The NYT maintains an informative U.S. case count map here.

Note: the mean time between exposure and first symptoms is 4-6 days. This means that the numbers we see today correspond to people who exhibit symptoms but were infected a week or more ago. 

Arizona

The state of Arizona continues to badly fail regarding testing, primarily because of the ever-incompetent and inept Trump administration. These numbers are therefore likely an egregious undercount—most especially the daily new cases and daily new deaths. It would be great news if the daily new cases curve really was turning over, but, unfortunately, instead it seems more likely that this is due to Trump’s (and Republicans’ generally) unconscionable failures of governance.

The log scale of the above plot is useful for showing that the increase in cases (and deaths) is still an exponential. However, to emphasize what that really means it is worth making the same plot but on a linear scale:

United States

Note: the United States continues to fail regarding testing, both at the state level and—especially—among the incompetent, inept, anti-science, know-nothing Trump administration. These numbers are therefore likely an egregious undercount. [Apr. 6: the U.S. is now admitting to “severe underreporting”.]

Here is the same plot but on a linear scale.

World

The doubling time of exponential growth

Suppose a measured quantity (such as the number of COVID-19 cases), call it $n(t)$, is growing exponentially:

\begin{equation}
n(t) = n_0 e^{c t} \label{exponential}
\end{equation}

where $c$ is a constant, the rate at which $n$ is increasing with time, and $n_0$ is the initial size of the quantity. This is a simplified—in fact, the simplest possible—description of an exponentially growing quantity. If we take the logarithm of eq. \eqref{exponential}, we have

\begin{equation}
\log{n(t)} = \log{n_0} + c t \label{logexp}
\end{equation}

We see that if we therefore plot $\log{n(t)}$ as a function of (linear) time, we should get a straight line with slope $c$. Indeed, this is a quick way to see if something is growing (or shrinking) exponentially.

In the plots above, the linear fits are least squares fits of a straight line to the indicated data.

A natural question is, “how long does it take to increase the quantity by a factor of 2 (or 10, or whatever)?” We can use eq. \eqref{exponential} to determine this. Suppose we want to know how long it takes to increase by a factor of $k$. We can write

\begin{eqnarray}
k \cdot n(t) &= & n_0 e^{c (t + \Delta t_k)} \\
&=& n_0 e^{c t} e^{c\,\Delta t_k} \\
&=& n(t) e^{c \Delta t_k}
\end{eqnarray}

where $\Delta t_k$ is the amount of time to increase by a factor of $k$. Thus,

\begin{equation}
\log k = c \Delta t_k
\end{equation}

or
\begin{equation}
\Delta t_k = \dfrac{\log k}{c} \label{k-period}
\end{equation}

With $k=2$, the doubling time is therefore

\begin{equation}
\Delta t_2 = \dfrac{\log 2}{c} \label{doubling-period}
\end{equation}

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$\pi$ is everywhere you look. It is even the case that there is $\pi$ in the sky. We need $\pi$ in order to live and function. These three observations are fundamental to the way our universe is put together.

2. The Value of $\pi$
     or, How Much is that Round Thing in the Window?

(with apologies to Patti Page) So, for fun, let’s calculate $\pi$ using Ramanujan’s famous infinite series formula, and check the error against a clever, arbitrary-precision algorithm for $\pi$, based on the Chudnovsky brothers’ improvement on Ramanujan’s series approximation, and which is correct to as many digits as we care to specify. While we’re at it, we’ll include just a straight-up implementation of the Chudnovsky brothers’ series approximation, too.

Ramanujan’s formula (see also here, and here):

\begin{equation}
\dfrac{1}{\pi} = \dfrac{2\sqrt 2}{9801}
\sum_{n=0}^{\infty} \dfrac{\left(4 n\right)!}{\left(n!\right)^4}
\dfrac{1103 + 26390\,n}{396^{4n}} \label{eq:ram}
\end{equation}

As mentioned here, the Chudnovsky brothers derived a Ramanujan-like formula that converges considerably faster(!) than Ramanujan’s original:

\begin{equation}
\dfrac{1}{\pi} = \dfrac{1}{53360\sqrt{640320}}
\sum_{n=0}^{\infty} \left(-1\right)^n
\dfrac{\left(6 n\right)!}{\left(n!\right)^3\left(3 n\right)!}
\dfrac{13591409 + 545140134\,n}{640320^{3n}} \label{eq:chud}
\end{equation}

We can take advantage of Python’s decimal module for exact arithmetic to as many digits of precision as we might want in calculating each term of the series. Doing so, we find the following errors after each successive iteration of the two series (note the exponents!):

      Ramanujan   Chudnovsky
  n   Rpi(n)-pi    Cpi(n)-pi
 --  ----------  -----------
  0    7.642E-8   -5.903E-14
  1   6.395E-16    3.078E-28
  2   5.682E-24   -1.721E-42
  3   5.239E-32        1E-56
  4   4.944E-40   -5.959E-71
  5   4.741E-48    3.609E-85
  6   4.599E-56   -2.212E-99
  7     4.5E-64   1.368E-113
  8   4.433E-72  -8.515E-128
  9   4.391E-80   5.331E-142
 10    4.37E-88  -3.353E-156
 11   4.364E-96   2.117E-170
 12  4.372E-104  -1.341E-184
 13  4.393E-112   8.513E-199
 14  4.424E-120   -5.42E-213

As we can see, Ramanujan’s formula, eq. \eqref{eq:ram}, gives eight orders of improvement (i.e., eight more digits of accuracy) per successive iteration, while the Chudnovsky formula, eq. \eqref{eq:chud}, yields fourteen orders of precision per iteration!

To illustrate, after fifteen Chudnovsky series terms, the difference between the series approximation and the actual value of $\pi$ is:

-0.00000 00000 00000 00000 00000 00000 00000 00000 00000 00000
   00000 00000 00000 00000 00000 00000 00000 00000 00000 00000
   00000 00000 00000 00000 00000 00000 00000 00000 00000 00000
   00000 00000 00000 00000 00000 00000 00000 00000 00000 00000
   00000 00000 00542...

Even just the first Chudnovsky term by itself (or just the first two Ramanujan terms) gives $\pi$ to almost machine precision ($2^{-52}\approx 2.22\!\times\!10^{-16}$) on a 64-bit computer.

For another perspective (thanks for the idea, Daniel Greenspan), let’s calculate (roughly!) the total number of atoms in the universe. As you might imagine, this will be a big number. We’ll break it down into two parts.

First, how many stars are in the universe? This is a number we can estimate from observations of galaxies and the amount of light that they emit. Essentially, since we can determine distances to galaxies, we just add it all up. Modern astronomical estimates for the equivalent number of solar-mass stars in our universe, based on the amount of light we detect coupled with the distances to the objects (galaxies) emitting that light, all come in at around

\begin{equation}
N_{stars} \approx 2\!\times\!10^{23} \label{eq:Nstars}
\end{equation}

This is equivalent to the mass of the visible universe divided by the mass of the Sun.

The amount of baryonic—that is, visible, or what we think of as “normal”—matter in the universe is only a small fraction of the total mass of the universe. Our universe, based on several different kinds of observations, is $68.3\%$ dark energy, $26.8\%$ dark matter, and $4.9\%$ ordinary matter. But that’s another story. We’ll just stick to the ordinary matter that we can detect via the electromagnetic radiation it emits.

Second, how many atoms are in a star the mass of our Sun? Now, the Sun has a measured mass of $M_{\odot} = 1.9884\!\times\!10^{30}$kg(†) and is composed of about $74.9\%$ hydrogen and $23.8\%$ helium by mass(‡). For this exercise, we will assume that the mass contributions of electrons and the other elements besides hydrogen and helium are negligible. The mass of a proton is $1.00784$amu, and the mass of a helium nucleus is $4.002602$amu. One amu (atomic mass unit) is $1.66053904\!\times\!10^{-27}$kg. The approximate number of atoms in the Sun, $N_{\odot}$, is then

\begin{equation}
N_{\odot} \approx
\dfrac{0.749 M_{\odot}}{1.00784\mathrm{amu}} +
\dfrac{0.238 M_{\odot}}{4.002602\mathrm{amu}}
\approx 9.6 \!\times\!10^{56} \mathrm{atoms} \label{eq:Nsuns}
\end{equation}

Hence, combining \eqref{eq:Nstars} and \eqref{eq:Nsuns}, the number of atoms in the universe, $N_{universe}$, is, roughly,

\begin{equation}
N_{universe} \approx N_{stars}\cdot N_{\odot}
\approx 1.9\!\times\!10^{80} \mathrm{atoms}
\end{equation}

Notice from the table above that the error of the Chudnovsky series after only the first six terms is about one part in $2.8\!\times\!10^{84}$—a number that is several orders of magnitude larger than the number of atoms in the entire universe!

3. A Short Introduction to Astrophysics
    or, Is there Really $\pi$ in the Sky?
    or, Here are the Footnotes

You might not have seen this coming. But here it is, wherein we demonstrate that, indeed, there is $\pi$ in the sky.

(†) We can determine the mass of the Sun by measuring the motions of the planets and asteroids in our Solar System, and then using Newton’s Law of Gravity. As Kepler discovered from Tycho Brahe’s meticulous observations, and Newton proved mathematically after he invented calculus and then turned his attention to the Moon’s motion, the orbital period $P$ of a body of mass $m$ and its mean distance $a$ from the Sun with mass $M_{\odot}$ are related by

\begin{equation}
P^2 = \dfrac{4 {\pi}^2}{G\left(M_{\odot}+m\right)} a^3
\end{equation}

Look at that: $\pi$ is in this equation that describes what we see in the sky.

(‡) We can determine the relative abundances of the elements that make up the Sun (and almost any star) by measuring, with a spectroscope, the amount of radiation absorbed by those elements in the atmosphere of the Sun (called the photosphere). Every element has its own discrete spectral signature in the form of absorption lines at specific sets of wavelengths. The amount of radiation absorbed by an element, relative to the other elements present, and in combination with the measured temperature, luminosity, and mass of a star, tells us what fraction of the star’s photosphere consists of that element. (We also need to know the distance to the star, but that’s a long story.)

Stars are, roughly speaking (i.e., ignoring the radiation absorbed by the elements in their photospheres), black body radiators. This means we can relate their luminosity (total radiated energy per unit time) to their radius $R$ and their effective surface temperature, $T_{eff}$. Simply put, the luminosity is the surface area of the star ($4\pi R^2$) times the amount of radiation emitted per unit surface area of the star:

\begin{equation}
L = 4\pi R^2 \sigma T_{eff}^4 \label{eq:L}
\end{equation}

where $\sigma = \dfrac{2\pi^5 k^4}{15 c^2 h^3}$ is the Stefan-Boltzmann constant, $k=1.38064852\!\times\!10^{−23}$ Joules per degree Kelvin ($J\cdot K^{-1}$) is the Boltzmann constant, $c$ is the speed of light in vacuum, and $h=6.62607015\!\times\!10^{−34} J\cdot s$ is the Planck constant from quantum mechanics. Eq. \eqref{eq:L} is a consequence of the physics of black body radiation.

Look at that: $\pi$ is integral to these relations that describe what we see in the sky, too.

(Don’t ask about the quantum mechanics connection. You can go down that rabbit hole by following the provided links. Quantum mechanics hurts my head.)

4. Full Disclosure
     or, So This is Where That Came From

The Python code that produces the Ramanujan and Chudnovsky results (table and plot) is:


import decimal
from decimal import Decimal as D
from utils import mutils
from utils import mplot as plt

prec = 300  # Set the number of digits of precision
            # for calculations.
decimal.getcontext().prec = prec 

def dfac(n):
    """ Arbitrary digits factorial. """
    m = D('1')
    for k in range(1,n+1):
        m *= k
    return m

def Rpi(n):
    """
    Calculate pi using n iterations of Ramanujan's
    formula.
    """
    s = D('0')
    for k in range(n+1):
        facterm = dfac(4*k)/dfac(k)**4
        num = D('1103') + D('26390')*k
        den = D('396')**(4*k)
        s += facterm*num/den
    s *= D('8').sqrt()/D('9801')
    return 1/s

def Cpi(n):
    """
    Calculate pi using n iterations of the Chudnovsky
    brothers' Ramanujan-like formula.
    """
    s = D('0')
    for k in range(n+1):
        facterm = dfac(6*k)/(dfac(k)**3*dfac(3*k))
        num = D('13591409') + D('545140134')*k
        den = D('640320')**(3*k)
        s += D('-1')**k*facterm*num/den
    s *= D('1')/(D('53360')*D('640320').sqrt())
    return 1/s

# Print a table of the error of n iterations
# of Ramanujan's formula.
print('     Ramanujan   Chudnovsky')
print(' n   Rpi(n)-pi    Cpi(n)-pi')
print('--  ----------  -----------')
fmt = '{:2d}  {:>10s}  {:>11s}'
c4  = decimal.Context(prec=4)
rerrs = []
cerrs = []
for n in range(15):
    exact_pi = D(mutils.pi_chudnovsky(prec))
    errR = Rpi(n) - exact_pi*D('1e-{:d}'.format(prec))
    errC = Cpi(n) - exact_pi*D('1e-{:d}'.format(prec))
    normerrR = errR.normalize(c4)
    normerrC = errC.normalize(c4)
    print(fmt.format(n, str(normerrR), str(normerrC)))
    rerrs.append(float(errR))
    cerrs.append(float(errC))

fig = plt.figure(figsize=(8.2, 5))
xlab = ['$\mathrm{number\ of\ series\ terms}\ n$', 12]
ylab = ['$\mid f(n) - \pi \mid$', 12]
pt = ['$\mathrm{\pi\ series\ approximation\ error}$',
      14]
labs = [['$f(n) = \mathrm{Ramanujan}$', 10],
        '$f(n) = \mathrm{Chudnovsky}$']
xticks = np.arange(15)
yticks = np.array([0.1**k for k in range(0, 240, 30)])
ylim = (1e-220, 1e-1)
plt.lineplot([np.array(rerrs), abs(np.array(cerrs))],
             np.arange(15), ['k-', 'r-'], [1, 1], 
             ylim=ylim, logy=True, xlab=xlab,
             ylab=ylab, xticks=xticks, yticks=yticks,
             doxticks='bottom', doyticklabels='both', 
             dolegend=True, labels=labs, plottitle=pt)
fname = (os.environ['PYTHONPATH'] +
         '/misc/Ramanujan pi.jpg')
plt.savefig(fname, dpi=300)


All images © 2019 Marc A. Murison: CC-BY-NC-ND

When you wake up, get up. When you get up, do something.
—one of my favorite quotes, from Viola Babbitt

Wake up.

Sit with the cat for a bit. Because Cat Time. “Is necessary, hooman. Sit.” Wait for cat to reach inevitable “bored now” condition…lick warning…condescending departure to stretch. Stretching is important. There is no moral analogy in this.

Get up.

Shuffle around the stretching kitty to the computer in the next room. Shuffle back to get your glasses from the nightstand. Shuffle around the feline a third time, oblivious to its withering stare.

Do something.

Start the computation job you were too tired run at 3am. Rely on the setup you already did. At 3am.

“Initializing”…?

Grogginess is measured by the amount of time it takes to realize one has done something unfortunate. It is an accurate measure. Being after the fact, though, it is not helpful towards the prevention of unfortunate acts. Lassitude can be assessed by the time needed to recognize that feeling in the pit of one’s stomach and say “Oh, hi again, you”. These are the same.

So herein lies the moral of our tale: Coffee. Do coffee first.

An orb weaver and its web. [image credit: San Diego Zoo]

What is the first wildlife you remember encountering in the actual wild (wild is broadly meant here–not a zoo)? Maybe don’t count squirrels. Or maybe do. I’m pretty sure mine was a squirrel. Or a pot gut up at Brighton.

Childhood memories. Such hole-y, but ever-tantalizing, constructs (say cognitive neuroscientists, whom I’ve no reason to question, although I’d like to).

My timeline ordering is necessarily diaphanous from this far vantage of almost five decades, but the set of vivid memories poking out from the gray mists comprises the following most likely non-boring contenders (for current purposes, deer = boring, although at first-sighting they, too, were exciting):

1. various SoCal orb weavers — hence my early and lasting fascination with and fondness of arachnids and the beguiling, interrupted symmetry of spider webs (was this the seed of my later captivation by mathematics?);
2. praying mantises — hence, subsuming arachnids, my early and lasting chitinous fascination with and fondness of arthropods generally;
3. a brown Desert tortoise, whom we named Fred, my brother and I, and who hung out in the grass and ivy of our suburban Los Angeles back yard one summer, happily devouring our fig tree droppings (Fred did not seem to care for fallen apricots) — hence, yada yada, reptiles;
   [Later, I had an alligator lizard, also named Fred. He was brown, too. Much later, my wife and I had a cat, also named Fred, whom we rescued from the wilds of our Washington, DC, suburb. Fred the cat was also brown. Fred is the name of all the best brown pets. One day, alligator Fred laid an egg. It had a soft, leathery shell, cream-white with brown stains. Not at all like a chicken egg from the grocery store. Fred with the hard shell and a fondness for brown-black figs laying in the sun did not lay an egg. Fred’s egg never hatched; not understanding why, I was crestfallen. (And cats, silly, don’t lay eggs. But an ancestor that cats deign to share in common with us laid eggs. I think that’s cool. I bet it was brown, that ancestor.)]
4. a hawk, perched on top of a tall party-line telephone pole (it seemed impossibly tall to an eight-or-nine-year-old kid from the L.A. suburbs, and the isolated farmhouses in eastern Washington wheat country were all connected by party lines, and church socials) across and down the draw — that’s what my grandmother always called it, The Draw, as if that was its name — a mile or so south and west from my grandparents’ farmhouse one hot summer day, which one of my ancestral uncles shot, from an impressive distance, with his childhood .22 rifle with the well-worn brown wood stock, the bird’s feathers suddenly exploding — a long time, it seemed, after the rifle hammer made the bullet’s gunpowder explode — and fluttering down against the hot still air, shimmering silhouettes arcing in and out of existence against an infinite deep blue sky (back then, deep blue skies were infinite, especially on hot summer days), the way blue and black dinosaur feathers do when they explode — hence my early and lasting dislike of firearms, and of immature, inconsiderate mentalities;
   [The hawk flew away, I would guess surprised at the sudden disappearance of tail feathers and angry at such a rude interruption of its respite. One of the feathers, like my favorite lucky agate, I kept for many years.]
5. and, indirectly, an American black bear, who, some time in the dead of an inky starlit night in the Sierras,
   1. quietly and neatly tore a perfectly square, hand-sized opening in the corner of a fellow Boy Scout member’s tent (pretty sure it was a blue tent, like mine, but it could have been green) and took his illicit stash of Butterfingers (yes, that kid, who had red hair and freckles and I’m sure a name, and whom I never did much like because he was kind of a bully, was a fucking idiot), and
   2. quietly and neatly tore the left rear door off a neighboring camper’s metallic-blue Toyota sedan and took the food from the subsequently undamaged blue and white Coleman cooler sitting under a blanket on the back seat (the sedan owner wasn’t too bright, either, though, unlike freckles the idiot, he had black horn rimmed glasses and was a really nice guy)

  — hence, my early and lasting appreciation for the wide range of manual dexterity among bears.

Images ©Marc A. Murison, CC BY-NC-ND.

Click on an image to enlarge; right-click to view at full resolution in a separate browser tab.

This first image was fairly early, when the Sun was just 6.8 degrees below the horizon (just past the end of civil twilight) and the twilight sky was still bright. Visually, you could just discriminate the reflected Earth light on the dark part of the Moon’s disk from the twilight sky background.

7:32pm, 1/4 sec, 125mm f/5.0, ISO 125

This next exposure was when the Sun was 8.8 degrees below the horizon, about halfway through nautical twilight. Right around this time, the lighting was at its most ethereal and, I think, beautiful. It was a wondrous sight.

7:42pm MST, 1 sec, 80mm f/4.5, ISO 125

This next one is a closeup of just the Moon, showing reflected Earth light from the non-sunlit part of the Moon’s disk. Reflected Earth light is visually striking when the Moon is a thin crescent — look for it the next time you see a thin crescent Moon. The faint arc to the left of the Moon is an internal camera lens reflection from the bright, sunlit sliver. The star on the left is 5 Tau, magnitude 4.1. The fainter one below-left is HD 21379, magnitude 6.25. The stars appear as short trails, and the features of the Moon are blurred, due to the Earth’s rotation during the 2 seconds that the camera shutter was open.

7:51pm, 2 sec, 600mm f/5.6, ISO 125

This final image shows the sky when the Sun was 11.8 degrees below the horizon — just before the end of nautical twilight and the beginning of astronomical twilight. The end of astronomical twilight, when the Sun is 18 degrees below the horizon, is when the sky is typically dark enough for astronomers to begin observing. Although you can still see a faint twilight glow near the western horizon even at the end of astronomical twilight, the sky is bright enough to interfere with observations only close to the horizon — where we almost never point telescopes anyway. (The usual rule-of-thumb limit is 30 degrees above the horizon. Lower than that, and you’re looking through too much of the Earth’s turbulent atmosphere for most types of observations to be scientifically useful.)

7:58pm MST, 8 sec, 115mm f/5.0, ISO 125

ABC-15, Phoenix, via Associated Press

A driverless car hit a woman who was walking her bicycle across a street at night. Numerous articles, some less useful than others, but also some more helpful to a better understanding, describe this incident. For example, in order of increasing technical detail,

Police In Arizona Release Dashcam Video Of Fatal Crash Involving Self-Driving Car

Self-Driving Uber Car Kills Pedestrian in Arizona, Where Robots Roam

How a Self-Driving Uber Killed a Pedestrian in Arizona

Uber Self-Driving Car Fatality Reveals the Technology’s Blind Spots

I have tried to find articles that do not center on or advance irresponsible histrionics, egregious ideological bias, or insulting, simplistic thinking. Critical thinking matters, especially when people’s lives are at stake.

Given the infancy of driverless vehicle technology, how significant is this incident? All knowledgeable observers have been expecting, and dreading, that something like this would happen. Now that it has, how do we put this in a context that makes sense? That is helpful? That is at least somewhat productive for society? Certainly, snap judgments and scapegoating are the opposite. So let’s first consider a few facts.

According to the CDC, in 2015 (latest official U.S. mortality data) the lower bound on the number of pedestrians killed by people-driven cars was 5,719. Another 17,008 were killed by ‘unspecified’ means involving motor vehicles, so the real number is probably higher than 5,719. Another 8,313 vehicle occupants died in accidents, cars killed 4,431 motorcyclists, 675 bicyclists, and 15 ‘other’. Of necessity, those are all lower bounds as well. The total number of motor vehicle deaths in the U.S. in 2015 was therefore 36,161. Further, 94% of traffic accidents are due to human error (i.e., insufficiently sound judgment).

However, the number of driverless cars on the road is as yet statistically minuscule, and the miles driven under realistic circumstances even more statistically minuscule. So any direct comparison is invalid in the absence of substantially further data — estimates put the number of miles necessary for statistically significant discrimination at hundreds of millions of miles or more — and better-discriminating data measures — such as number of fatal accidents per mile driven — with which to compare. The significance of this incident, for now, lies elsewhere.

In the immediate aftermath of this unfortunate and terrible accident, many of us have several questions in common: what might have gone wrong? Where should we look for correctable fault? With whom should we lay blame? Is attempting to lay blame on someone even useful, to anybody? I would argue that the answer to that last question, at least, is easy and should be obvious: no. The situation is complicated in several arenas (technical, political, psychological, among probably others), with many confounding factors. So, based on experience if nothing else, we know it is likely there are no easy or quick answers. Certainly, simplistic thinking is neither productive nor beneficial.

With regard to the political context around this particular event in this particular state: As so often has happened in the past couple of decades, we could again be looking at the sad, unnecessary consequences of conservative values enacted as irresponsible and reckless public policy: private profits tend to matter more than people’s lives. But that is an entirely separate, infuriating, and thoroughly expletive-laden subject, at least for me. Here, let us choose not to go there.

Could we also, or maybe instead, be looking at technology advance pursued at the knowing expense of public safety? Of people’s lives? The implication being that tech advance, and science more generally, is pursued by heartless elites who disregard public concerns? In short: no. This is an ignorant, flagrantly dishonest strawman. No. This is not that. Just no. It is never, in the real world, that. The people who manufacture and push this false cultural meme on us are dishonest; they seek to steer an often unknowing society toward their own self-serving agenda — an agenda that cannot survive the light of rational, fair, ethical examination, hence their sly dishonesty. The real world is never this simple-minded, one-dimensional fiction. In the real world, scientists and technologists are not evil, are of necessity if nothing else the opposite of dishonest, have neither time nor inclination for shady, conspiratorial ulterior motives. That story, however superficially enticing, is pure fiction. Deep down, under whatever emotions, ideological bent, and biased noise might be pummeling our conscious minds, I think we all know this.

With regard to sensors, the Wikipedia article on driverless cars sheds some light but, surprisingly, not much:

Typical sensors include lidar, stereo vision, GPS and IMU.[42] Visual object recognition uses machine vision including neural networks.

The last two journalistic articles linked above are more helpful. In this context, “vision” can mean optical or infrared or both (and/or even some other wavelength range, such as radar). I would have been utterly shocked if these systems did not use both IR and optical sensors. It’d surely be the height of both stupidity and irresponsibility if they did not. It is worth pointing out that engineers are not stupid, and rarely irresponsible, while corporate upper management sometimes is — and politicians almost certainly can be counted on being — both stupid and the epitome of irresponsibility. Given the current apparently poor regulation in this area, I suppose stupidity and irresponsibility are therefore potentially viable likelihoods, despite the no doubt multiple layers of safety protocols that smart people, down at the technical levels inside the companies, have nevertheless managed to put in place despite upper-level idiots (if any).

If this particular car was outfitted with both visible and IR sensors (lidar would necessarily be IR in this context), then the fault must be either 1) with the chosen sensor sensitivities, the chosen sensor ranges, or the chosen sensor fields of view, or some combination thereof; or 2) within the AI decision assessment of the filtered and cleaned input signals; or 3) some combination of both. That’s it; those are the options. Yes, it’s complicated.

(I’m assuming there was not a sensor failure, which would be both a manufacturing testing and reliability fault as well as a redundancy failure in the design. This is a different topic I won’t address here, but at this point it’s also a possibility.)

Indeed, as we learn from the articles above, this vehicle used optical, IR, and radar sensors, as one would reasonably and correctly think should be the case. Whether or not the production design had been recklessly limited to just optical wavelengths — I find it unthinkably unlikely that any significant U.S. company would be that self-destructively reckless, but suppose so anyway — then you must arguably add, and otherwise could reasonably add, a fourth potential likelihood: the fault lies foremost with a cost/safety trade-off decision or series of decisions that some fucking idiot in a position of overriding power might have made — undoubtedly (if this even happened) over the vehement objections of the design engineers and other technical experts. Trade off decisions are ultimately subjective assessments, whether wisely made or not, therefore a thick morass of difficulties and ambiguities. (As an aside, optical-only would be, and many other potential technical shortcomings could be, a direct consequence of allowing privatization in the absence of regulation of technological advances, in spite of all the obvious public risks. Ayn Rand was a fucktard and an awful human.)

But here’s the point I wish to make: either way, the woman who was struck and killed was NOT at fault, no matter how careless or distracted she may or may not have been in that terrible moment. Keep in mind that we can never really know her state of mind or level of distractedness anyway. The available data are insufficient for attempting any such assessment, even if it could have been useful (it is not). But even so that judgment, however tempting for some people when emotions run high, is not relevant. In addition, it is not relevant that she was not at a cross walk; those exist to primarily to counter (or, rather, partly contain) human driver error. Nor is it relevant that it was dark instead of broad daylight: a paucity of optical photons does not affect two of the three sensor wavelength ranges, while optical sensors that are extremely sensitive at low light levels are a well-established and inexpensive technology. It even is not relevant that the safety driver was provably distracted and then failed to react and take control. None of these are relevant if a fatal fault lies further up the precedence hierarchy.

I watched the video clip of the incident, compiled from on-board optical cameras, several times. It is crystal clear to me that this accident should never have happened, because first and foremost something either is or went wrong (or is at least insufficiently comprehensive, therefore still wrong) with the sensor design, or the sensor data analysis software (this is where noise filtering takes place), or the software decision module (the machine learning and AI part), or — and this I think is most likely — some combination of those. Even if every reasonable precaution had been allowed — and we all know that every precaution engineers deemed necessary and reasonable very likely was not allowed, for a variety of reasons (some good, some bad) — this still is where the fault lies that we should all care most about right now. It is a complex system operating in a complex, time-variable setting, and somewhere therein lies a problem, a bug, or an oversight that unambiguously takes precedence over whatever may or may not have been happening with that poor, tragically unlucky woman.

We don’t know yet what, exactly, happened. But we know where we should be looking. It behooves all of us — society — to allow unhindered gathering of all possibly relevant data around the event and give the experts the access, resources, and time that, according to them, they need to hunt down the real cause and find ways to fix it. The technical problem, wherever it lies, is the only valid first priority. Setting up red herrings (“it’s her fault!”, “no, it’s Uber’s fault!”, “no, it’s capitalism’s fault!”, “no, it’s the idiot Arizona legislators’ faults for once again valuing corporate profit over human lives!”, etc.) only uselessly distracts from — and maybe even will prevent — tracking down the technical problem.

Complex systems can behave in unexpected ways. Probably several things — perhaps even each one innocuous in isolation — had to combine for this accident to have happened. It is likely every single-fault failure mode was identified and mitigated by the design and test engineers. Those failure points are relatively straightforward to deal with, and, again, engineers are far from stupid. However, multiple chained events leading to unexpected, even unpredictable, behavior in a complex system can be extremely difficult and time-consuming to debug, both before and after the fact — especially in the presence of insufficient testing or design resources. This is every engineer’s absolute, hands down, most disturbing nightmare. It is THE thing that keeps engineers up at night.

Yet even if every conceivable precaution had been (allowed to be) taken, dependent multimode failures can and do still happen anyway. It is inherent in the very natures of technological advance and sophisticated systems.

Sometimes, nobody is to blame.

The best anybody can ever do — and therefore the most anybody can ever ask — is to implement every reasonably knowable precaution and perform every reasonably knowable relevant test, and iterate sufficiently many of these design-test-correct cycles to satisfy everyone’s most important misgivings, in order to ferret out all (you hope) of the important gotchas you undoubtedly didn’t or couldn’t, for whatever reasons, think of beforehand. Even then, surprising shit will happen. To further compound things, in the real world you very rarely (as in: never) can afford to fully implement these to everybody’s satisfaction, which just makes bad unexpected events all the more unavoidably likely.

Nobody wants it to happen, or even to be a possibility, but bad shit is very likely going to happen no matter what. You do your best — and, even in the real world, almost everybody in this line of work does — given the current context, available knowledge, and available resources, and you hope that you’ve managed to mitigate the severity of the consequences of the inevitable but unpredictable bad things to a sufficient extent that in the end you have navigated this unavoidable minefield without anybody getting hurt or killed.

That didn’t happen, this time. But nobody should be surprised. Nor should anybody be at all quick in pronouncing — nor is anybody entitled to pronounce — ill-informed judgments.

This is the way it is. You learn from the problems that, despite all your efforts, hit you and your team; you fix them; you become wiser (if more saddened); and despite the paralyzing pit in your stomach you move on.

My dad was an aircraft flight test instrumentation engineer, so I grew up seeing him and his friends live this process, several times. As a kid I did not fully comprehend what had happened, and he had a tendency for understatement in place of emphasis in serious events. One of the incidents occupies a vividly chiseled volume in my brain. In a freak accident, a chase plane — a standard safety measure — was nicked on the wing by a test helicopter’s main rotor blade. As it happened this time, Dad was not on board the helicopter but directing the chorus of test data from the ground. The helicopter pilot managed, somehow, to recover control, but the chase jet went down, hitting the ocean surface off the California coast (another standard safety precaution — never expose the public to even the slightest possible risk). Out of long experience, the seasoned test pilot habitually flew with his harness buckled but loosened. This habit saved his life. His copilot was a different story.

The stunned test engineers, as my Dad later relayed to me, listened to this event unfold, from routine start to grisly finish, on the comms radio, while pitiless equipment monitored pilot life signs data. The test pilot and his copilot both survived the crash. The cockpit canopy had automatically released upon impact. As the plane sank the pilot immediately unbuckled and escaped. The descent of the plane-turned-anchor was too rapid for any hope of him to turning around and helping his copilot (he tried anyway). The apparent size of the crippled jet shrank with distance, and it disappeared from sight, his friend and colleague methodically struggling in the dark, finger-numbing cold and the exponentially rising pressure. (Test pilots are a uniquely cool-headed lot in dire circumstances.) The well liked twenty-five year old, ironically trapped by his snugly tightened safety harness, sank with his plane, fighting to release himself until he lost consciousness and died in the unforgiving abyss.

Imagine the pilot having to witness this. Imagine the test team looking up from their instruments, in dawning horror informed by knowing dread, to each other’s faces, one after another, hoping for some sign from the more experienced team leaders — in this case my Dad — that, despite what their rational brains were telling them, things really would be okay, that nobody was really going to die. For any engineer, nothing is more inexpressibly awful, nor more dreaded, sitting there in the backs of their minds, nor more devastating when it one day happens, than  to have to live through — and, afterwards, continue to live with — somebody getting killed. But it happens. It will happen. And sometimes, nobody is to blame.