Dr. Michael L. Larsen – Research Overview

Effects of Finite Sampling and Dead-Time on Statistical Inference

 

Basic Background

 

Many particle probes seek to detect the arrival of individual particles in some sort of sensing volume.  When a particle arrives in this volume, the time is recorded, usually some measurement is made on the particle to determine its size, velocity, composition, orientation, and/or optical properties and the values of all the measured variables are recorded.

 

The time it takes an instrument to make all of these measurements is usually short (depending on what is being measured, often on the order of a few millionths of a second), but it is not 0.  It takes time to measure something and record it.  Also, because of how these variables are often measured, usually you can only measure a single particle at a time.  So, in addition to taking time to measure each particle, you have to wait until one particle is done being measured and leaves the sensing volume of the probe before the next particle can be detected and measured.

 

So what happens if two particles show up in the sensing volume at essentially the same time?  The sensing volumes are made small and the time to do the measurements is very short to try and keep this from happening often, but simultaneous (or nearly-simultaneous) arrivals do happen.

 

To prevent a second particle arriving very shortly after an initial particle from throwing off the measurements, most instruments have some sort of mechanism for ignoring any subsequent particles for a little while during the detection and measurement of the first particle.  This brief “turned-off” period is called the dead-time of the instrument.

 

The error introduced in estimating how many particles arrived in the sensing volume due to the turned-off time periods is often called “dead-time error” or, if the turning-off is due to mutual occupation of the sensing volume instead of electronic reset time, it is sometimes called “coincidence error”.

 

Atmospheric particulate probes are most definitely not the only instruments that have to cope with this phenomena.  Most of the theory associated with dead-time was developed over a half-century ago by people who were working on nuclear decay detectors (e.g. Geiger counters).  There is a long, involved history associated with accounting for the “missed” particles.  The methods used depend on the specific characteristics of the instrument used to detect the particles, but there are general techniques that are considered well-known and used in both the nuclear detection and atmospheric particulate communities.

 

 

So what’s the Problem?

 

By definition, the probe is turned off during the “dead-time” and – by definition of “turned off” – we don’t know for certain how many particles were missed.  The theoretical corrections discussed in the above section nearly always assume that there are no correlations between particles, the particle arrivals follow a Poisson distribution, and generally speaking the statistics governing the system follows that of perfect randomness.

 

The implicit assumption of perfect randomness for particulate arrivals is in very serious doubt.  Consequently, the correction mechanisms above can be in error.  If we assume, for the sake of argument, that particles exhibit positive pair-correlations (statistically, they show some proclivity to clump) – then the standard dead-time correction formulas underestimate the number of particles missed and, hence, the total number of particles in the cloud.

 

The problem is at least a 2-edged sword.  To determine whether there is a statistical deviation from perfect randomness, we tend to use tools that are themselves subject to dead-time.  To correct the dead-time errors in these tools, we need to know the magnitude of the deviation from perfect randomness.

 

That’s a sticky situation.

 

As far as I see, the only way out of the problem is to use tools that are not subject to dead-time to determine the statistical deviation from perfect randomness.  Then we can go back to the other tools and fix their dead-time errors and reinforce the conclusion from the tools without dead-time.  The problem – such tools without dead-time (for the most part) don’t exist yet.

 

 

 

 

What is Dr. Larsen doing to try and Solve the Problem?

 

Rather than just sitting around and waiting for the right tools, there are other (less ideal) ways to try and progress.  One method that I am trying to work on essentially is a method of successive approximations.  If we assume the general “statistical shape” of the deviations from perfect randomness, we can then predict how an instrument with dead-time would modify that shape.  If we then compare this modified shape to what we see when the instrument with dead-time really does measure the system, we can attempt to infer whether or not our original assumed “statistical shape” had any chance of being right.  Unfortunately, this is an inverse problem and consequently sports non-unique solutions.  As a first step, this can be helpful.

 

 

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