flat clustering

“Do my samples fall into definable clusters?”

kmeans

One of the questions we’ve been asking is “which of my samples are most closely related?”. We’ve been answering that question using clustering. However, now that we know how to run principal components analyses, we can use another approach. This alternative approach is called k-means, and can help us decide how to assign our data into clusters. It is generally desirable to have a small number of clusters, however, this must be balanced by not having the variance within each cluster be too big. To strike this balance point, the elbow method is used. For it, we must first determine the maximum within-group variance at each possible number of clusters. An illustration of this is shown in A below:

One we know within-group variances, we find the “elbow” point - the point with minimum angle theta - thus picking the outcome with a good balance of cluster number and within-cluster variance (illustrated above in B and C.)

Let’s try k-means using runMatrixAnalysis. For this example, let’s run it on the PCA projection of the alaska lakes data set. We can set analysis = "kmeans". When we do this, an application will load that will show us the threshold value for the number of clusters we want. We set the number of clusters and then close the app. In the context of markdown document, simply provide the number of clusters to the parameters argument:

alaska_lake_data_pca <- runMatrixAnalysis(
    data = alaska_lake_data,
    analysis = c("pca"),
    column_w_names_of_multiple_analytes = "element",
    column_w_values_for_multiple_analytes = "mg_per_L",
    columns_w_values_for_single_analyte = c("water_temp", "pH"),
    columns_w_additional_analyte_info = "element_type",
    columns_w_sample_ID_info = c("lake", "park")
)

alaska_lake_data_pca_clusters <- runMatrixAnalysis(
    data = alaska_lake_data_pca,
    analysis = c("kmeans"),
    parameters = c(5),
    column_w_names_of_multiple_analytes = NULL,
    column_w_values_for_multiple_analytes = NULL,
    columns_w_values_for_single_analyte = c("Dim.1", "Dim.2"),
    columns_w_sample_ID_info = "sample_unique_ID"
)

alaska_lake_data_pca_clusters <- left_join(alaska_lake_data_pca_clusters, alaska_lake_data_pca) 

We can plot the results and color them according to the group that kmeans suggested. We can also highlight groups using geom_mark_ellipse. Note that it is recommended to specify both fill and label for geom_mark_ellipse:

alaska_lake_data_pca_clusters$cluster <- factor(alaska_lake_data_pca_clusters$cluster)
ggplot() +
  geom_point(
    data = alaska_lake_data_pca_clusters,
    aes(x = Dim.1, y = Dim.2, fill = cluster), shape = 21, size = 5, alpha = 0.6
  ) +
  geom_mark_ellipse(
    data = alaska_lake_data_pca_clusters,
    aes(x = Dim.1, y = Dim.2, label = cluster, fill = cluster), alpha = 0.2
  ) +
  theme_classic() +
  coord_cartesian(xlim = c(-7,12), ylim = c(-4,5)) +
  scale_fill_manual(values = discrete_palette) 

dbscan

There is another method to define clusters that we call dbscan. In this method, not all points are necessarily assigned to a cluster, and we define clusters according to a set of parameters, instead of simply defining the number of clusteres, as in kmeans. In interactive mode, runMatrixAnalysis() will again load an interactive means of selecting parameters for defining dbscan clusters (“k”, and “threshold”). In the context of markdown document, simply provide “k” and “threshold” to the parameters argument:

alaska_lake_data_pca <- runMatrixAnalysis(
    data = alaska_lake_data,
    analysis = c("pca"),
    column_w_names_of_multiple_analytes = "element",
    column_w_values_for_multiple_analytes = "mg_per_L",
    columns_w_values_for_single_analyte = c("water_temp", "pH"),
    columns_w_additional_analyte_info = "element_type",
    columns_w_sample_ID_info = c("lake", "park")
) 

alaska_lake_data_pca_clusters <- runMatrixAnalysis(
    data = alaska_lake_data_pca,
    analysis = c("dbscan"),
    parameters = c(4, 0.45),
    column_w_names_of_multiple_analytes = NULL,
    column_w_values_for_multiple_analytes = NULL,
    columns_w_values_for_single_analyte = c("Dim.1", "Dim.2"),
    columns_w_sample_ID_info = "sample_unique_ID"
)

alaska_lake_data_pca_clusters <- left_join(alaska_lake_data_pca_clusters, alaska_lake_data_pca)

We can make the plot in the same way, but please note that to get geom_mark_ellipse to omit the ellipse for NAs you need to feed it data without NAs:

alaska_lake_data_pca_clusters$cluster <- factor(alaska_lake_data_pca_clusters$cluster)
ggplot() +
  geom_point(
    data = alaska_lake_data_pca_clusters,
    aes(x = Dim.1, y = Dim.2, fill = cluster), shape = 21, size = 5, alpha = 0.6
  ) +
  geom_mark_ellipse(
    data = drop_na(alaska_lake_data_pca_clusters),
    aes(x = Dim.1, y = Dim.2, label = cluster, fill = cluster), alpha = 0.2
  ) +
  theme_classic() +
  coord_cartesian(xlim = c(-7,12), ylim = c(-4,5)) +
  scale_fill_manual(values = discrete_palette) 

summarize by cluster

One more important point: when using kmeans or dbscan, we can use the clusters as groupings for summary statistics. For example, suppose we want to see the differences in abundances of certain chemicals among the clusters:

alaska_lake_data_pca <- runMatrixAnalysis(
  data = alaska_lake_data,
  analysis = c("pca"),
  column_w_names_of_multiple_analytes = "element",
  column_w_values_for_multiple_analytes = "mg_per_L",
  columns_w_values_for_single_analyte = c("water_temp", "pH"),
  columns_w_additional_analyte_info = "element_type",
  columns_w_sample_ID_info = c("lake", "park")
)

alaska_lake_data_pca_clusters <- runMatrixAnalysis(
  data = alaska_lake_data_pca,
  analysis = c("dbscan"),
  parameters = c(4, 0.45),
  column_w_names_of_multiple_analytes = NULL,
  column_w_values_for_multiple_analytes = NULL,
  columns_w_values_for_single_analyte = c("Dim.1", "Dim.2"),
  columns_w_sample_ID_info = "sample_unique_ID",
  columns_w_additional_analyte_info = colnames(alaska_lake_data_pca)[6:18]
) 

alaska_lake_data_pca_clusters <- left_join(alaska_lake_data_pca_clusters, alaska_lake_data_pca)

alaska_lake_data_pca_clusters %>%
  select(cluster, S, Ca) %>%
  pivot_longer(cols = c(2,3), names_to = "analyte", values_to = "mg_per_L") %>%
  drop_na() %>%
  group_by(cluster, analyte) -> alaska_lake_data_pca_clusters_clean

plot_2 <- ggplot() + 
  geom_col(
    data = summarize(
      alaska_lake_data_pca_clusters_clean,
      mean = mean(mg_per_L), sd = sd(mg_per_L)
    ),
    aes(x = cluster, y = mean, fill = cluster),
    color = "black", alpha = 0.6
  ) +
  geom_errorbar(
    data = summarize(
      alaska_lake_data_pca_clusters_clean,
      mean = mean(mg_per_L), sd = sd(mg_per_L)
    ),
    aes(x = cluster, ymin = mean-sd, ymax = mean+sd, fill = cluster),
    color = "black", alpha = 0.6, width = 0.5, size = 1
  ) +
  facet_grid(.~analyte) + theme_bw() +
  geom_jitter(
    data = alaska_lake_data_pca_clusters_clean,
    aes(x = cluster, y = mg_per_L, fill = cluster), width = 0.05,
    shape = 21
  ) +
  scale_fill_manual(values = discrete_palette) 

plot_1<- ggplot() +
  geom_point(
    data = alaska_lake_data_pca_clusters,
    aes(x = Dim.1, y = Dim.2, fill = cluster), shape = 21, size = 5, alpha = 0.6
  ) +
  geom_mark_ellipse(
    data = drop_na(alaska_lake_data_pca_clusters),
    aes(x = Dim.1, y = Dim.2, label = cluster, fill = cluster), alpha = 0.2
  ) +
  theme_classic() + coord_cartesian(xlim = c(-7,12), ylim = c(-4,5)) +
  scale_fill_manual(values = discrete_palette)

plot_1 + plot_2

further reading

http://www.sthda.com/english/wiki/wiki.php?id_contents=7940

https://ryanwingate.com/intro-to-machine-learning/unsupervised/hierarchical-and-density-based-clustering/

https://ryanwingate.com/intro-to-machine-learning/unsupervised/hierarchical-and-density-based-clustering/hierarchical-4.png

https://www.geeksforgeeks.org/dbscan-clustering-in-r-programming/

exercises

For this set of exercises, please use the dataset hawaii_aquifers or tequila_chemistry, available after you run the source() command. Do the following:

1. Run a PCA analysis on the data set and plot the results.

2. Create an ordination plot and identify one analyte that varies with Dim.1 and one analyte that varies with Dim.2 (these are your “variables of interest”).

3. Run kmeans clustering on your PCA output. Create a set of clusters that seems to appropriately subdivide the data set.

4. Use the clusters defined by kmeans as groupings on which to run summary statistics for your two variables of interest.

5. Create a plot with three subpanels that shows: (i) the PCA analysis (colored by kmeans clusters), (ii) the ordination analysis, and (iii) the summary statistics for your two variables of interest within the kmeans groups. Please note that subpanel plots can be created by sending ggplots to their own objects and then adding those objects together. Please see the subsection in the data visualization chapter on subplots.

6. Run dbscan clustering on your PCA output. Create a set of clusters that seems to appropriately subdivide the data set.

7. Use the clusters defined by dbscan as groupings on which to run summary statistics for your two variables of interest.

8. Create a plot with three subpanels that shows: (i) the PCA analysis (colored by dbscan clusters), (ii) the ordination analysis, and (iii) the summary statistics for your two variables of interest within the dbscan groups.