In Chapter 9: Physics we
took the falling mass physics example and built around it a physics
simulation, a detector simulation, and an experiment analysis tool,
all in one big program.. Then in Chapter
10: Physics we split this into two parts: (1) an experiment
simulation program, which contained the physics simulation and the
detector simulation, and (2) an analysis program.
The experiment simulator wrote the data for the events
(i.e. the drops) into a file that could be read by the analysis
program. Data from a real experiment would go into files with the
same format and examined by the analysis program in the same way.
Variations between the two types of data would then highlight problems
with the simulation or with the experiment or with the analysis.
Calibrations
Difference between simulated and real data could be due to instrument
imperfections that produce fixed but significant variations from
the "true" values. For example, an analog-to-digital conveter
(ADC) converts an analog voltage to a numerical value. So a 1.0
volt input might give, say, a value of 255 for an 8-bit ADC. Perhaps,
however, a 0.0 volt input doesn't produce a 0 output but a value
of 5. Then this 5 is an instrumental offset that should be subtracted
from the ADC's output. An ADC module might have, say, a dozen ADC
inputs, or channels, and each could have an offset that varies somewhat
from each other, e.g. 3 for one, 8 for another, etc.
There might also be variations in the slope of the analog -to-digital
conversion. That is, say that the conversion goes as
N = C + S * V
where V is the input voltage, c
is the constant offset, and N is the digital
output. The slope S might vary slightly
from one channel to another. This nonlinearity variation would also
have to be removed before the data could be analyszed.
This correction of the data for known instrument offsets and channel
variations is carried out in the calibration phase. This also refers
to converting the scale of the instrument output to the units of
interest. For example, the 0-255 range of our 8-bit ADC would need
to be converted to the 0-1V scale.
Typically, you will carry out special "runs" with your
instrument to determine the calibration. That is, you put in exact,
known input values and then compare these with the outputs. For
example, with our ADC we could put in a series of values stepping
from 0V up to 1V and use the outputs to determine our offset and
slope corrections.
Systematic Errors
Differences between a simulation and the real data might also be
due to some aspect of the experiment that varied unexpectedly or
because of an incorrect assumption about the instrument. This kind
of uncertainity falls under the systematic error category.
This differs from random errors (sometimes referred to as
the statistical error), which are due to the fluctuations
when the number of measurements is less than infinite.
In lab courses such systematic errors come up in the context of
explaining the difference between accuracy and precision.
A ruler, for example, might have very finely graded markings that
allow you to read a measurement out to a fraction of a millimeter,
but if you had not noticed that the lower end of the ruler had been
worn down by a few millimeters, the measurements would be precise
but inaccurate.
A famous case of this is the primary mirror for the Hubble
Telescope. Its surface was ground down to a curvature that
was extremely precise (1/20 of the wavelength of light). However,
due to an incorrectly calibrated device used to measure the curvature,
it was the wrong curvature.
It might seem that a systematic error would simply be fixed once
it is found or calibrated out of the data. However, there are several
situations where such solutions don't apply:
- The experiment data was already taken and it's impractical or
impossible to redo the experiment.
- There are so many different possible systematic effects, it
isn't practical to remove them all or calibrate them all out of
the data.
- The underlying physics isn't perfectly understood and different
simulations of the physics and the interaction with the experimental
apparatus system lead to different results.
To overcome these problems, the simulation of the experiment allows
you to estimate the systematic effects. You can vary different aspects
of an experiment in the simulation and see what affect this has
on the calculated results.
Case 3 above is common in high energy particle physics where one
needs a simulation to correct for the areas around a collision point
that are not covered by the detector system. These acceptance
corrections would be required, for example, when calculating the
total cross-section for a reaction of some sort. The assumptions
on the loss of scattered particles down the beampipe might vary
slightly from one simulation model to the next and so the resulting
cross-section calculations might vary slightly but significantly.
Different models and simulations are used and the variation on the
final cross-section value would be determined.
Typically, a systematic error is shown separately from the random
error as in
x = 2.34 +/-0.05 +/-0.10
where the +/-0.05 is the statistical error and the +/-0.10 is the
systematic error, which would typically be the combined error of
several systematic effects.
Note that another source of systematic variations might be due
to differences in experimental appraratus and technique. A phenomena
measured by different types of instruments, perhaps by different
experimentalists in different parts of the world, might show that
instruments of one type obtain different results for some unknown
reason. An analyst trying to combine the results from several different
independent experiments might put the instrument variations into
the systematic error.
Note that if the experimental results are especially sensitive
to a particular system parameter, one might want to redo the experiment
and closely monitor that system parameter to insure that remains
within its acceptable range.
Similarly, if you are using a simulation to design an experiment,
the study of systematics will help you to decide what aspects of
the system need to be controlled and monitored most closely. Conversely,
you might find that even big variations in a particular parameter
don't affect the result very much and so can get by without a complex
and/or expensive system to control that parameter.
Demo Simulation
In the demo programs discussed on the following pages, we try to
illuminate the above topics by adding instrument offsets, calibration
runs, and systematic errors to our falling mass experiment simulation.
References & Web Resources
Most recent update: Nov. 14, 2005
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