# Using SNPknock in R

#### Matteo Sesia (msesia@stanford.edu)

#### 2019-05-16

This vignette illustrates the usage of the `SNPknock` package for creating knockoffs of discrete Markov chains and hidden Markov models (Sesia, Sabatti, and Candès 2019). For simplicity, we will use synthetic data.

The `SNPknock` package also provides a simple interface to the genotype imputation software `fastPhase`, which can be used to fit hidden Markov models for genotype data. Since `fastPhase` is not available as an R package, this particular functionality of `SNPknock` cannot be demonstrated here. A tutorial showing how to use a combination of `SNPknock` and `fastPhase` to create knockoffs of genotype data can be found here.

## Knockoffs of discrete Markov chains

First, we verify that the `SNPknock` can be loaded.

    library(SNPknock)

We define a Markov chain model with 50 variables, each taking one of 5 possible values. We specify a uniform marginal distribution for the first variable in the chain and create 49 transition matrices with randomly sampled entries.

    p=50; # Number of variables in the model
    K=5;  # Number of possible states for each variable
    # Marginal distribution for the first variable
    pInit = rep(1/K,K)
    # Create p-1 transition matrices
    Q = array(stats::runif((p-1)*K*K),c(p-1,K,K))
    for(j in 1:(p-1)) {
      Q[j,,] = Q[j,,] + diag(rep(1,K))
      Q[j,,] = Q[j,,] / rowSums(Q[j,,])
    }

We can sample 100 independent observations of this Markov chain using the `SNPknock` package.

    set.seed(1234)
    X = sampleDMC(pInit, Q, n=100)
    print(X[1:5,1:10])

    ##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    ## [1,]    0    0    3    2    3    3    2    0    0     2
    ## [2,]    3    3    3    4    2    3    1    2    3     2
    ## [3,]    3    1    1    1    3    3    1    3    2     3
    ## [4,]    3    0    3    3    3    0    1    1    1     2
    ## [5,]    4    1    1    4    3    3    2    2    2     3

Above, each row of `X` contains an independent realization of the Markov chain.

A knockoff copy of `X` can be sampled as follows.

    Xk = knockoffDMC(X, pInit, Q)
    print(Xk[1:5,1:10])

    ##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    ## [1,]    3    3    3    3    3    3    2    0    0     2
    ## [2,]    3    3    3    3    0    0    2    3    2     2
    ## [3,]    0    1    1    2    3    1    1    4    3     2
    ## [4,]    3    3    3    4    0    1    1    1    1     1
    ## [5,]    0    0    1    2    3    3    2    2    4     2

## Knockoffs of hidden Markov models

We define a hidden Markov chain model with 50 variables, each taking one of 3 possible values, and 50 latent variables distributed as a Markov chain, each taking one of 5 values. We specify a uniform marginal distribution for the first variable in the chain and create 49 transition matrices with randomly sampled entries. We also define an emission distributions over the 3 possible emission states, for each of the 50 latent variables and the 5 latent states.

    p=200; # Number of variables in the model
    K=5;  # Number of possible states for each variable
    M=3;  # Number of possible emission states for each variable
    # Marginal distribution for the first variable
    pInit = rep(1/K,K)
    # Create p-1 transition matrices
    Q = array(stats::runif((p-1)*K*K),c(p-1,K,K))
    rho = stats::runif(p-1, min = 1, max = 50)
    for(j in 1:(p-1)) {
      Q[j,,] = Q[j,,] + rho[j] * diag(rep(1,K))
      Q[j,,] = Q[j,,] / rowSums(Q[j,,])
    }
    # Create p emission matrices
    pEmit = array(stats::runif(p*M*K),c(p,M,K))
    for(j in 1:p) { pEmit[j,,] = sweep(pEmit[j,,],2,colSums(pEmit[j,,]),`/`) }

We can sample 100 independent observations of this hidden Markov model using the `SNPknock` package.

    set.seed(1234)
    X = sampleHMM(pInit, Q, pEmit, n=100)
    print(X[1:5,1:10])

    ##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    ## [1,]    0    2    2    2    0    1    1    2    0     0
    ## [2,]    2    1    2    0    1    2    0    0    0     0
    ## [3,]    2    2    0    1    0    1    2    0    1     2
    ## [4,]    1    0    2    2    0    0    2    0    1     0
    ## [5,]    1    0    2    2    2    1    2    2    2     1

Above, each row of `X` contains an independent realization of the hidden Markov model.

A knockoff copy of `X` can be sampled as follows.

    Xk = knockoffHMM(X, pInit, Q, pEmit)
    print(Xk[1:5,1:10])

    ##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    ## [1,]    0    0    2    2    0    2    1    1    2     0
    ## [2,]    0    0    1    0    0    2    1    0    2     1
    ## [3,]    1    1    2    1    0    1    1    0    2     2
    ## [4,]    1    0    2    2    0    1    0    1    1     1
    ## [5,]    1    0    1    2    2    2    2    2    2     2

## See also

If you want to see how to use `SNPknock` to create knockoffs of genotype data, see the genotypes vignette.

If you want to learn about `SNPknock` for large genome-wide association studies (Sesia et al. 2019), see https://msesia.github.io/knockoffzoom/

Sesia, M., E. Katsevich, S. Bates, E. Candès, and C. Sabatti. 2019. “Multi-Resolution Localization of Causal Variants Across the Genome.” bioRxiv. Cold Spring Harbor Laboratory. https://doi.org/10.1101/631390.

Sesia, M., C. Sabatti, and E. J. Candès. 2019. “Gene Hunting with Hidden Markov Model Knockoffs.” Biometrika 106:1–18. https://doi.org/10.1093/biomet/asy033.