Projects | Ian Arriaga-MacKenzie

Binomial Distribution of Allele Data

Sat, 01 May 2021 00:00:00 +0000

Summary population allele data can be modeled using a binomial mixture distribtion with homogeneous reference populations. Here we show an example of this using the gnomAD V2.1 African/African-American data set, and homogeneous African and European reference panels from 1000 Genomes. We adopt the following model:

$$ P \left( n \vert N, \Theta \right) = \mbox{Binom} \left( n \bigg\vert N, \sum_{k=1}^{K} \pi_k \theta_k \right) $$ $$ \ell( \Theta ) = ln \mathcal{L} (\Theta \vert X) = \sum_{i=1}^S ln \left[ \mbox{Binom} \left( n \bigg\vert N, \sum_{k=1}^{K} \pi_k \theta_k \right) \right] $$

where

S is the set of SNPs

K are ancestries

$\pi_k$ are ancestry proportions for k

$n_i$ is the Allele Count for that SNP

$N_i$ is the Allele Number for that SNP

$\theta_k$ is the Allele Frequency for that SNP

This leads to the above image which shows a maximation of the log likelihood at the following values:

AFR: 0.8277273 EUR: 0.1722727

These values are consistent with known admixture within the gnomAD sample, and are confirmed with other estimation methods (Summix, ADMIXTURE). There are several ways to maximize the log-likelihood including grid-search and Expecation-Maximization algorithms. The binomial distribution can also be inverted and solved using gradient descent methods, such as Sequential Quadratic Programming.

EM Algorithm for Mixture Model

Fri, 01 Jan 2021 00:00:00 +0000

We can generate a population for a binomial mixture under the following assumptions: That we have a number n of true cases out of a possible number N total, and that these have a binomial distribution with $\pi_i$ probability of being from each distribution.

This is easier to visualize with a ‘coins in a pot’ example. Say we have a pot filled with two types of coins, each type of coin having its own probability of heads. We pull a number of coins S from the pot, flip each coin N times and record the number of n heads. Under these assumptions, the probability of seeing n heads for each coin would be:

$$ P \left( n \vert N, \Theta \right) = \pi_1 \mbox{Binom} \left( n \vert N, \theta_1 \right) + \pi_2 \mbox{Binom} \left( n \vert N, \theta_2 \right) $$

or more generally for K different types of coins

$$ P \left( n \vert N, \Theta \right) = \sum_{k=1}^{K} \pi_k \mbox{Binom} \left( n \vert N, \theta_k \right) $$

where $\theta_k$ is the probability of heads for each type of coin. Therefor the log-likelihood for our parameters is:

$$ \mathcal{L} \left( \Theta \vert X, Z \right) = ln P \left( X, Z \vert \Theta \right) $$

Where $\Theta_k$ represents the set of $\pi_k, \theta_k$, X represents the set of heads and total coin flips, and Z represents the set of coin proportions and heads proportions. So the Auxiliary Function is:

$$ Q(\Theta, \Theta_o) = E \left[ ln \mathcal{L} \left( \Theta \vert X, Z \right) \vert X, \Theta_o \right] $$

and our expectation is:

$$ E \left[ ln P \left( X, Z, \vert \Theta \right) \vert X, \Theta_o \right] = \sum_{z_i = 1}^K ln P \left( n_i, z_i \vert \Theta \right) \cdot P \left( z_i \vert n_i, \Theta_o \right) $$

where

$$ P \left( z_i = k \vert n_i , \Theta_o \right) = \dfrac{P \left( z_i = k , n_i \vert \Theta_o \right) }{P \left( z_i , n_i \vert \Theta_o \right)} = \dfrac{\pi_{k,o} \mbox{Binom} \left( n_i \vert N_i , \theta_{k,o} \right) }{\sum_{l=1}^{K} \pi_{l,o} \mbox{Binom} \left( n_i \vert N_i , \theta_{l,o} \right) } $$

We can then use the following expressions to update $\pi$ and $\theta$ until convergence.

$$ \pi_m = \dfrac{1}{S} \sum_{i=1}^S P \left( z_i = m \vert n_i , \Theta_{o} \right) $$ $$ \theta_{m,S} = \dfrac{\sum_{i=1}^S n_i \cdot P \left( z_i = m \vert n_i , \Theta_{o} \right)}{\sum_{j=1}^S N_j \cdot P \left( z_j = m \vert n_j , \Theta_{o} \right)} $$

Signal Decomposition of EEG Data

Fri, 01 Nov 2019 00:00:00 +0000

Above is a 10 second measurement from an A1-A2 electroencephalogram (EEG) lead, taken from a sleeping patient. We examine this data with a motivating example and application, using a Fast Fourier Transform (FFT), which is a computationally efficient and quick method to decompose sinusoidal signals into their component frequencies. We can show this using a series of signals at 1, 5, 10, 20 and 45 Hz shown below:

Figure 1. Each of the 5 individual sinusoidal signals.

These 5 signals can be combined into a signal signal, with a gaussian random sample added to simulate noise. This combined signal can then be put through a finite impulse response (FIR) band-pass (3-30 Hz) filter to further reduce noise and target specific frequencies for signal decomposition.

Figure 2. Combined figures, with noise and band-pass filter.

Using a multi-taper power spectral density method of estimation, we can show the original frequencies used to create the combined signal, both before and after a band-pass filter is applied.

Figure 3. Frequency Estimation, no FIR filter.

Figure 4. Frequency Estimation, 3-30 Hz FIR filter.

We also apply multitaper PSD estimation to our EEG sample. In particular (because the patient was asleep) we focus on the lower frequencies (0.5 Hz - 40 Hz) as we expect there to be a large concentration of lower frequency brain waves.

Figure 5. PSD Estimation of EEG Lead.

While this result may seem surprising at first, it does seem consistent with the data and highlights several features. Within EEG data, there are several defined frequency bandwidths which are associated with various states of consciousness. As this patient is asleep, we see high concentrations of Delta (0.5 - 4 Hz) and Theta (4 - 8 Hz) brain waves, with minimal higher brain waves. It also is visually consistent with the plotted EEG lead, as there are large sinusoidal waves in the reading.