Re: [Phys-L] Timing Statistic

[John Denker <jsd@av8n.com>]

On 11/9/18 11:31 AM, Paul Nord wrote:

> https://drive.google.com/open?id=1oHf0zcq3Q_yWS7452ETSrXdVXfCWLM8p

Cool!  I downloaded the spreadsheet and played with it.

> The model reliably predicts the position of the disk within 0.03 radians
> for most of the sample data.

The first data point is off by a mile, but the rest are
within 0.02 radians, as shown here:
  https://www.av8n.com/physics/img48/spindown-theta.png
a zoomed-in view is here:
  https://www.av8n.com/physics/img48/spindown-theta-z1.png

> Feel free to critique that model.

The model is 100% appropriate for present purposes.
Perhaps the time will come when fancier models are
needed, but we've got muuuuch bigger fish to fry at
the moment.

> This model doesn't seem to predict the velocity well.

The model is fine.  The data is what's screwed up,
as will become obvious in a moment.

> Or suggest that the sensor is reporting something
> other than the real angular velocity.

I started with the position data, and from that I calculated
my own instantaneous velocity, Δ(angle)/Δ(time).  Plain
vanilla, no monkey business.

The velocity reported by Vernier is different.  They have
cooked the data somehow.  It is not worth my time to figure
out what they did, for reasons that will become obvious in
a moment.

I fitted a straight line to my velocity data.  This is
a two-parameter (first-order) model.  It is similar in
spirit to the three-parameter (second-order) Paul used
for the position data, although I did my own independent
fit and got slightly different coefficients.

See here:
  https://www.av8n.com/physics/img48/spindown-omega.png

When you see stripy residuals like that, you instantly
know we're being killed by roundoff errors.  Somebody
didn't keep enough guard digits.  To find out more about
guard digits, there is a notorious paper on the subject:
  http://doi.org/10.1119/1.5064563
or equivalently
  https://www.av8n.com/reprint/guard.pdf

But wait, you might say, the angle is inherently quantized,
because the device just counts the number of times the
marker moves past the sensor.  So the angular roundoff
error is inherent and irreducible.

I reply yes, the angle is quantized, but time is not,
and we're being killed by roundoff error in the *time*
variable.  In such a situation the magic word is
*timestamp*.  The device should timestamp each detection
event to high accuracy, with a bajillion guard digits.
The data points will be evenly spaced in angle, because
that's the physics of the encoder, but there is no law
that says the timestamps need to be evenly spaced.
Setting up the device to take data evenly spaced in
time is bad crazy.

You can approximate timestamp behavior by taking data
that is 100× or 1000× more finely spaced in the t direction.
Then find the points when detection events occur.  Keep
those points.  Throw away the rest, because they're not
telling you anything real;  either they are plain duplicates,
or they are disguised duplicates that Vernier is generating
by interpolation and guesswork.

Either that or reach out to Vernier.  Slap them upside
the head with a wet fish and get them to stop screwing
with the data and just report the timestamps.

====

To see exactly what I did and play with it:
  https://www.av8n.com/physics/spindown.gnumeric
  https://www.av8n.com/physics/spindown.xls

I don't at the moment have a machine that runs excel, so
I can't guarantee that the .xls file gets the graphics
exactly right, but it should be close, and the non-graphics
stuff (formulas and such) will be OK.

================

What's the lesson here?

1) Well, ever since the industrial revolution, physics has
lived and died by the instrumentation.  Before you can
study the physics you care about, you have to study your
instrumentation.  You have to beat the instruments into
submission.

2) The original suspicion was that the residuals were not
IID (independent and identically distributed) and not
normally distributed.  This suspicion turns out to be
exceedingly well-founded.