Author: Justin Silverman

# Non-linear models with fido::basset

#### Justin Silverman

If you have not already done so, I would read through the pibble vignette before this one.

# fido::basset, the model

fido can be used for non-linear modeling. Here we are going to go through one such model which is based on multivariate Gaussian processes. As Gaussian processes are a type of lazy learner, I have named this model after a lazy dog: a Basset Hound. Hence this model is called basset. The basset model can be written as \[ \begin{align} Y_j & \sim \text{Multinomial}\left(\pi_j \right) \\ \pi_j & = \phi^{-1}(\eta_j) \\ \eta &\sim N(\Lambda[X], \Sigma, I_N) \\ \Lambda[X] &\sim \textsf{GP}(\Theta[X], \Sigma, \Gamma[X]) \\ \Sigma &\sim W^{-1}(\Xi, \upsilon) \end{align} \] where we now label quantities interpreted as function with square brackets. In particular, we use the notation that \(\Lambda[X]\) is a function that maps am \(Q \times N\) matrix of covariates \(X\) to a matrix \(\eta\). \(\Lambda\) can be interpreted as a smooth function that relates covariates \(X\) to composition \(\eta\) in a potentially non-linear manner. \(\Theta[X]\) is the mean function evaluated at \(X\), \(\Gamma[X]\) is a covariance matrix (or gram matrix) formed by evaluating a positive semi-definite kernel function \(K\) at each set of points \(i\) and \(j\), i.e., \(\Gamma_{ij} = K(X_i, X_j)\). Many different possible kernel functions makes this a very flexible and powerful model.

# Non-linear time-series modeling

To demonstrate basset I am going to reanalyze the mallard1 dataset from Silverman et al. (2018) which is provided as a phyloseq object in fido. This dataset features 4 artificial gut vessels sampled both daily and hourly for 1 month with many technical replicates.

To accord with prior analyses (and to make it easy to visualize results) I am going to just analyze those bacterial families that are frequently observed. Note: the full dataset (at the sequence variant level without preprocessing) is available if you run `data(mallard)`. For simplicity we will just look at samples from vessel 12.

    library(fido)
    library(dplyr)
    library(tidyr)
    library(ggplot2)
    data(mallard_family)

    # Just take vessel 1
    sample.ids <- mallard_family$sample_data[mallard_family$sample_data$Vessel == 1,]
    # Just take hourly samples
    sample.ids <- sample.ids[(sample.ids$time > "2015-11-20 15:00:00 UTC") & (sample.ids$time < "2015-11-25 16:00:00 UTC"),]

    # Subsetting the sample data and OTU data
    subset.sample_data <- mallard_family$sample_data[mallard_family$sample_data$X.SampleID %in% sample.ids$X.SampleID,]

    subset.otu_table <- mallard_family$otu_table[rownames(mallard_family$otu_table) %in% sample.ids$X.SampleID,]

    # Order samples - to make plotting easy later
    o <- order(subset.sample_data$time)
    subset.otu_table <- subset.otu_table[o,]
    subset.sample_data <- subset.sample_data[o,]

    # Extract Data / dimensions from Phyloseq object
    Y <- t(as(subset.otu_table, "matrix"))
    D <- nrow(Y)
    N <- nrow(subset.sample_data)

    # X in hours
    X <- as.numeric(subset.sample_data$time)
    X <- t((X-min(X)) / 3600)

The function basset is a wrapper around some fido internals that make fitting the above model easy. basset is patterned after the function pibble but requires that `Theta` and `Gamma` be given as matrix functions rather than as matrices. That is `Theta` must be a function that given `X` outputs a \(D-1 \times N\) matrix (just like you would give to pibble) and `Gamma` must be a function that given `X` outputs a \(N \times N\) covariance matrix (e.g., covariance over the samples). There are a few kernel functions that meet the requirements for `Gamma` in fido already. Here we will use a Squared Exponential (SE) kernel for `Gamma` and will set `Theta` to be the zero function. The SE kernel will give us non-linear smoothing of the observed time-series.

Important Note: Currently, `Theta` must output a matrix represented in the default coordinate system (\(ALR_D\)). This may be generalized in the future.

Here we will just specify the Kernel parameters manually, more generally these parameters can be chosen by cross-validation.

    # Specify Priors
    Gamma <- function(X) SE(X, sigma=5, rho=10) # Create partial function
    Theta <- function(X) matrix(0, D-1, ncol(X))
    upsilon <- D-1+3
    Xi <- matrix(.4, D-1, D-1)
    diag(Xi) <- 1

    # Now fit the model
    fit <- fido::basset(Y, X, upsilon, Theta, Gamma, Xi)
    #> No rows of X were specified. Using all rows...

It turns out that basset is really just creating a pibblefit object that is a little special because it represents a posterior over non-linear functions. The benefit of this is that many of the methods available for pibblefit objects work for bassetfit objects. For example, we can use the same transformation methods:

    fit.clr <- to_clr(fit)

    # Plot Sigma in CLR
    plot(fit.clr, par="Sigma", focus.coord=c("clr_seq_6", "clr_seq_5", "clr_seq_2"))

Really basset shows its power when you use it to smooth.

    # predict not just missing days but also forecast into future
    X_predict <- t(1:(max(X)))
    predicted <- predict(fit.clr, X_predict, jitter=1)

Now I am going to create a visual that shows the observed data in CLR coordinates (to do that I will need to add a pseudo-count) along with the smoothed estimates.

    family_names <- as(mallard_family$tax_table$Family, "vector")
    Y_clr_tidy <- clr_array(Y+0.65, parts = 1) %>%
      gather_array(mean, coord, sample) %>%
      mutate(time = X[1,sample],
             coord = paste0("CLR(", family_names[coord],")"))

    predicted_tidy <- gather_array(predicted, val, coord, sample, iter) %>%
      mutate(time = X_predict[1,sample]) %>%
      filter(!is.na(val)) %>%
      group_by(time, coord) %>%
      summarise_posterior(val, na.rm=TRUE) %>%
      ungroup() %>%
      mutate(coord = paste0("CLR(", family_names[coord],")"))

    ggplot(predicted_tidy, aes(x = time, y=mean)) +
      geom_ribbon(aes(ymin=p2.5, ymax=p97.5), fill="darkgrey", alpha=0.5) +
      geom_ribbon(aes(ymin=p25, ymax=p75), fill="darkgrey", alpha=0.9)+
      geom_line(color="blue") +
      geom_point(data = Y_clr_tidy, alpha=0.5) +
      facet_wrap(~coord, scales="free_y") +
      theme_minimal()+
      theme(axis.title.y = element_blank(),
            axis.title.x = element_blank(),
            axis.text.x = element_text(angle=45))

# Updated Functionality: Additive Gaussian Processes

Michelle Nixon

Starting with fido version 1.1.0, the basset model has been updated. The new model is a more flexible extension of the old model as it allows \(\eta\) to be modeled as an additive Gaussian process model:

\[\begin{align*} Y_j &\sim \mathrm{Multinomial}(\pi_j)\\ \pi_j &= \phi(\eta_j)\\ \eta_j &\sim N(\Lambda(X), \Sigma, I)\\ \Lambda(X) &= \Lambda_1 X + \Lambda_2(X) + ... + \Lambda_p(X)\\ \Lambda_1 &\sim N(\Theta_1, \Sigma, \Gamma_1)\\ \Lambda_2(X) &\sim GP(\Theta_2(X), \Sigma, \Gamma_2(X))\\ & \cdots\\ \Lambda_p(X) &\sim GP(\Theta_p(X), \Sigma, \Gamma_p(X))\\ \Sigma &\sim IW(\Xi, \upsilon). \end{align*}\]

There is a great deal of flexibility in how \(\Theta_i(X)\) and \(\Gamma_i(X)\) can be specified for \(i = 1,...,p\). To invoke the additive Gaussian process, the inputs Theta and Gamma can now be a list of length \(p\) with each element of both lists corresponding to the prior specification for a given additive component. The elements of the list can either be functions over the correct dimensions (see documentation) or an (optional) matrix of correct size. If the matrix is used for component \(i\), then a linear model is fit for that component, and \(\Lambda_i(X)\) returns estimated parameters similar to the fido model. If a function is used for component \(i\), then that component works similarly to the old basset model in both input and returned results. Note: If the \(i^{th}\) list item for Theta is a function, then the \(i^{th}\) list item for Gamma must also be a function. If the \(i^{th}\) list item for Theta is a matrix, then the \(i^{th}\) list item for Gamma must also be a matrix.

Now, a simple simulation study to show how the updated functionality works. First, we are going to simulate our data. Here, we will simulate three taxa (\(D=3\)) over 100 observations (\(N=100\)).

    library(fido)
    library(MCMCpack)
    library(LaplacesDemon)

    set.seed(2024)
    D <- 3 # number of taxa
    N <- 100 #number of observation
    X <- seq(0,10, length.out = N) # specifying the values of X

We are going to simulate our data such that \(\Lambda(X) = \Lambda_1(X) + \Lambda_2(X)\). The first component will be using a radial basis function kernel. The second component will be using a linear kernel.

    # RBF kernel function
    rbf <- function(x,l,sigma) {
      K <- outer(x,x, FUN = function(x,y, sigma,l){sigma^2 * exp(- (x - y)^2 / (2 * l^2))}, sigma = sigma, l = l)
      identity_matrix <- diag(ncol(K))
      K  <-  K  + 1e-8 * identity_matrix
      return(K)
    }

    # Linear kernel function
    linear <- function(x,sigma,c) {
      K <- outer(x,x, FUN = function(x,y, sigma,c){ (sigma^2 + (x-c) * (y-c))}, sigma = sigma, c=c)
      identity_matrix <- diag(ncol(K))
      K  <-  K  + 1e-8 * identity_matrix
      return(K)
    }

First, we will simulate \(\Sigma\), then \(\Lambda_1(X)\) and \(\Lambda_2(X)\) (and thus \(\Lambda(X)\)). Then, we will simulate \(\eta\), transform to proportions, and, finally, simulate \(Y\).

    ## Step 1: Sigma ~ IW(upsilon,Xi)
    Sigma <- rinvwishart(D-1, 0.5*diag(D-1))

    ## Step 2a: Lambda_1 ~ GP(0,Sigma,K)
    K <- rbf(X,1,1)
    Theta <- matrix(0, D-1, N)
    x <- matrix(rnorm(N*(D-1)),nrow = D-1)
    l1 <- Theta+ t(chol(Sigma))%*%x%*%(chol(K))

    ## Step 2b: Lambda_2 ~ GP(0, Sigma, K2)
    K2 <- linear(X,.1,0)
    Theta <- matrix(0, D-1, N)
    x <- matrix(rnorm(N*(D-1)),nrow = D-1)
    l2 <- Theta+ t(chol(Sigma))%*%x%*%(chol(K2))

    l <- l1 + l2

    ## Step 3: eta ~ N(Lambda,Sigma, I)
    eta <- matrix(rnorm(N*(D-1)),nrow = D-1, byrow = TRUE)
    eta <- l + t(chol(Sigma))%*%eta

    ## Step 4: transform to proportions
    pai <- t(alrInv(t(eta)))

    ## Step 5: Simulate Y.
    Y <- matrix(0, D, N)
    for (i in 1:N) Y[,i] <- rmultinom(1, 5000, prob = pai[,i])

Phew! Now, to analyze the data. First, we want to specify our priors for \(\Gamma_1(X)\) and \(\Gamma_2(X)\). Since we know how the data were simulated, we will use the ground truth as our priors:

    rbf <- function(x,l,sigma) {
      x <- x[1,]
      K <- outer(x,x, FUN = function(x,y, sigma,l){sigma^2 * exp(- (x - y)^2 / (2 * l^2))}, sigma = sigma, l = l)
      identity_matrix <- diag(ncol(K))
      K  <-  K  + 1e-8 * identity_matrix
      return(K)
    }

    # Linear kernel function for Gaussian process
    linear <- function(x,sigma,c) {
      x <- x[1,]
      K <- outer(x,x, FUN = function(x,y, sigma,c){ (sigma^2 + (x-c) * (y-c))}, sigma = sigma, c=c)
      identity_matrix <- diag(ncol(K))
      K  <-  K  + 1e-8 * identity_matrix
      return(K)
    }

    Gamma <- list(function(X) rbf(X,1,1), function(X) linear(X,.1,0))

Next, we need to specify our prior for \(\Theta_1(X)\) and \(\Theta_2(X)\):

    Theta <- list(function(X) matrix(0, D-1, ncol(X)), function(X) matrix(0, D-1, ncol(X)))

And, finally, we can fit the model:

    mmX <- model.matrix(~X-1)
    mod <- basset(Y, t(mmX), Theta = Theta, Gamma = Gamma)
    #> No rows of X were specified. Using all rows...

Now, let’s look at the results. You’ll notice that the parameter objects for \(\Lambda\) is now a list. It behaves as you largely would expect: the \(i^{th}\) element of the list for Lambda corresponds to the \(i^{th}\) component as specified by Theta and Gamma.

    typeof(mod$Lambda)
    #> [1] "list"

The model object should largely work as it has before. For example, we can find predictions:

    ## predictions
    preds <- predict(mod)

    ## summarizing predictions over posterior samples
    summary_preds <- apply(preds,MARGIN = c(1,2), FUN = mean)

And, finally, we can plot the predictions:

    ## plotting
    data.frame("points" = c(eta[1,], summary_preds[1,]), "group" = c(rep("Truth", length(eta[1,])), rep("Predicted", length(summary_preds[1,]))), "Index" = c(1:N, 1:N)) %>%
      ggplot(aes(x=Index, y= points, color = group)) +
      geom_line() +
      theme_bw() +
      scale_color_manual(values = c("red", "black")) +
      theme(legend.title=element_blank()) +
      ylab("Eta")

Note that since the second component was linear, we could have specified the priors using matrices as well. This might be more helpful if, for example, the estimated individual coefficients for the linear terms are of interest.

    Theta <- list(matrix(0, D-1, ncol(mmX)), function(X) matrix(0, D-1, ncol(X)))
    Gamma <- list(diag(ncol(mmX)), function(X) rbf(X,1,1))

    mmX <- model.matrix(~X-1)
    mod <- basset(Y, t(mmX), Theta = Theta, Gamma = Gamma)
    #> No rows of X were specified. Using all rows...

Finally, predicting and plotting:

    ## predictions
    preds <- predict(mod)

    ## summarizing predictions over posterior samples
    summary_preds <- apply(preds,MARGIN = c(1,2), FUN = mean)

    ## plotting
    data.frame("points" = c(eta[1,], summary_preds[1,]), "group" = c(rep("Truth", length(eta[1,])), rep("Predicted", length(summary_preds[1,]))), "Index" = c(1:N, 1:N)) %>%
      ggplot(aes(x=Index, y= points, color = group)) +
      geom_line() +
      theme_bw() +
      scale_color_manual(values = c("red", "black")) +
      theme(legend.title=element_blank()) +
      ylab("Eta")

An updated vignette will be available soon.

# References

Silverman, Justin D, Heather Durand, Rachael J Bloom, Sayan Mukherjee, and Lawrence A David. 2018. “Dynamic Linear Models Guide Design and Analysis of Microbiota Studies Within Artificial Human Guts.” bioRxiv. https://doi.org/10.1101/306597.

1. I have a long history of coming up with ridiculous names for statistical models and software.↩︎

2. In the future I may make a vignette discussing how we can model all the vessels jointly using basset. Note: it’s just a different kernel function that also relates the covariance between samples as a function of which vessel they were taken from.↩︎