{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" }, "tags": [] }, "source": [ "# Counting Photons" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "In this notebook, we will study the statistics of counting photons, which is described by a Poisson process. The idea is that photoreceptors detect light by absorbing photons, which are discrete events. Since the arrivals of photons are random in time, the number of such events within a short period can fluctuate. We would like to study the statistics of those events." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "This random process of receiving photons can be described by an average rate of photon arrival, $k$. By definition, it means that over a *long* period $T$ ($T \\gg 1/k$), there will be $k T$ events. However, those events would be distributed randomly and independently over the period $T$. Let us first simulate such a process." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import scipy.stats as st" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "k = 3.\n", "T = 10000\n", "\n", "N = int(k*T) # number of events during the period T\n", "events = np.random.rand(N) * T # draw N random numbers uniformly distributed between 0 and T\n", "# events.sort() # sort events in increasing order in time" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" }, "tags": [] }, "source": [ "## Statistics of the number of events" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If we look at *short* periods, of length $\\Delta t$, then the number of events within each of those periods would vary from time to time. To see this, let us pick many short windows of length $\\Delta t$ and count the events within each of them. Since the events are independent and uniformly distributed, it does not matter where we put the windows. So we might as well divide the long period $T$ into consecutive short windows. Counting the number of events within every time window can be done using the `histogram` function." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "outputs": [], "source": [ "dt = 1\n", "bins = np.arange(0, T+dt, dt) # specify edges of the bins, including left- and right-most edges\n", "counts = np.histogram(events, bins=bins)[0] # count number of values within every bin (we did not have to sort `events`)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "Let us plot the actual time of events and their counts within the time windows." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "T1 = 100 # due to space limit, only plot up to time T1\n", "events1 = events[events" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.figure()\n", "plt.bar(np.arange(len(dist)), dist)\n", "plt.xlabel(r'number of events within $\\Delta t$')\n", "plt.ylabel('probability distribution')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "We can calculate the mean and variance of this distribution." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "mean number of events within dt = 3.0\n", "variance = 3.0016\n" ] } ], "source": [ "mean = np.mean(counts)\n", "print(f'mean number of events within dt = {mean}')\n", "var = np.var(counts)\n", "print(f'variance = {var}')" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "source": [ "The mean number of events is simply $k \\Delta t$, as expected. We are more interested in the variance, which turns out to be very close to the mean." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "source": [ "This is the hallmark of a *Poisson distribution*. In fact, when we said in the beginning that the random process is described by a Poisson process, it basically means that the distribution of the number of events should be a Poisson distribution. The shape of such a distribution is completely determined by its mean, $\\mu$, and can be written as:\n", "\\begin{equation}\n", "P(n) = \\frac{\\mu^n}{n!} \\, \\mathrm{e}^{-\\mu}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "We may plot this distribution function and overlay it on top of the empirically found distribution above. In our case, $\\mu = k \\, \\Delta t$." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "mu = k*dt\n", "n_array = np.arange(len(dist))\n", "p_array = st.poisson.pmf(n_array, mu) # use pre-defined Poisson distribution from `scipy.stats` package\n", "\n", "plt.figure()\n", "plt.bar(n_array, dist, label='data')\n", "plt.plot(n_array, p_array, color='orange', label='Poisson')\n", "plt.xlabel(r'number of events within $\\Delta t$')\n", "plt.ylabel('probability distribution')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "What happens if we choose a longer time window? We expect the mean number of events within each window to increase linearly with $\\Delta t$, and so does the variance. Let us check if this is the case. We need to repeat the calculations above for different $\\Delta t$." ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "outputs": [], "source": [ "dt_list = np.arange(1, 10) # different sizes for the time window\n", "mean_list = [] # collect means\n", "var_list = [] # collect variances\n", "\n", "for dt in dt_list:\n", " bins = np.arange(0, T/2, dt) # only use first half of total time; otherwise the mean will be exactly k\n", " counts = np.histogram(events, bins=bins)[0]\n", " mean = np.mean(counts)\n", " var = np.var(counts)\n", " mean_list.append(mean)\n", " var_list.append(var)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots(1,2, figsize=(16,6))\n", "ax[0].plot(dt_list, mean_list, 'o', label='data')\n", "ax[0].plot(dt_list, k*dt_list, label='theory')\n", "ax[0].set_xlabel(r'$\\Delta t$')\n", "ax[0].set_ylabel('mean number of events')\n", "ax[0].legend()\n", "ax[1].plot(dt_list, var_list, 'o', label='data')\n", "ax[1].plot(dt_list, k*dt_list, label='theory')\n", "ax[1].set_xlabel(r'$\\Delta t$')\n", "ax[1].set_ylabel('variance')\n", "ax[1].legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "We can also plot the variance versus the mean. A Poisson distribution will have them equal to each other." ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.figure(figsize=(6,6))\n", "plt.plot(mean_list, var_list, 'o', label='data')\n", "plt.plot(mean_list, mean_list, label='Poisson')\n", "plt.xlabel('mean')\n", "plt.ylabel('variance')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" }, "tags": [] }, "source": [ "## Probability of having more than $K$ events" ] }, { "cell_type": "markdown", "metadata": { "tags": [] }, "source": [ "In the experiment of (Hecht, Shlaer, & Pirenne, 1942), they sent repeated impulses of light to an observer and recorded how often he reported seeing the light. They then varied the intensity of the light and measured how the frequency of seeing the light depended on the amount of light in each impulse. The light intensity is proportional to the rate of photon arrival, and the exposure time is like the short time window in the analysis above. Therefore, for a given intensity and exposure time of the light, the number of photons in an impulse follows a Poisson distribution $P(n)$. If we assume that the observer can see the light only if the eye absorbed $K$ photons or more, then the \"frequency of seeing\" is given by the probability that there are $n \\geq K$ photons in the impulse, i.e., $P(n \\geq K) = \\sum_{n=K}^{\\infty} P(n)$. (In reality not all photons that arrive at the eye are absorved, as there is a finite efficiency by which the eye absorbs photons. But this would not change the math, as we can simply think of $n$ as the number of photons that *are absorbed* rather than arrive.)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "Let us simulate this process to find the frequency of seeing the light. Suppose our threshold is $K = 7$. We have to choose an exposure time for each impulse, i.e., the size of our time window, such as $\\Delta t = 2$. For the photon arrival rate $k = 3$ used above, each window would have $\\mu = k \\Delta t = 6$ photons on average. We would like to know how many windows have more than $K = 7$ photons. For that, we will first recount the number of events in each window and then screen for windows with $n \\geq K$." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "mean number of events in each window = 6.0\n", "frequency of having 7 events or more = 0.3986797359471894\n" ] } ], "source": [ "dt = 2 # window size\n", "mu = k * dt # mean number of events in each window\n", "bins = np.arange(0, T, dt)\n", "counts = np.histogram(events, bins=bins)[0]\n", "\n", "K = 7 # threshold\n", "above = np.sum(counts >= K) # number of windows with >=K events\n", "freq = above / len(counts) # frequency of having >=K events\n", "print(f'mean number of events in each window = {mu}')\n", "print(f'frequency of having {K} events or more = {freq}')" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "Now we would like to see how this frequency depends on the light impulse. Instead of varying the intensity (rate $k$), we can equivalently vary the exposure time (time window $\\Delta t$). We will plot the frequency of seeing the light as a function of the mean number of photons, $\\mu = k \\Delta t$." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [], "source": [ "dt_list = np.arange(0.5, 10, 0.5) # different sizes for the time window\n", "mean_list = [] # list to store mean number of events\n", "freq_list = [] # list to store frequency of having >= K events\n", "\n", "for dt in dt_list:\n", " mean = k * dt\n", " bins = np.arange(0, T, dt)\n", " counts = np.histogram(events, bins=bins)[0]\n", " above = np.sum(counts >= K)\n", " freq = above / len(counts)\n", " mean_list.append(mean)\n", " freq_list.append(freq)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.figure()\n", "plt.plot(mean_list, freq_list, '.')\n", "plt.xscale('log') # use log scale\n", "plt.xlabel('mean number of events')\n", "plt.ylabel(r'frequency of having $n \\geq K$ events')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "Let us compare to the expected frequency, which is given by the probability $P(n \\geq K)$. To calculate the latter, note that it is simply 1 minus the cumulative distribution function, $C(K-1) = \\sum_{n=0}^{K-1} P(n)$. We can compute this function by calling `scipy.stats.poisson.cdf()`." ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "outputs": [], "source": [ "mu_array = np.geomspace(0.5, 50, 101) # like `linspace()` but points are uniform in log scale\n", "PK_array = [] # list to store values of P(n>=K) for each mu\n", "\n", "for mu in mu_array:\n", " PK = 1 - st.poisson.cdf(K-1, mu) # use Poisson CDF to calculate P(n>=K)\n", " PK_array.append(PK) # store the value of P(n>=K) for the given mu" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.figure()\n", "plt.plot(mean_list, freq_list, '.', label='data') # plot simulated data\n", "plt.plot(mu_array, PK_array, label=f'K={K}') # plot theoretical curve\n", "plt.xscale('log')\n", "plt.xlabel('mean number of events')\n", "plt.ylabel(r'frequency of having $n \\geq K$ events')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "What if we have the data but do not know the actual value of $K$ that generated the data? This is the situation in the experiment of (Hecht, Shlaer, & Pirenne, 1942). What we can do is to plot theoretical curves for different values of the threshold $K$, then compare to the data to find which value fits better." ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "K_list = [3, 5, 7, 9, 12] # list of K values to try\n", "\n", "plt.figure()\n", "plt.plot(mean_list, freq_list, '.', label='data') # plot simulated data\n", "for K in K_list: # for each K value, repeat the steps above\n", " mu_array = np.geomspace(0.5, 50, 101)\n", " PK_array = []\n", " for mu in mu_array:\n", " PK = 1 - st.poisson.cdf(K-1, mu)\n", " PK_array.append(PK)\n", " plt.plot(mu_array, PK_array, label=f'K={K}')\n", "plt.xscale('log')\n", "plt.xlabel('mean number of events')\n", "plt.ylabel(r'frequency of having $n \\geq K$ events')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "Note that, in the experiment, they did not know what was the mean number of photons absorbed by the eye, but only that it should be proportional to the light intensity. In other words, the x-axis values of the data is multiplied by an unknown factor. To overcome that problem, they plotted the data in log scale on the x-axis (which is what we did above), so that an unknown factor amounts to shifting the data horizontally. We can equivalently shift the theoretical curves horizontally, for example by making them all cross at the point $x = 6, y = 0.39$. Since different values of $K$ result in different slopes in the rising part of the curves, by comparing the slopes to the data we can find the best $K$ value." ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "K_list = [3, 5, 7, 9, 12] # list of K values to try\n", "\n", "plt.figure()\n", "plt.plot(mean_list, freq_list, '.', label='data') # plot simulated data\n", "for K in K_list: # for each K value, repeat the steps above\n", " mu_array = np.geomspace(0.5, 50, 101)\n", " PK_array = []\n", " for mu in mu_array:\n", " PK = 1 - st.poisson.cdf(K-1, mu)\n", " PK_array.append(PK)\n", " i = np.argmin((np.array(PK_array) - 0.39)**2) # find the point on the curve closest to y=0.39\n", " mu_array = mu_array / mu_array[i] * 6 # shift the curve along x-axis so that this point is moved to x=6\n", " plt.plot(mu_array, PK_array, label=f'K={K}')\n", "plt.xscale('log')\n", "plt.xlim(1, 30)\n", "plt.xlabel('mean number of events')\n", "plt.ylabel(r'frequency of having $n \\geq K$ events')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "source": [ "It can be seen that $K \\sim 7$ gives the best fit. In the experiment, with less accurate data, they were still able to narrow $K$ down to around 5-7, which means our eyes are sensitive enough to detect as few as several photons!" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" }, "tags": [] }, "source": [ "## Statistics of waiting time" ] }, { "cell_type": "markdown", "metadata": { "tags": [] }, "source": [ "So far we have studied the statistics of the number of events within a given amount of time. Now let us switch our attention to the time interval between two consecutive events, or the \"waiting time\" before the next event." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" }, "tags": [] }, "source": [ "We can calculate such waiting times from the same dataset that we simulated above." ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [], "source": [ "events.sort() # here we must sort the events first\n", "intervals = np.diff(events) # this numpy function calculates the difference between consecutive array elements" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "Let us plot the distribution of those waiting times." ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "nbins = 50 # number of bins\n", "bins = np.linspace(0, 2, nbins+1) # create bins between 0 and 2\n", "wait_dist = np.histogram(intervals, bins=bins, density=True)[0] # make histogram of waiting times\n", "\n", "plt.figure()\n", "plt.bar(bins[:-1], wait_dist, width=np.diff(bins))\n", "plt.xlabel('waiting time')\n", "plt.ylabel('distribution')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "This looks like an exponentially decaying function. To check, let us plot the y-axis in log scale and fit a line." ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "x_data = (bins[:nbins//2] + bins[1:nbins//2+1]) / 2 # center of the bins; use only first half of data to avoid noisy tail\n", "y_data = np.log(wait_dist[:nbins//2]) # height of the bins in log scale\n", "fit = np.polyfit(x_data, y_data, 1) # this numpy function fits polynomial of degree 1, i.e., a line\n", "val = np.polyval(fit, x_data) # this numpy function evaluates the fitted function at given points\n", "\n", "plt.figure()\n", "plt.bar(bins[:-1], wait_dist, width=np.diff(bins), label='data')\n", "plt.plot(x_data, np.exp(val), color='orange', label=f'slope={fit[0]:.3f}')\n", "plt.yscale('log')\n", "plt.xlabel('waiting time')\n", "plt.ylabel('distribution')\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "tags": [] }, "source": [ "We see that the line fit is pretty good, which suggests that the distribution of waiting times is exponential. Indeed, it can be mathematically shown that the waiting time, $\\tau$, of a Poisson process follows the exponential distribution:\n", "\\begin{equation}\n", "P(\\tau) = k \\, \\mathrm{e}^{-k \\tau}\n", "\\end{equation}\n", "where $k$ is the event rate as defined earlier. Therefore the slope in the log plot is expected to be equal to $-k$, as borne out by our numerical results." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.8.10" } }, "nbformat": 4, "nbformat_minor": 4 }