Detailed Walkthrough of Harmony Algorithm

Korsunsky et al.: Fast, sensitive, and accurate integration of single cell data with Harmony

Motivation

This notebook breaks down the Harmony algorithm and model in the context of a simple real-world dataset.

After reading this, the user should have a better understanding of how

  1. the equations connect to the algorithm
  2. the algorithm works on real data
  3. to access the different parts of the Harmony model from R

Prerequisites

For this vignette we are going to use harmony among other tools that will help with the visualization and inspection of the algorithm intermediate states. Also, we provide a few helper functions that can be found in the source block below.

## Source required libraries
library(data.table)
library(tidyverse)
library(ggthemes)
library(ggrepel)
library(harmony)
library(patchwork)
library(tidyr)

## Useful util functions

cosine_normalize <- function(X, margin) {
    if (margin == 1) {
        res <- sweep(as.matrix(X), 1, sqrt(rowSums(X ^ 2)), '/')
        row.names(res) <- row.names(X)
        colnames(res) <- colnames(X)        
    } else {
        res <- sweep(as.matrix(X), 2, sqrt(colSums(X ^ 2)), '/')
        row.names(res) <- row.names(X)
        colnames(res) <- colnames(X)
    }
    return(res)
}

onehot <- function(vals) {
    t(model.matrix(~0 + as.factor(vals)))
}


colors_use <- c(`jurkat` = rgb(129, 15, 124, maxColorValue=255),
                `t293` = rgb(208, 158, 45, maxColorValue=255),
                `half` = rgb(0, 109, 44, maxColorValue=255))


do_scatter <- function(umap_use, meta_data, label_name, no_guides = TRUE, do_labels = TRUE, nice_names, 
                       palette_use = colors_use,
                       pt_size = 4, point_size = .5, base_size = 10, do_points = TRUE, do_density = FALSE, h = 4, w = 8) {
    umap_use <- umap_use[, 1:2]
    colnames(umap_use) <- c('X1', 'X2')
    plt_df <- umap_use %>% data.frame() %>% 
        cbind(meta_data) %>% 
        dplyr::sample_frac(1L) 
    plt_df$given_name <- plt_df[[label_name]]
    
    if (!missing(nice_names)) {
        plt_df %<>%
            dplyr::inner_join(nice_names, by = "given_name") %>% 
            subset(nice_name != "" & !is.na(nice_name))

        plt_df[[label_name]] <- plt_df$nice_name        
    }
        
    plt <- plt_df %>% 
        ggplot(aes(X1, X2, colour = .data[[label_name]], fill = .data[[label_name]])) + 
        theme_tufte(base_size = base_size) + 
        theme(panel.background = element_rect(fill = NA, color = "black")) + 
        guides(color = guide_legend(override.aes = list(stroke = 1, alpha = 1, shape = 16, size = 4)), alpha = FALSE) +
        scale_color_manual(values = palette_use) + 
        scale_fill_manual(values = palette_use) +    
        theme(plot.title = element_text(hjust = .5)) + 
        labs(x = "UMAP 1", y = "UMAP 2") 
    
    if (do_points) 
        plt <- plt + geom_point(size = 0.2)
    if (do_density) 
        plt <- plt + geom_density_2d()    
        

    if (no_guides)
        plt <- plt + guides("none")
    
    if (do_labels) 
        plt <- plt + geom_label_repel(data = data.table(plt_df)[, .(X1 = mean(X1), X2 = mean(X2)), by = label_name], label.size = NA,
                                      aes(label = .data[[label_name]]), color = "white", size = pt_size, alpha = 1, segment.size = 0) + 
        guides(col = FALSE, fill = FALSE)
    return(plt)
}

Cell line data

This dataset is described in figure 2 of the Harmony manuscript. We downloaded 3 cell line datasets from the 10X website. The first two (jurkat and 293t) come from pure cell lines while the half dataset is a 50:50 mixture of Jurkat and HEK293T cells. We inferred cell type with the canonical marker XIST, since the two cell lines come from 1 male and 1 female donor.

We library normalized the cells, log transformed the counts, and scaled the genes. Then we performed PCA and kept the top 20 PCs. We begin the analysis in this notebook from here.

data(cell_lines)
V <- cell_lines$scaled_pcs
V_cos <- cosine_normalize(V, 1)
meta_data <- cell_lines$meta_data

To get a feel for the data, let’s visualize the cells in PCA space. The plots below show the cells’ PC1 and PC2 embeddings. We color the cells by dataset of origin (left) and cell type (right).

do_scatter(V, meta_data, 'dataset', no_guides = TRUE, do_labels = TRUE) + 
    labs(title = 'Colored by dataset', x = 'PC1', y = 'PC2') +
do_scatter(V, meta_data, 'cell_type', no_guides = TRUE, do_labels = TRUE) + 
    labs(title = 'Colored by cell type', x = 'PC1', y = 'PC2') +
NULL

Initialize a Harmony object

The first thing we do is initialize a Harmony object. We pass 2 data structures:

  1. V: the PCA embedding matrix of cells.
  2. meta_data: a dataframe object containing the variables we’d like to Harmonize over.

The rest of the parameters are described below. A few notes:

set.seed(1)
harmonyObj <- harmony::RunHarmony(
    data_mat = V, ## PCA embedding matrix of cells
    meta_data = meta_data, ## dataframe with cell labels
    theta = 1, ## cluster diversity enforcement
    vars_use = 'dataset', ## variable to integrate out
    nclust = 5, ## number of clusters in Harmony model
    max_iter = 0, ## stop after initialization
    return_object = TRUE ## return the full Harmony model object
)
## Transposing data matrix
## Initializing state using k-means centroids initialization

By initializing the object, we have prepared the data in 2 ways. First, we’ve scaled the PCA matrix to give each cell unit length. Second, we’ve initialized cluster centroids with regular kmeans clustering on these scaled data. We’ll dig into these two steps below.

L_2 scaling to induce cosine distance

A key preprocessing step of Harmony clustering is L2 normalization. As shown in Haghverdi et al 2018, scaling each cell to have L2 norm equal to 1 induces a special property: Euclidean distance of the scaled cells is equivalent to cosine distance of the unscaled cells. Cosine distance is a considerably more robust measure of cell-to-cell similarity (CITE Martin and Vlad). Moreover, it has been used in clustering analysis of high dimensional text datasets (CITE NLP spherical kmeans).

\(L_2\) Normalization of cell \(i\):

\(\hat{Z}_{\cdot, i} \leftarrow \frac{\hat{Z}_{\cdot, i}}{||{\hat{Z}_{\cdot, i}}||_{2}}\)

TL;DR Harmony clustering uses cosine distance. By normalizing each cell to have unit length, we can directly visualize the cosine distances between cells (right). These relationships are not obvious in Euclidean space (left).

In the Harmony object, we now have 2 copies of the cell embeddings. The first, \(Z_{orig}\) is the original PCA matrix (PCs by cells). The second, \(Z_{cos}\) is the new \(L_2\) scaled matrix. Since this scaling projects cells into a unit hypersphere, cells appear pushed away from the origin (0,0).

do_scatter(t(harmonyObj$Z_orig), meta_data, 'dataset', no_guides = TRUE, do_labels = TRUE) + 
    labs(title = 'Z_orig', subtitle = 'Euclidean distance', x = 'PC1', y = 'PC2') +
do_scatter(t(harmonyObj$Z_cos), meta_data, 'dataset', no_guides = TRUE, do_labels = TRUE) + 
    labs(title = 'Z_cos', subtitle = 'Induced Cosine distance', x = 'PC1', y = 'PC2')

In the \(Z_{cos}\) scatterplot (right), cells that are nearby have a high cosine similarity. Although it is not obvious in this example, cells closeby in Euclidean space do not always have a high cosine similarity!

Above, we only visualize the first two PCs. In this simple example with cell lines, this is sufficient to visualize most of the variation. Note, however, that all clustering and correction in Harmony uses all the PCs. For completeness, we can visualize the quantiles of PCA embeddings for all 20 PCs, colored by original dataset.

harmonyObj$Z_cos %>% t %>% data.frame() %>% 
    cbind(meta_data) %>% 
    tidyr::gather(key, val, X1:X20) %>% 
    ggplot(aes(reorder(gsub('X', 'PC', key), as.integer(gsub('X', '', key))), val)) + 
        geom_boxplot(aes(color = dataset)) + 
        scale_color_manual(values = colors_use) + 
        labs(x = 'PC number', y = 'PC embedding value', title = 'Z_cos (unit scaled PCA embeddings) for all 20 PCs') + 
        theme_tufte(base_size = 10) + geom_rangeframe() + 
        theme(axis.text.x = element_text(angle = 45, hjust = 1))

Initial clustering

Initializing the Harmony object also triggered initialization of all the clustering data structures. Harmony currently uses regular kmeans, with 10 random restarts, to find initial locations for the cluster centroids. Let’s visualize these centroids directly! We can do this by accessing the Y matrix in the Harmony object. This is a matrix with \(d=20\) rows and \(K=5\) columns, so each column represents one 20-dimensional centroid.

Remember that we set the number of clusters to 5 above, so there are now 5 clusters below.

cluster_centroids <- harmonyObj$Y

do_scatter(t(harmonyObj$Z_cos), meta_data, 'dataset', no_guides = FALSE, do_labels = FALSE) + 
    labs(title = 'Initial kmeans cluster centroids', subtitle = '', x = 'PC1', y = 'PC2') +
    geom_point(
        data = data.frame(t(cluster_centroids)), 
        color = 'black', fill = 'black', alpha = .8,
        shape = 21, size = 6
    ) +
NULL

Based on these cluster centroids, we also assigned probabilistic cluster memberships to each cell. In the algorithm, this is done using the formula below.

\(R_{ki} \propto \exp(\frac{-||Z_i - Y_k||^2_2}{\sigma})\)

Above, \(R_{ki}\) is a value from \(0\) to \(1\) and denotes the probability that cell \(i\) is assigned to cluster \(k\). Accordingly, the squared distance \(||Z_i - Y_k||^2_2\) is the distance between cell \(i\) and the centroid of cluster \(k\). Because we’re using cosine distance (i.e. cells and centroids have unit length), we can simplify the distance computation:

\(R_{ki} \propto \exp(\frac{-(2(1 - Y^TZ))}{\sigma})\)

Finally, the \(\propto\) symbol means that we will normalize R to form a proper probability distribution for each cell:

\(\sum_k R_{ki} = 1\)

Let’s take a look at these initial cluster assignments. We can find these assignments in the \(K\) row by \(N\) column matrix \(R\).

cluster_assignment_matrix <- harmonyObj$R

The plots below color each cell by cluster membership, from 0 (grey) to 1 (blue). For clarity, each column is a different dataset. Each row is one of the 5 clusters.

t(harmonyObj$Z_cos) %>% data.frame() %>%
    cbind(meta_data) %>% 
    tibble::rowid_to_column('id') %>% 
    dplyr::inner_join(
        cluster_assignment_matrix %>% t() %>% data.table() %>% 
            tibble::rowid_to_column('id') %>%
            tidyr::gather(cluster, r, -id) %>% 
            dplyr::mutate(cluster = gsub('V', 'Cluster ', cluster)), 
        by = 'id'
    ) %>% 
    dplyr::sample_frac(1L) %>% 
    ggplot(aes(X1, X2, color = r)) + 
        geom_point(size=0.2) + 
        theme_tufte(base_size = 10) + theme(panel.background = element_rect()) + 
        facet_grid(cluster ~ dataset) + 
        scale_color_gradient(low = 'lightgrey', breaks = seq(0, 1, .1)) + 
        labs(x = 'Scaled PC1', y = 'Scaled PC2', title = 'Initial probabilistic cluster assignments')

Evaluating initial cluster diversity

A key part of clustering in Harmony is diversity. We can evaluate the initial diversity of clustering by aggregating the number of cells from each batch assigned to each cluster. For this, we need two data structures:

  1. \(\phi\) (B rows, N columns): the one-hot encoded design matrix.

  2. \(R\) (K rows, N columns): the cluster assignment matrix.

The cross product \(R\phi^T\) gives us a matrix of the number of cells from batch b (columns) that are in cluster k (rows). Note that since cluster assignment is probabilistic, the observed counts don’t have to be integer valued. For simplicity, we round the values to their closest integers.

observed_counts <- with(harmonyObj, R %*% t(as.matrix(Phi)))
round(observed_counts)
##      [,1] [,2] [,3]
## [1,]  158    0  295
## [2,]    3  419    0
## [3,]    8  399    0
## [4,]  248    0  405
## [5,]  429    6    0

In fact, this information is already stored in the Harmony model object! The observed cluster by batch counts are stored in the \(O\) matrix. The expected counts are in the \(E\) matrix. We can check that the observed counts matrix has exactly the same values we computed above.

## observed counts
round(harmonyObj$O)
##      [,1] [,2] [,3]
## [1,]  158    0  295
## [2,]    3  419    0
## [3,]    8  399    0
## [4,]  248    0  405
## [5,]  429    6    0
## observed counts
round(harmonyObj$E)
##      [,1] [,2] [,3]
## [1,]  162  158  134
## [2,]  151  147  125
## [3,]  145  141  120
## [4,]  233  227  193
## [5,]  155  151  129

It looks like clusters 2, 4, and 5 are not very diverse, with most cells coming from a single dataset. However, clusters 1 and 3 look pretty well mixed already! Cluster 1 has 900 cells from batch \(b=1\) (half dataset) and 1574 cells from batch \(b=3\) (t293 dataset). As we move into the maximum diversity clustering, we should see the clusters getting more and more mixed!

In this benchmark, we also have some ground truth cell types. In the same way that we evaluated the cluster diversity, we can evaluate the cluster accuracy. Since we didn’t tell Harmony what the ground truth cell types are, we need to first construct a cell-type design matrix (shown below). We want these columns to be as mutually exclusive as possible. It looks like the initial clustering is fairly accurate. The only mistakes are the \(n=2\) jurkat cells clustered with the \(n=2472\) 293t cells in cluster \(k=1\) and \(n=12\) jurkat cells clustered with \(n=1952\) t293 cells in cluster \(k=3\).

phi_celltype <- onehot(meta_data$cell_type) 
observed_cell_counts <- harmonyObj$R %*% t(phi_celltype)
round(observed_cell_counts)
##      as.factor(vals)jurkat as.factor(vals)t293
## [1,]                     2                 452
## [2,]                   422                   0
## [3,]                   406                   0
## [4,]                     0                 652
## [5,]                   435                   0

Maximum-diversity soft-clustering

In the previous section, we initialized the Harmony object. At this point, we have some initial cluster assignments (\(R\), \(Y\)), scaled PC embeddings (\(Z_{cos}\)), and statistics about cluster diversity (\(O\), \(E\)). Now we’re going to do some Harmony clustering to find more diverse clusters!

We do this by calling the cluster() function defined in the Harmony package. This will perform a few rounds of clustering, defined by the parameter max_iter_kmeans. In each round, we iterate between two steps: centroid estimation and cluster assignment. We dig into both of these in more detail in the subsections below.

harmonyObj$max_iter_kmeans
## [1] 20
## we can specify how many rounds of clustering to do
harmonyObj$max_iter_kmeans <- 10
harmonyObj$cluster_cpp()
## [1] 0

Now that we’ve done some maximum diversity clustering, how have the clusters changed? Let’s first look at the observed counts matrix \(O\).

In contrast to the \(O\) matrix we started with above, this one looks much more diverse!

round(harmonyObj$O)
##      [,1] [,2] [,3]
## [1,]  176    0  324
## [2,]    7  401    0
## [3,]   13  408    0
## [4,]  230    0  376
## [5,]  420   15    0

While clusters 1 and 3 were already diverse in the initial clustering, it seems that clusters 2, 4, and 5 are now considerably more mixed as well. Let’s see how these assignments have changed in space.

new_cluster_assignment_matrix <- harmonyObj$R

t(harmonyObj$Z_cos) %>% data.frame() %>%
    cbind(meta_data) %>% 
    tibble::rowid_to_column('id') %>% 
    dplyr::inner_join(
        new_cluster_assignment_matrix %>% t() %>% data.table() %>% 
            tibble::rowid_to_column('id') %>%
            tidyr::gather(cluster, r, -id) %>% 
            dplyr::mutate(cluster = gsub('V', 'Cluster ', cluster)), 
        by = 'id'
    ) %>% 
    dplyr::sample_frac(1L) %>% 
    ggplot(aes(X1, X2, color = r)) + 
        geom_point(shape = '.') + 
        theme_tufte(base_size = 10) + theme(panel.background = element_rect()) + 
        facet_grid(cluster ~ dataset) + 
        scale_color_gradient(low = 'lightgrey', breaks = seq(0, 1, .1)) + 
        labs(x = 'Scaled PC1', y = 'Scaled PC2', title = 'New probabilistic cluster assignments')

Of course, it is equally important to make sure that our clusters do not mix up different cell types. Recall that in this benchmark, we have access to these ground truth labels.

phi_celltype <- onehot(meta_data$cell_type)
observed_cell_counts <- harmonyObj$R %*% t(phi_celltype)
round(observed_cell_counts)
##      as.factor(vals)jurkat as.factor(vals)t293
## [1,]                     2                 498
## [2,]                   408                   0
## [3,]                   421                   0
## [4,]                     0                 606
## [5,]                   435                   0

Initially, the largest error we had was in cluster 1 with 12 out of 1952 cells misclustered. So our initial error rate was at most 0.6%. Let’s take a look at the error rates in our maximum diversity clustering (shown below). Applying the same kind of error analysis, we see that we have <0.6% error across all the clusters.

It is worth noting that in the original clustering, clusters 2, 4, and 5 had 0% error. But they also had almost no diversity. These clusters have incurred a non-zero error but gained substantial diversity. This trade-off between accuracy and diversity is present in all integration settings.

round(apply(prop.table(observed_cell_counts, 1), 1, min) * 100, 3)
## [1] 0.425 0.000 0.000 0.019 0.000

Diverse cluster assignment

Now let’s re-assign cells to cluster centroids. We did this above, when we assigned cells during the Harmony initialization step. The difference is that we want to assign cells to clusters that are both nearby and will increase diversity.

In the algorithm, this assignment is defined by

\(R_{ki} \propto \exp(\frac{-(2(1 - Y^TZ))}{\sigma}) (\frac{E}{O})^\theta \phi\)

Let’s see what this looks like in code. Then we’ll break down the formula to see what it does.

with(harmonyObj, {
    distance_matrix <- 2 * (1 - t(Y) %*% Z_cos)
    distance_score <- exp(-distance_matrix / as.numeric(sigma))
    diversity_score <- sweep(E / O, 2, theta, '/') %*% as.matrix(Phi)
    ## new assignments are based on distance and diversity
    R_new <- distance_score * diversity_score  
    ## normalize R so each cell sums to 1
    R_new <- prop.table(R_new, 2)    
})

So how does the formula we used above help to create more diverse cluster assignment?

The diversity penalty is encoded in the new term: \((\frac{E}{O})^\theta \phi\). This has some familiar data structures: \(O\) for observed counts, \(E\) for expected counts, and \(\phi\) for the design matrix. \(\theta\) is a new term. \(\theta\) decides how much weight to give diversity versus distance to cluster centroids.

With \(\theta=0\), there is no penalty and each cluster gets a score of 1.

## with theta = 0
with(harmonyObj, {
    (E / O) ^ 0
})
##      [,1] [,2] [,3]
## [1,]    1    1    1
## [2,]    1    1    1
## [3,]    1    1    1
## [4,]    1    1    1
## [5,]    1    1    1

As we increase \(\theta\), let’s see what happens (shown below). Recall that in cluster \(k=1\), batches 1 and 3 were well represented. Below, note that in that cluster (\(k=1\)), the penalties for batches 1 and 3 are relatively low (0.98 and 0.47). On the other hand, batch 2 gets a penalty score of 30914. This means that cells from batches 1 and 3 will be encouraged to move into cluster \(k=1\). On the other hand, cluster \(k=2\) is replete with batch 2. The penalty for batch 2 in cluster \(k=2\) is relatively low, and noticeably smaller than the penalty score for batch 2 in cluster \(k=1\). Thus, cells from batch 2 will be discouraged from moving into cluster \(k=1\), as this cluster has a higher penalty score for cells from batch 2 compared to other clusters (such as \(k=1\)).

## with theta = 1
with(harmonyObj, {
    round((E / O) ^ 1, 2)
})
##       [,1]       [,2]         [,3]
## [1,]  1.01 2702150.70         0.46
## [2,] 20.88       0.35 353142040.63
## [3,] 11.58       0.36  12842105.70
## [4,]  0.94 1109875.94         0.48
## [5,]  0.37      10.06 938593575.72

We should always be wary of setting \(\theta\) too high, since the diversity scores can go to \(\infty\). Below, we set \(\theta\) to 1 million. We do not recommend setting \(\theta\) to 1 million!

## as theta approach infinity
with(harmonyObj, {
    round((E / O) ^ 1e6, 2)
})
##      [,1] [,2] [,3]
## [1,]  Inf  Inf    0
## [2,]  Inf    0  Inf
## [3,]  Inf    0  Inf
## [4,]    0  Inf    0
## [5,]    0  Inf  Inf

Finally, it is important to note that we cannot re-assign cells independently as we did above. Why not? As soon as we re-assign one cell, the diversity counts in the \(O\) and \(E\) matrices change. Thus, the assignment formula for all other cells is different! For this reason, we need to assign one cell at a time and update the \(O\) and \(E\) as we go. In practice, we can update some chunk of cells (e.g. 5%), update the \(O\) and \(E\) matrices, and update another chunk of cells.

Cluster centroid estimation

In the previous step, we re-assigned cells to maximize diversity within the clusters. With this new assignment, we need to update the cluster centroids. In this step, we’ll use the cell positions \(Z_{cos}\) and the cluster assignments (\(R\)) to re-position cluster centroids to be close to their assigned cells.

\(Y \leftarrow Z_{cos}R^T\)
Y_unscaled <- with(harmonyObj, Z_cos %*% t(R))

We then scale Y to make each centroid unit length:

\(Y \leftarrow \frac{Y}{\sum_{d}Y_d}\)
Y_new <- cosine_normalize(Y_unscaled, 2)

Correction

In the previous section, we performed clustering in order to identify shared groups of cells between batches. Now we make use of these groups in order to correct the data in a sensitive way. To run the correction step, we call the function moe_correct_ridge() from the Harmony package. First, let’s see what happens to the cells. In the subsections that follow, we’ll look deeper into how we got there.

harmonyObj$moe_correct_ridge_cpp()
do_scatter(cosine_normalize(t(harmonyObj$Z_orig), 1), meta_data, 'dataset', no_guides = TRUE, do_labels = TRUE) + 
    labs(title = 'Z_cos before MoE', x = 'PC1', y = 'PC2') +
do_scatter(t(harmonyObj$Z_cos), meta_data, 'dataset', no_guides = TRUE, do_labels = TRUE) + 
    labs(title = 'Z_cos after MoE', x = 'PC1', y = 'PC2')

We can see the the jurkat cells are starting to come together on the right (purple and green). There is also more local mixing of the 293T cells on the left (yellow and green). What happened to actually get them there?

For each cell, we estimate how much its batch identity contributes to its PCA scores. We then subtract this contribution from that cell’s PCA scores. That’s it!

Very importantly, this correction factor is not in the unit scaled space (i.e. \(Z_{cos}\))! The data in \(Z_{cos}\) have been projected onto a hypersphere. This makes the cells easier to cluster but the space is no longer closed under linear transformations! In other words, if we push a cell over a bit by adding 10 to PC1, that cell is no longer on the hypersphere.

To query the Harmony model object, we need to introduce another variable: \(Z_{corr}\). \(Z_{corr}\) contains the cells’ PCA embeddings post correction. However, we never scale cells in \(Z_{corr}\) to have unit length. After we compute \(Z_{corr}\), we immediately update \(Z_{cos}\) as the unit scaled version of \(Z_{corr}\). The plot below shows all three of Harmony’s data structures that contain PCA embeddings. To summarize:

do_scatter(t(harmonyObj$Z_orig), meta_data, 'dataset', no_guides = TRUE, do_labels = TRUE) + 
    labs(title = 'Z_orig', subtitle = 'Original PCA embeddings', x = 'PC1', y = 'PC2') +
do_scatter(t(harmonyObj$Z_corr), meta_data, 'dataset', no_guides = TRUE, do_labels = TRUE) + 
    labs(title = 'Z_corr', subtitle = '= Z_orig - correction_factors', x = 'PC1', y = 'PC2') +
do_scatter(t(harmonyObj$Z_cos), meta_data, 'dataset', no_guides = TRUE, do_labels = TRUE) + 
    labs(title = 'Z_cos', subtitle = '= Unit_scaled(Z_corr)', x = 'Scaled PC1', y = 'Scaled PC2') +
NULL

Let’s take a look a closer look at these cell specific correction factors. For exposition, let’s focus on PC1 and compare each cell’s position before (from \(Z_{orig}\)) and after (from \(Z_{corr}\)) correction.

The plots below show the PC1 value before (x-axis) and after (y-axis) correction for each cell. The black line is drawn at \(y=x\) to represent the level curve of no change.

plt <- data.table(PC1_After = harmonyObj$Z_corr[1, ], PC1_Before = harmonyObj$Z_orig[1, ]) %>% 
    cbind(meta_data) %>% 
    dplyr::sample_frac(1L) %>% 
    ggplot(aes(PC1_Before, PC1_After)) + 
        geom_abline(slope = 1, intercept = 0) + 
        theme_tufte(base_size = 10) + geom_rangeframe() + 
        scale_color_tableau() + 
        guides(color = guide_legend(override.aes = list(stroke = 1, alpha = 1, shape = 16, size = 4))) + 
        NULL

plt + geom_point(shape = '.', aes(color = dataset)) + 
        labs(x = 'PC1 before correction', y = 'PC1 after correction', 
             title = 'PC1 correction for each cell', subtitle = 'Colored by Dataset') + 
plt + geom_point(shape = '.', aes(color = cell_type)) + 
        labs(x = 'PC1 before correction', y = 'PC1 after correction', 
             title = 'PC1 correction for each cell', subtitle = 'Colored by Cell Type') + 
NULL

We can see a few interesting things from these plots.

Mixture of Experts model

The theory behind this algorithm is based on the Mixture of Experts model. This is a natural extension of linear modeling, in which each cluster is deemed an expert and is assigned its own linear model.

We model each PC coordinate with a combination of linear factors.

\(Z_{d} = \sum_k \beta_{0,k} + \beta_{1,k} \mathbb{1}_{(dataset = jurkat)} + \beta_{2,k} \mathbb{1}_{(dataset = half)} + \beta_{3,k} \mathbb{1}_{(dataset = 293T)}\)

In the model above, each cluster gets 4 \(\beta\) terms: \(\beta_{0,k}\) is the intercept term. This term is independent of which dataset a cell comes from. Therefore, it represents the contribution of cell type or cell state to the PC score. The other three \(\beta\) terms are accompanied by an indicator variable. This means that a cell from dataset half will have \(\mathbb{1}_{(dataset = half)}\) equal to 1 and the rest 0.

Following this cell from dataset half half, we can write rewrite the MoE equation above as

\(Z_{di} = \sum_k \beta_{0,k} + \beta_{2,k} \mathbb{1}_{(dataset = half)}\)

Estimate MoE model parameters

We estimate the matrix of linear regression terms using the formula described in the manuscript:

\(W_k \leftarrow (\phi^* diag(R_k) \phi^{*T} + \lambda I)^{-1} \phi^* diag(R_k)Z_{orig}^T\)

The matrix above contains linear regression terms for the the intercept \(W_k[0] = \beta_{0,k}\) and the batch terms:

\(W_k[1] = \beta_{1, k} \mbox{ (for dataset half)}\)

\(W_k[2] = \beta_{2, k} \mbox{ (for dataset jurkat)}\)

\(W_k[3] = \beta_{3, k} \mbox{ (for dataset 293T)}\)

W <- list()
## Convert sparse data structures to dense matrix
Phi.moe <- as.matrix(harmonyObj$Phi_moe)
lambda <- diag(c(harmonyObj$lambda))
## Get beta coeeficients for all the clusters
for (k in 1:harmonyObj$K) {
    W[[k]] <- solve(Phi.moe %*% diag(harmonyObj$R[k, ]) %*% t(Phi.moe) + lambda) %*% (Phi.moe %*% diag(harmonyObj$R[k, ])) %*% t(harmonyObj$Z_orig)
}

Let’s take a look at how these regression terms relate to the data. Recall that the mixture of experts model is trying to estimate the contribution of intercept and batch to cell’s positions in space. So first we’ll take a look at the positions of each batch and each cluster in the original PCA embeddings. The color below represents soft cluster membership learned using the maximum diversity clustering above.

cluster_assignment_matrix <- harmonyObj$R

t(harmonyObj$Z_orig) %>% data.frame() %>%
    cbind(meta_data) %>% 
    tibble::rowid_to_column('id') %>% 
    dplyr::inner_join(
        cluster_assignment_matrix %>% t() %>% data.table() %>% 
            tibble::rowid_to_column('id') %>%
            tidyr::gather(cluster, r, -id) %>% 
            dplyr::mutate(cluster = gsub('V', 'Cluster ', cluster)), 
        by = 'id'
    ) %>% 
    dplyr::sample_frac(1L) %>% 
    ggplot(aes(X1, X2, color = r)) + 
        geom_point(shape = 0.2) + 
        theme_tufte(base_size = 10) + theme(panel.background = element_rect()) + 
        facet_grid(cluster ~ dataset) + 
        scale_color_gradient(low = 'grey', breaks = seq(0, 1, .2)) + 
        labs(x = 'PC1', y = 'PC2', title = 'Cluster assigned in original PCA space (Z_orig)')

Now let’s draw the \(\beta\) terms into this space. For each cluster, we expect the sum of the intercept plus the batch terms to land squarely in the center of each batch:cluster. The arrows below represent the intercept (in black) and batch (colored) offsets.

plt_list <- lapply(1:harmonyObj$K, function(k) {
    plt_df <- W[[k]] %>% data.frame() %>% 
        dplyr::select(X1, X2)
    ## Append n
    plt_df <- plt_df %>% 
        cbind(
            data.frame(t(matrix(unlist(c(c(0, 0), rep(plt_df[1, ], 3))), nrow = 2))) %>% 
                dplyr::rename(x0 = X1, y0 = X2) 
        ) %>%
        cbind(type = c('intercept', unique(meta_data$dataset)))
    plt <- plt_df %>% 
        ggplot() + 
            geom_point(aes(X1, X2),
                       data = t(harmonyObj$Z_orig) %>% data.frame(),
                       size = 0.5,
                       color = 'grey'
            ) + 
            geom_segment(aes(x = x0, y = y0, xend = X1 + x0, yend = X2 + y0, color = type), linewidth=1) + 
            scale_color_manual(values = c('intercept' = 'black', colors_use)) + 
            theme_tufte(base_size = 10) + theme(panel.background = element_rect()) + 
            labs(x = 'PC 1', y = 'PC 2', title = sprintf('Cluster %d', k))
    plt <- plt + guides(color = guide_legend(override.aes = list(stroke = 1, alpha = 1, shape = 16)))    
    # if (k == harmonyObj$K) {
    # } else {
    #     plt <- plt + guides(color = FALSE)
    # }
    plt
})
Reduce(`+`, plt_list) + 
  patchwork::plot_annotation(title = 'Mixture of experts beta terms before correction (Z_orig)') + 
  plot_layout(ncol = 2)

After correction, we remove the batch specific terms (colored arrows above). We can see the result in the corrected linear space (\(Z_{corr}\)). Notice that now, the cells are centered around the tips of the black arrows, which represent the intercept term. This is because we’ve removed the effect of the batch terms (colored arrows).

plt_list <- lapply(1:harmonyObj$K, function(k) {
    plt_df <- W[[k]] %>% data.frame() %>% 
        dplyr::select(X1, X2)

    plt_df <- plt_df %>% 
        cbind(
            data.frame(t(matrix(unlist(c(c(0, 0), rep(plt_df[1, ], 3))), nrow = 2))) %>% 
                dplyr::rename(x0 = X1, y0 = X2) 
        ) %>%
        cbind(type = c('intercept', unique(meta_data$dataset))) 

    plt <- plt_df %>% 
        ggplot() + 
            geom_point(aes(X1, X2),
                data = t(harmonyObj$Z_corr) %>% data.frame(),
                shape = '.', 
                color = 'grey'
            ) + 
            geom_segment(aes(x = x0, y = y0, xend = X1 + x0, yend = X2 + y0, color = type), linewidth=1) + 
            scale_color_manual(values = c('intercept' = 'black', colors_use)) + 
            theme_tufte(base_size = 10) + theme(panel.background = element_rect()) + 
            labs(x = 'PC 1', y = 'PC 2', title = sprintf('Cluster %d', k))
    plt <- plt + guides(color = guide_legend(override.aes = list(stroke = 1, alpha = 1, shape = 16)))
    plt
})
Reduce(`+`, plt_list) + 
  patchwork::plot_annotation(title = 'Mixture of experts beta terms after correction (Z_corr)') + 
  plot_layout(ncol = 2)