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Visualize differential expression markers

Source: R/markersplot.R
MarkersPlot.Rd

Visualize differential expression (DE) results — typically the output of Seurat::FindMarkers() or Seurat::FindAllMarkers() — across a variety of plot types. MarkersPlot() bridges the gap between DE testing and visualization by providing a unified interface for both summary-level DE visualizations (volcano, jitter, heatmap, and dot plots of fold changes and significance) and expression-level visualizations (violin, box, bar, ridge, heatmap, and dot plots of actual expression values from a Seurat object).

The function handles two broad categories of plots:

  • DE summary plots (no object required): visualize the DE statistics themselves — log2 fold change, percentage difference, p-values, and adjusted p-values — across groups or comparisons.

    • "volcano" / "volcano_log2fc" — Volcano plot with log2 fold change on the x-axis and \(-log_{10}(p)\) on the y-axis. Genes passing the cutoff are highlighted and top genes are labeled. Ideal for overview of effect size vs. significance.

    • "volcano_pct" — Volcano plot with percentage-point difference (pct.1 - pct.2) on the x-axis. Useful when the biological question is about detection rate rather than expression magnitude.

    • "jitter" / "jitter_log2fc" — Jitter plot of log2 fold changes across groups (defined by subset_by). Dot size encodes \(-log_{10}(p)\). Reveals distribution of effect sizes per cluster or condition.

    • "jitter_pct" — Jitter plot of percentage-point differences across groups.

    • "heatmap_log2fc" — Heatmap of log2 fold changes (genes × groups). Cells can be marked for significance via cutoff and sig_mark.

    • "heatmap_pct" — Heatmap of percentage-point differences (genes × groups). Same significance-marking support.

    • "dot_log2fc" — Dot plot of log2 fold changes (genes × groups). Dot size encodes \(-log_{10}(p)\).

    • "dot_pct" — Dot plot of percentage-point differences (genes × groups). Dot size encodes \(-log_{10}(p)\).

  • Expression plots (object required): visualize the actual expression values of the selected marker genes in the context of the original Seurat object. These are useful for validating DE results by inspecting the underlying expression distributions.

    • "heatmap" — Expression heatmap of selected marker genes.

    • "violin" — Violin plots of expression per gene.

    • "box" — Box plots of expression per gene.

    • "bar" — Bar plots of mean expression per gene.

    • "ridge" — Ridge plots of expression distribution per gene.

    • "dot" — Dot plot of expression (fraction expressing × mean expression) per gene.

Usage

MarkersPlot(
  markers,
  object = NULL,
  plot_type = c("volcano", "volcano_log2fc", "volcano_pct", "jitter", "jitter_log2fc",
    "jitter_pct", "heatmap_log2fc", "heatmap_pct", "dot_log2fc", "dot_pct", "heatmap",
    "violin", "box", "bar", "ridge", "dot"),
  subset_by = NULL,
  subset_as_facet = FALSE,
  comparison_by = NULL,
  p_adjust = TRUE,
  cutoff = NULL,
  show_labels = FALSE,
  sig_mark = "*",
  order_by = NULL,
  select = ifelse(plot_type %in% c("volcano", "volcano_log2fc", "volcano_pct",
    "jitter", "jitter_log2fc", "jitter_pct"), 5, 10),
  ...
)

Arguments

markers

A data frame of differential expression results, typically the output of Seurat::FindMarkers() or Seurat::FindAllMarkers(). Must contain columns "gene" (or gene symbols as rownames), "p_val", and "avg_log2FC". For percentage-based plots (volcano_pct, jitter_pct, heatmap_pct, dot_pct), columns "pct.1" and "pct.2" are also required.

object

A Seurat object. Required for expression-based plot types: "heatmap", "violin", "box", "bar", "ridge", and "dot". Not used for DE summary plot types. Default: NULL.

plot_type

The type of plot to generate. One of "volcano", "volcano_log2fc", "volcano_pct", "jitter", "jitter_log2fc", "jitter_pct", "heatmap_log2fc", "heatmap_pct", "dot_log2fc", "dot_pct", "heatmap", "violin", "box", "bar", "ridge", or "dot". See Description for details on each type.

subset_by

A column name in markers indicating the grouping from which each marker was identified (e.g., the cluster column from FindAllMarkers()). Supports the "marker_column:metadata_column" syntax for linking to Seurat object metadata (see Metadata column mapping section). For jitter and DE heatmap/dot plot types, subset_by is required and defines the x-axis or column groups. For expression plot types, subset_by controls faceting or splitting. Default: NULL.

subset_as_facet

Logical. If TRUE, facet the plot by subset_by groups instead of splitting into separate plots. Most useful for expression plot types. For volcano plots, controls whether faceting or split_by dispatch is used. Default: FALSE.

comparison_by

A column name in markers indicating the comparison (e.g., "g1:g2" for a pairwise comparison, or a single group name for one-vs-rest). Required for expression-based plot types ("heatmap", "violin", "box", "bar", "ridge", "dot"). Supports the "marker_column:metadata_column" syntax (see Metadata column mapping section). If the comparison values contain a colon (e.g., "G2M:G1"), the two groups on either side of the colon are used to subset the object. If only a single group is present, a one-vs-other comparison is assumed. Default: NULL.

p_adjust

Logical. If TRUE (default), use adjusted p-value (p_val_adj column) for significance calculations and y-axis transformations. If FALSE, use raw p-value (p_val column).

cutoff

Numeric. The p-value (or adjusted p-value, depending on p_adjust) threshold for labeling significance. For volcano plots, sets y_cutoff. For heatmap-based DE plots (heatmap_log2fc, heatmap_pct), controls which cells receive significance marks. Default: NULL (no cutoff; defaults to 0.05 for volcano plots).

show_labels

Logical. For heatmap_log2fc and heatmap_pct plot types only. If TRUE, display numeric values in heatmap cells. When combined with cutoff, both values and significance marks are shown. Default: FALSE.

sig_mark

Character. The symbol or compound mark used to annotate statistically significant cells in heatmap_log2fc and heatmap_pct plots. Must be a valid ComplexHeatmap mark: single characters ("-", "|", "+", "/", "\\", "x", "o") or compound marks ("[*]", "<*>", "(*)", "{*}"). Note that "*" conflicts with show_labels = TRUE because both use the label layer — use a compound mark instead. Default: "*".

order_by

A string expression to order markers by (evaluated with dplyr::arrange()). Can reference columns in markers as well as columns from the object metadata (when object is provided and subset_by enables merging). Only the first value of merged metadata columns is used. Example: "desc(avg_log2FC)". The ordering affects which markers are selected when select is numeric. Default: NULL.

select

How to select markers for labeling or display. See Marker selection and filtering section for full details.

  • Numeric: Top N markers per subset_by group (default: 5 for volcano/jitter types, 10 for others).

  • Character expression: Filter condition for dplyr::filter().

  • Character vector: Multiple filter expressions; those containing the subset_by column name filter the overall data, others filter within remaining data.

...

Additional arguments passed to the underlying plotting function, depending on plot_type:

For volcano, volcano_log2fc, volcano_pct

Passed to plotthis::VolcanoPlot(). Common arguments: x_cutoff, x_cutoff_name, label_by, color_by, nlabel, flip_negative.

For jitter, jitter_log2fc, jitter_pct

Passed to plotthis::JitterPlot(). Common arguments: add_hline, shape, size_by, nlabel.

For heatmap_log2fc, heatmap_pct, dot_log2fc, dot_pct

Passed to plotthis::Heatmap(). Common arguments: show_row_names, show_column_names, values_fill, palette, cluster_rows, cluster_columns, add_reticle.

For heatmap, violin, box, bar, ridge, dot

Passed to FeatureStatPlot. Common arguments: name, palette, ncol, nrow, stack, columns_split_by.

Value

A ggplot object (from plotthis::VolcanoPlot() or plotthis::JitterPlot()), a Heatmap object (from plotthis::Heatmap()), or a ggplot/patchwork object (from FeatureStatPlot). When split_by or faceting generates multiple plots and combine = TRUE (default), a combined patchwork object is returned; when combine = FALSE, a list of individual plots is returned.

Note

  • subset_by is required for jitter plots ("jitter", "jitter_log2fc", "jitter_pct") and DE heatmap/dot plots ("heatmap_log2fc", "heatmap_pct", "dot_log2fc", "dot_pct"). Without it, there is no grouping axis.

  • comparison_by is required for expression-based plot types ("heatmap", "violin", "box", "bar", "ridge", "dot") — it tells the function which comparison groups to extract from the object.

  • When object is provided and subset_by maps to a metadata column, the markers data frame is left-joined with the object metadata. Only the first row per group is kept for non-key columns, which is sufficient for most annotation purposes but can cause issues if per-cell metadata is needed.

  • For expression-based heatmap and dot plots, when subset_by_2 is available (i.e., the metadata column is mapped), genes are automatically grouped by subset_by via columns_split_by, and group_by is set to NULL.

  • The function calculates \(-log_{10}(p)\) (or \(-log_{10}(p_{adj})\)) internally and stores it in a temporary neg_log10_p column. This column is available for use in order_by.

  • When the comparison involves only a single group (one-vs-rest), cells not in the comparison group are labeled "Other" in the object metadata.

Metadata column mapping

Both subset_by and comparison_by support a special "marker_column:metadata_column" syntax for linking columns in the markers data frame to columns in the Seurat object's metadata.

  • The part before the colon refers to a column in markers.

  • The part after the colon refers to a column in object@meta.data.

  • If only one name is provided (no colon), it is used for both the markers column and the metadata column (if a matching metadata column exists).

  • Example: subset_by = "cluster:RNA_snn_res.0.8" maps the cluster column in the DE results to the RNA_snn_res.0.8 column in the Seurat metadata.

When the markers data frame and object metadata are merged via subset_by, only the first value of each non-key column within each group is retained — this is by design to avoid duplication.

Marker selection and filtering

The select argument supports three modes:

  • Numeric — Select the top N markers (ordered by order_by) within each group defined by subset_by. For volcano and jitter plots, all markers are plotted but only the top N per group are labeled. For other plot types, only the selected markers are shown.

  • Single expression — A filter expression string evaluated by dplyr::filter(). For example, "p_val_adj < 0.05 & avg_log2FC > 1". All markers matching the condition are retained across all groups.

  • Multiple expressions (character vector) — Each element is evaluated independently. Expressions that mention the subset_by column filter the overall data (removing groups); other expressions filter within the remaining data. For example, select = c("cluster %in% c('0', '1')", "p_val_adj < 0.05") first restricts to clusters 0 and 1, then keeps only significant markers. A numeric string like "5" among the expressions is treated as a top-N selection.

Default select: 5 for volcano and jitter plot types, 10 for all other plot types.

Significance marking in heatmaps

For heatmap_log2fc and heatmap_pct, the cutoff and sig_mark arguments control how statistically significant cells are annotated in the heatmap:

  • When cutoff is set and show_labels = FALSE, cells with p-value (or adjusted p-value) below the cutoff are marked with sig_mark using ComplexHeatmap's mark system. Valid sig_mark values include "-", "|", "+", "/", "\\", "x", "o", and compound marks like "[*]", "<*>", "(*)", "{*}".

  • When cutoff is set and show_labels = TRUE, both numeric values and significance marks are displayed (cell_type = "label+mark"). Note that sig_mark = "*" does not work with show_labels = TRUE — use compound marks instead.

  • When cutoff = NULL and show_labels = TRUE, all cells are labeled with their numeric values.

See also

plotthis::VolcanoPlot(), plotthis::JitterPlot(), plotthis::Heatmap(), FeatureStatPlot, Seurat::FindMarkers(), Seurat::FindAllMarkers()

Examples

# \donttest{
data(pancreas_sub)
markers <- Seurat::FindMarkers(pancreas_sub,
 group.by = "Phase", ident.1 = "G2M", ident.2 = "G1")
#> For a (much!) faster implementation of the Wilcoxon Rank Sum Test,
#> (default method for FindMarkers) please install the presto package
#> --------------------------------------------
#> install.packages('devtools')
#> devtools::install_github('immunogenomics/presto')
#> --------------------------------------------
#> After installation of presto, Seurat will automatically use the more 
#> efficient implementation (no further action necessary).
#> This message will be shown once per session
allmarkers <- Seurat::FindAllMarkers(pancreas_sub)  # seurat_clusters
#> Calculating cluster 0
#> Calculating cluster 1
#> Calculating cluster 2
#> Calculating cluster 3
#> Calculating cluster 4
#> Calculating cluster 5
#> Calculating cluster 6

MarkersPlot(markers)
#> Warning: no non-missing arguments to min; returning Inf
#> Warning: no non-missing arguments to max; returning -Inf

MarkersPlot(markers, x_cutoff = 2)
#> Warning: no non-missing arguments to min; returning Inf
#> Warning: no non-missing arguments to max; returning -Inf

MarkersPlot(allmarkers,
    subset_by = "cluster", ncol = 2, subset_as_facet = TRUE)

MarkersPlot(markers, plot_type = "volcano_pct", flip_negative = TRUE)
#> Warning: no non-missing arguments to min; returning Inf
#> Warning: no non-missing arguments to max; returning -Inf


MarkersPlot(allmarkers, plot_type = "jitter", subset_by = "cluster")

MarkersPlot(allmarkers, plot_type = "jitter_pct",
    subset_by = "cluster", add_hline = 0, shape = 16)


MarkersPlot(allmarkers, plot_type = "heatmap_log2fc", subset_by = "cluster")

MarkersPlot(allmarkers, plot_type = "heatmap_log2fc", subset_by = "cluster",
    label = scales::label_number(accuracy = 0.01),
    cutoff = 0.05, show_labels = TRUE, sig_mark = '{}')

MarkersPlot(allmarkers, plot_type = "heatmap_pct", subset_by = "cluster",
    cutoff = 0.05)


MarkersPlot(allmarkers, plot_type = "dot_log2fc", subset_by = "cluster",
    add_reticle = TRUE)


MarkersPlot(allmarkers, object = pancreas_sub, plot_type = "heatmap",
   columns_split_by = "CellType",
   comparison_by = "cluster:seurat_clusters")
#> Warning: Layer counts isn't present in the assay object; returning NULL


# Suppose we did a DE between g1 and g2 in each cluster
allmarkers$comparison <- "g1:g2"
MarkersPlot(allmarkers, object = pancreas_sub, plot_type = "heatmap",
   comparison_by = "Phase", subset_by = "cluster:seurat_clusters")
#> Warning: Layer counts isn't present in the assay object; returning NULL

MarkersPlot(allmarkers, object = pancreas_sub, plot_type = "dot",
   comparison_by = "Phase", subset_by = "cluster:seurat_clusters")
#> Warning: Layer counts isn't present in the assay object; returning NULL


MarkersPlot(allmarkers, object = pancreas_sub, plot_type = "violin", select = 3,
   comparison_by = "Phase", subset_by = "cluster:seurat_clusters")
#> Warning: Layer counts isn't present in the assay object; returning NULL


# select markers with a custom condition, e.g.,
# significant markers in cluster 0, 1, and 2 with pct.2 - pct.1 > 0.6
# Note that other clusters are still included in the plot
MarkersPlot(allmarkers, object = pancreas_sub, plot_type = "violin", subset_by = "cluster",
   select = c('cluster %in% c("1", "2", "0") & pct.2 - pct.1 > 0.6'),
   comparison_by = "cluster:seurat_clusters",
   cutoff = 0.05)
#> Warning: [MarkersPlot] `subset_by` 'cluster' is ignored, since it is not found in the object's metadata. Set `subset_by` to 'cluster:<object_metadata_column>' to make it work.
#> Warning: Layer counts isn't present in the assay object; returning NULL


# To exclude other clusters, you can separate the filtering conditions into
# multiple expressions
MarkersPlot(allmarkers, object = pancreas_sub, plot_type = "violin", subset_by = "cluster",
   select = c('cluster %in% c("1", "2", "0")', 'pct.2 - pct.1 > 0.6'),
   comparison_by = "cluster:seurat_clusters",
   cutoff = 0.05)
#> Warning: [MarkersPlot] `subset_by` 'cluster' is ignored, since it is not found in the object's metadata. Set `subset_by` to 'cluster:<object_metadata_column>' to make it work.
#> Warning: Layer counts isn't present in the assay object; returning NULL


MarkersPlot(allmarkers, object = pancreas_sub, plot_type = "box", select = 3,
   comparison_by = "Phase", subset_by = "cluster:seurat_clusters")
#> Warning: Layer counts isn't present in the assay object; returning NULL


MarkersPlot(allmarkers, object = pancreas_sub, plot_type = "ridge", select = 2,
   comparison_by = "Phase", subset_by = "cluster:seurat_clusters",
   ncol = 2)
#> Warning: Layer counts isn't present in the assay object; returning NULL
#> Picking joint bandwidth of 0.222
#> Picking joint bandwidth of 0.155
#> Picking joint bandwidth of 0.32
#> Picking joint bandwidth of 0.331
#> Picking joint bandwidth of 0.118
#> Picking joint bandwidth of 0.32
#> Picking joint bandwidth of 0.322
#> Picking joint bandwidth of 0.105
#> Picking joint bandwidth of 0.265
#> Picking joint bandwidth of 0.232
#> Picking joint bandwidth of 0.326
#> Picking joint bandwidth of 0.31
#> Picking joint bandwidth of 0.094
#> Picking joint bandwidth of 0.306
#> Picking joint bandwidth of 0.0355
#> Picking joint bandwidth of 0.116
#> Picking joint bandwidth of 0.33
#> Picking joint bandwidth of 0.278
#> Picking joint bandwidth of 0.186
#> Picking joint bandwidth of 0.102
#> Picking joint bandwidth of 0.251
#> Picking joint bandwidth of 0.261
#> Picking joint bandwidth of 0.0735
#> Picking joint bandwidth of 0.0714
#> Picking joint bandwidth of 0.202
#> Picking joint bandwidth of 0.267
#> Picking joint bandwidth of 0.256
#> Picking joint bandwidth of 0.0755
#> Picking joint bandwidth of 0.234
#> Picking joint bandwidth of 0.201
#> Picking joint bandwidth of 0.381
#> Picking joint bandwidth of 0.478
#> Picking joint bandwidth of 0.573
#> Picking joint bandwidth of 0.515
#> Picking joint bandwidth of 0.381
#> Picking joint bandwidth of 0.419
#> Picking joint bandwidth of 0.174
#> Picking joint bandwidth of 0.132
#> Picking joint bandwidth of 0.368
#> Picking joint bandwidth of 0.356
#> Picking joint bandwidth of 0.461
#> Picking joint bandwidth of 0.259
#> Picking joint bandwidth of 0.327
#> Picking joint bandwidth of 0.031
#> Picking joint bandwidth of 0.351
#> Picking joint bandwidth of 0.385
#> Picking joint bandwidth of 0.145
#> Picking joint bandwidth of 0.327
#> Picking joint bandwidth of 0.264
#> Picking joint bandwidth of 0.26
#> Picking joint bandwidth of 0.0669
#> Picking joint bandwidth of 0.0739
#> Picking joint bandwidth of 0.187
#> Picking joint bandwidth of 0.211
#> Picking joint bandwidth of 0.282
#> Picking joint bandwidth of 0.255
#> Picking joint bandwidth of 0.229
#> Picking joint bandwidth of 0.254
#> Picking joint bandwidth of 0.0798
#> Picking joint bandwidth of 0.0639
#> Picking joint bandwidth of 0.291
#> Picking joint bandwidth of 0.193
#> Picking joint bandwidth of 0.188
#> Picking joint bandwidth of 0.144
#> Picking joint bandwidth of 0.267
#> Picking joint bandwidth of 0.185
#> Picking joint bandwidth of 0.0543
#> Picking joint bandwidth of 0.0514
#> Picking joint bandwidth of 0.18
#> Picking joint bandwidth of 0.196
#> Picking joint bandwidth of 0.0658
#> Picking joint bandwidth of 0.0491
#> Picking joint bandwidth of 0.317
#> Picking joint bandwidth of 0.215
#> Picking joint bandwidth of 0.204
#> Picking joint bandwidth of 0.0904
#> Picking joint bandwidth of 0.241
#> Picking joint bandwidth of 0.295
#> Picking joint bandwidth of 0.0677
#> Picking joint bandwidth of 0.0845
#> Picking joint bandwidth of 0.468
#> Picking joint bandwidth of 0.282
#> Picking joint bandwidth of 0.174
#> Picking joint bandwidth of 0.173
#> Picking joint bandwidth of 0.0428
#> Picking joint bandwidth of 0.0847
#> Picking joint bandwidth of 0.117
#> Picking joint bandwidth of 0.343
#> Picking joint bandwidth of 0.318
#> Picking joint bandwidth of 0.25
#> Picking joint bandwidth of 0.51
#> Picking joint bandwidth of 0.378
#> Picking joint bandwidth of 0.0588
#> Picking joint bandwidth of 0.0605
#> Picking joint bandwidth of 0.0428
#> Picking joint bandwidth of 0.032
#> Picking joint bandwidth of 0.259
#> Picking joint bandwidth of 0.285
#> Picking joint bandwidth of 0.222
#> Picking joint bandwidth of 0.155
#> Picking joint bandwidth of 0.32
#> Picking joint bandwidth of 0.331
#> Picking joint bandwidth of 0.118
#> Picking joint bandwidth of 0.32
#> Picking joint bandwidth of 0.322
#> Picking joint bandwidth of 0.105
#> Picking joint bandwidth of 0.265
#> Picking joint bandwidth of 0.232
#> Picking joint bandwidth of 0.326
#> Picking joint bandwidth of 0.31
#> Picking joint bandwidth of 0.094
#> Picking joint bandwidth of 0.306
#> Picking joint bandwidth of 0.0355
#> Picking joint bandwidth of 0.116
#> Picking joint bandwidth of 0.33
#> Picking joint bandwidth of 0.278
#> Picking joint bandwidth of 0.186
#> Picking joint bandwidth of 0.102
#> Picking joint bandwidth of 0.251
#> Picking joint bandwidth of 0.261
#> Picking joint bandwidth of 0.0735
#> Picking joint bandwidth of 0.0714
#> Picking joint bandwidth of 0.202
#> Picking joint bandwidth of 0.267
#> Picking joint bandwidth of 0.256
#> Picking joint bandwidth of 0.0755
#> Picking joint bandwidth of 0.234
#> Picking joint bandwidth of 0.201
#> Picking joint bandwidth of 0.381
#> Picking joint bandwidth of 0.478
#> Picking joint bandwidth of 0.573
#> Picking joint bandwidth of 0.515
#> Picking joint bandwidth of 0.381
#> Picking joint bandwidth of 0.419
#> Picking joint bandwidth of 0.174
#> Picking joint bandwidth of 0.132
#> Picking joint bandwidth of 0.368
#> Picking joint bandwidth of 0.356
#> Picking joint bandwidth of 0.461
#> Picking joint bandwidth of 0.259
#> Picking joint bandwidth of 0.327
#> Picking joint bandwidth of 0.031
#> Picking joint bandwidth of 0.351
#> Picking joint bandwidth of 0.385
#> Picking joint bandwidth of 0.145
#> Picking joint bandwidth of 0.327
#> Picking joint bandwidth of 0.264
#> Picking joint bandwidth of 0.26
#> Picking joint bandwidth of 0.0669
#> Picking joint bandwidth of 0.0739
#> Picking joint bandwidth of 0.187
#> Picking joint bandwidth of 0.211
#> Picking joint bandwidth of 0.282
#> Picking joint bandwidth of 0.255
#> Picking joint bandwidth of 0.229
#> Picking joint bandwidth of 0.254
#> Picking joint bandwidth of 0.0798
#> Picking joint bandwidth of 0.0639
#> Picking joint bandwidth of 0.291
#> Picking joint bandwidth of 0.193
#> Picking joint bandwidth of 0.188
#> Picking joint bandwidth of 0.144
#> Picking joint bandwidth of 0.267
#> Picking joint bandwidth of 0.185
#> Picking joint bandwidth of 0.0543
#> Picking joint bandwidth of 0.0514
#> Picking joint bandwidth of 0.18
#> Picking joint bandwidth of 0.196
#> Picking joint bandwidth of 0.0658
#> Picking joint bandwidth of 0.0491
#> Picking joint bandwidth of 0.317
#> Picking joint bandwidth of 0.215
#> Picking joint bandwidth of 0.204
#> Picking joint bandwidth of 0.0904
#> Picking joint bandwidth of 0.241
#> Picking joint bandwidth of 0.295
#> Picking joint bandwidth of 0.0677
#> Picking joint bandwidth of 0.0845
#> Picking joint bandwidth of 0.468
#> Picking joint bandwidth of 0.282
#> Picking joint bandwidth of 0.174
#> Picking joint bandwidth of 0.173
#> Picking joint bandwidth of 0.0428
#> Picking joint bandwidth of 0.0847
#> Picking joint bandwidth of 0.117
#> Picking joint bandwidth of 0.343
#> Picking joint bandwidth of 0.318
#> Picking joint bandwidth of 0.25
#> Picking joint bandwidth of 0.51
#> Picking joint bandwidth of 0.378
#> Picking joint bandwidth of 0.0588
#> Picking joint bandwidth of 0.0605
#> Picking joint bandwidth of 0.0428
#> Picking joint bandwidth of 0.032
#> Picking joint bandwidth of 0.259
#> Picking joint bandwidth of 0.285

# }