Markov Process Sampling

One of the major strategies employed by single molecule techniques is to model particle traces with a Hidden Markov Model (HMM), assuming some number of states typically optimized by a Bayesian Information Criteria (BIC). These experiments are typically probing macroscopic fluctuations of molecular conformations by watching FRET (Forster Resonance Energy Transfer) trajectories. These trajectories are recorded, typically, with a fast CCD or perhaps a PMT (for non-wide-field studies). In the former case, integration times are on time scale of milliseconds.

I was curious about the nature of how these HMM react to undersampling. It's well known that undersampling a signal causes well known effects, like aliasing unless one takes care to sample below the Nyquist frequency. If the relevant underlying fluctuations of your system are faster than 500 microseconds, you're no longer above the Nyquist frequency for a CCD with a millisecond integration time. How does that affect the estimation of HMM parameters? It seems obvious to me that the transition matrix / emission matrix are going to be the only things affected, assuming you observe the system long enough that you see all possible states. But how exactly will the transition matrix be affected? Can it be compensated for?

It is pretty standard knowledge that undersampling causes the signal to be aliased to a lower frequency. It make intuitive sense to me that lower frequencies in HMMs are represented by lower transition probabilities, assuming the system has constant rates (which means that state duration can be modeled by an exponential distribution with expectated life time equal to the inverse of the transition probability). The reasoning here is that if the inverse of the state duration is the transition probability, then short state durations would result in high transition probabilities and long state durations will result in low transition probabilities.

To test this hypothesis, I decided to run a simulation in Matlab. I generated a 3-state sequence based off a known Markov Model with 1 million data points. I computed the upperbound of the 3 expected state durations and then went back and sampled the simulated sequence at a rate equal to 10 times the maximum expected state duration, to simulate under sampling. Sampling was periodic, with a small time variance for each point, and I took 300 data points for each "experiment". I then went back and estimated the transition probabilities for each of these "experiments" with a HMM algorithm. I averaged 50 of the resulting transition matrices. The results were surprising. Name, the exact opposite happened: the probabilities increased on undersampling. At certain degrees of undersampling (like 10x the maximum expected state duration), they seemed to converge to a multiple of the real transition probabilities related to the multiple of the inverse of the real transition probability. Other times, there didn't seem to be an obvious correlation.


I have yet to do much literature research on this particular problem, preferring to just jump in and see what I could come up with in an hour of coding. But it's obvious to me now that we're not going to be able to use standard signal analysis theory for sampling problems like this. For one, Markov sequences are non-periodic, so in retrospect, aliasing doesn't have much basis. I refuse (as of yet) to believe that one cannot predict the effects of undersampling (at least qualitatively), simply on the basis that no matter how I tweaked the parameters of my simulation the probabilities were always higher than the real ones, as long as I under sampled.


~~~~~~~~~~~~~Edit~~~~~~~~~~~~~~

On some additional thought, I think I can explain why the TM (transition matrix) is exhibiting an increase in probabilities. Since the sequence was generated from a first order Markov system, later time points are, by definition, not probabilistically linked to any point beyond the one previous to it, though they are weakly linked by conditional probabilities, which decrease rapidly in magnitude as one goes further. At some point, the coupling between observed events will be so weak that one may as well just be sampling a uniform distribution. I thought, originally, I was seeing a conservation of the relative ratios of the TM elements compared to the real TM. More experimentation proved this initial hope wrong. Here's a summary of those results:

1. Looking at the internally relative probabilistic ratios of the TMs, I am seeing fairly large fluctuation in how the observed and fit ratios compare to the ratios used to generate the data. It wasn't until I collected and averaged 1000 traces that I started to see reasonable convergence on the difference between observed and actual internal probability ratios in the TM. This has significant practical implications for experimental study of markov processes in a possibly undersampled regime.

2. I discovered a problem in my previous simulations such that the generating model's TM was ill defined (rows didn't sum to 1). I fixed this. The result was that the averaged, observed TM appears to be converging towards [0.3 0.3 0.3; 0.3 0.3 0.3; 0.3 0.3 0.3] (for a 3 state system), which is consistent with my earlier assertion that undersampling a Markov process might result in merely sampling a uniform distribution. I haven't tested how sampling rate affects this property yet, I've just made sure that I was under sampling the sequence such that I would sample on the time scale of ten times the off-diagonal transition characteristic time scale.

EMP Device

Apart from working in an optics lab at the moment, I'm not sure what brought this line of inquiry on. But here it is, in its unpolished glory. I make no promises on the veracity of any ideas presented herein. These are my very hastily researched ideas.

The Motivation/General Idea:

• Create a rapid, high amplitude oscillation of charged particles (electrons). Possibly high enough amplitude to generate a destructive impulse, similar to the electromagnetic impulse generated by a nuclear warhead. There do exist such non-nuclear devices and are already employed by the military. (Source).
• Exploit current laser technology to shorten the time-width of the EM impulse from hundreds of microseconds to tens of nanoseconds.
• Have this impulse be generated without the use of explosives (making it cleaner).
• Preferably make this device inexpensive and ruggedly operable.
  

Preliminary Concepts:

• Stimulated Emission
• Cerenkov Radiation
• CO2 laser
• Photo-Electric Effect
• Frank-Tamm Formula
• Smith-Purcell Effect
  

Method Outline:

1. The photo-electric effect gives us a method to liberate electrons from a metallic surface.
2. The work function of the metallic surface acts as a high-pass filter for photon frequencies. High pass in the sense that only photons of sufficiently high frequency will liberate electrons, the remainder are reflected.
3. CO2 lasers output in the infrared. They commonly use gold plated reflectors, the work function for which is around 5 eV (Source). The work function can be reduced by altering the E-field across the metal. You can use this fact, coupled with a light source of known wavelength, to measure the work function of a metal.
4. CO2 lasers can be "Q-switched" to produce a pulsed laser with peak pulse energies in the Gigawatt region (commercially). The duration of these pulses can be as small as 10s of nano-seconds.
5. There are a number of ways to pump CO2 lasers and I'm not sure if this is the best means, but I thought it would be cool if we used the Cerenkov effect to provide a broad-band photon source. This would give us an incentive to make the lasing cavity double as a weak linear particle accelerator. Negatively charge the cavity walls to center an electron beam onto a Smith-Purcell grating. So long as some of the photons coming off the Cerenkov cell induce stimulated emission in the CO2, we're okay. As an added bonus, if the frequency content of the light is sufficiently high to overcome the already negatively biased cavity walls, it will cause electrons showers to either excite the Cerenkov device further or excite Nitrogen oscillations (which play a role in the CO2 lasing effect). Importantly, the Cerenkov pump produces radiation amplitude that is dependent on the length of the medium and velocity of the moving particle, which means it can be easily scaled up to provide large quantities of pump photons.
   
6. We can have the Q-switched output at the negatively charged end of the cavity. Beyond the aperature we could put a high-surface area, metallic receptor (that could also double as the cathode to provide the chamber's E-field.)
   
7. When we want to create a pulse, we'll allow the laser to output a pulse via active Q-switching. Simultaneously, we'll increase the negative bias on the metallic receptor. Provided the beam is diffuse enough to cover the entire receptor, and the receptor is sufficiently biased to loose its electrons from infrared excitation, then we should observe a rapid (on the time scale of the pulse), depopulation of electrons from the cathode. Provided the transient depopulation is strong enough (coupled perhaps with a loss of voltage bias on the cathode), the recently released electrons should fly back towards the now very positive cathode. And so we've produced an oscillation of electrons, as desired.

Discussion:

• If you're familiar with photoelectric effect capacitors (see here, for example), then this idea isn't terribly novel. All I'm suggesting is that we see what would happen in the extreme case of exciting such a capacitor with an enormous impulse from a CO2 pulsed laser.
• I wasn't thinking of the photoelectric effect capacitor from the outset. My initial goal was to explore how to use Cerenkov radiation in a laser. Perhaps with my idea of using both photoelectric effect and standard lasing, you could increase the efficiency of output light energy to input energy.
• This device should be fairly "rugged", as rugged as CO2 lasers are, at any rate. It would likely require cooling, a high voltage power supply, a small vacuum near the cathode, and a flow-through gas tube -- along with the other laser components.

Some Brief Calculations:

• A commercial Q-switched CO2 laser GEM Q-400 has a pulse energy of 1 mJ and a FWHM pulse width of 150 ns at an output wavelength of 9.25 uM. Lets say that you tune the cathode to eject photons with zero KE at 9.25 uM, how many electrons would you eject in the 150 ns of the pulse? [Energy per electron = hv. v = 2*pi*f = 2*pi*(c/wavelength). Ee = 6.6E-34*2*pi*(1E8/1E-5) = 4E-20 J/electron. So 1 mJ would liberate about 2E16 electrons. That's much less than a mol of electrons so I think it's safe to say that a moderately sized cathode could donate this many electrons.