Now that we know what an ECDF is, we can plot it." ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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measured. Given the above definition of the ECDF, it is defined for all real $x$. So, formally, the ECDF is a continuous function (with discontinuous derivatives at each data point). So, it could be plotted like a staircase according to the formal definition." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "
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one than the other.\n", "\n", "Let us now plot the percent correct data as an ECDF, both with dots and as a staircase. 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missing\");\n }\n }\n }, 10, root)\n }\n})(window);", "application/vnd.bokehjs_exec.v0+json": "" }, "metadata": { "application/vnd.bokehjs_exec.v0+json": { "id": "p1747" } }, "output_type": "display_data" } ], "source": [ "p = iqplot.ecdf(df, \"percent correct\")\n", "p = iqplot.ecdf(df, \"percent correct\", p=p, style='dots', marker_kwargs=dict(color=\"orange\"))\n", "\n", "bokeh.io.show(p)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The circles are on the concave corners of the staircase." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Computing environment" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Python implementation: CPython\n", "Python version : 3.13.7\n", "IPython version : 9.5.0\n", "\n", "numpy : 2.2.6\n", "polars : 1.33.1\n", "bokeh : 3.7.3\n", "iqplot : 0.3.7\n", "jupyterlab: 4.4.7\n", "\n" ] } ], "source": [ "%load_ext watermark\n", "%watermark -v -p numpy,polars,bokeh,iqplot,jupyterlab" ] } ], "metadata": { "kernelspec": { "display_name": "default", "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": "3.13.7" } }, "nbformat": 4, "nbformat_minor": 4 }