|
| 1 | +About |H2MM| |
| 2 | +============ |
| 3 | + |
| 4 | +What is |H2MM|? |
| 5 | +--------------- |
| 6 | + |
| 7 | +|H2MM| is an extension of hiddenm Markov modeling, a broad set of methods for finding underlying behavior in noisy systems. |
| 8 | +The basic assumptions of hidden Markov models are: |
| 9 | + |
| 10 | +#. The system is described by a Markvov model |
| 11 | + a. The data describes a system that transitions between a set of states. |
| 12 | + b. When the system is in a given state, it has a distinct set of probabilities to transition to each of the other states |
| 13 | + c. The system is memoryless, that is the probability to transition to another state is only influenced by the current state, and not any of the previous states |
| 14 | +#. The model is hidden, meaning |
| 15 | + a. The states cannot be directly observed, rather |
| 16 | + b. Each state has a certain probability to produce certain observable results, with each state having a different set of probabilities |
| 17 | + |
| 18 | +The earliest applications of HMM were towards language processing, for which it was not very successful, but has found much greater use in signal processing. |
| 19 | +HMM has also found extensive use in analyzing TIRF based smFRET trajectories. |
| 20 | +All of these applications however, assume that there is a constant data rate, for a camera with a set frame rate so that there is an image every ms. |
| 21 | + |
| 22 | +This is not the case for single molecule confocal based data using single photon detectors. |
| 23 | +Here, data comes in sparsely, as individual photons, with varying interphoton times. |
| 24 | +|H2MM| extends the HMM algorithm to accept these variable interphoton times, allowing application of the HMM machinery with confocal data without implementing an external time bin. |
| 25 | + |
| 26 | +HMM methods use the Baum-Welch algorithm in an optimization process which finds the model of a set number of states that best describes the data. |
| 27 | +Now since the number of states is fixed, other optimizations must be conducted with different numbers of states. |
| 28 | +Then the different optimized models must be compared, and the best one choosen, and the ones with too many (overfit) and too few (underfit) states regected. |
| 29 | + |
| 30 | +A final thing to understand about |H2MM|, is the use of indeces. |
| 31 | +In |H2MM|, data comes in a set of detetector chanels, in the first iterations, these were exclusively the |DD| and |DA| channels, but with mp |H2MM|, this was extended to the |AA| channel, and was even suggested to be able to include the parralel and perpendicular channels in anisotropy based measurments. |
| 32 | +A comparison to the original application of HMM methods is apt here: originally HMM was developed to anlayze word paterns, so each word was a unique index, and each state had a given probability to produce each word. |
| 33 | +So in |H2MM| each photon can be compared to a word, each photon recieves a set of a limited nubmer of indeces. |
| 34 | + |
| 35 | + |
| 36 | +A Brief History of |H2MM| |
| 37 | +------------------------- |
| 38 | + |
| 39 | +Application to confocal single molecule data started with `Gopich and Szabo 2009 <https://doi.org/10.1021/jp903671p>`_ who established the maximum likelihood estimator to calculated the likelihood of a model of transition rates and emission probabilities for a set of data. |
| 40 | +`Pirchi and Tsukanov et. al. 2016 <https://doi.org/10.1021/acs.jpcb.6b10726>`_ then integrated the Baum-Welch algorithm, with some reformulation of Gopich and Szabo's original equations, which allowed for an optimization procedure ensuring that the likelihood of each iteration improves. |
| 41 | +This made finding the ideal model a feasible undertaking. |
| 42 | +However, discrimination between over and underfit models (models with too many or too few states), `Lerner et. al. 2018 <https://doi.org/10.1063/1.5004606>`_ introduced the first attempt at this, with the modified Bayes Information Criterion, and finally the Integrated Complete Likelihood was introduced in `Harris et. al. 2022 <https://doi.org/10.1038/s41467-022-28632-x>`_, which proved a more reliable statistical discriminator. |
| 43 | +`Harris et. al. 2022 <https://doi.org/10.1038/s41467-022-28632-x>`_ also introduced the multiparameter approach, where the |AA| stream was integrated allowing discrimination of photophysical and FRET dynamics. |
| 44 | + |
| 45 | +.. |H2MM| replace:: H\ :sup:`2`\ MM |
| 46 | +.. |DD| replace:: D\ :sub:`ex`\ D\ :sub:`em` |
| 47 | +.. |DA| replace:: D\ :sub:`ex`\ A\ :sub:`em` |
| 48 | +.. |AA| replace:: A\ :sub:`ex`\ A\ :sub:`em` |
0 commit comments