Visualize differential expression markers

Visualize differential expression (DE) results — typically the output of Seurat::FindMarkers() or Seurat::FindAllMarkers() — across a variety of plot types. You can also compose the DE results from other tools into a data frame with the required columns and use this function to visualize them.

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 each). 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"),
  group_by = NULL,
  each = NULL,
  facet_each = FALSE,
  p_adjust = TRUE,
  cutoff = NULL,
  show_labels = FALSE,
  sig_mark = "*",
  order_by = "desc(abs(avg_log2FC))",
  select = ifelse(plot_type %in% c("volcano", "volcano_log2fc", "volcano_pct",
    "jitter", "jitter_log2fc", "jitter_pct"), 5, ifelse(plot_type %in% c("heatmap",
    "violin", "box", "bar", "ridge", "dot") && !is.null(each) && !grepl("^\\s*:",
    each), 5, 10)),
  flatten_markers = FALSE,
  ...
)

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.

group_by

Used only for expression-based plot types (ignored for DE summary plot types). A column in the Seurat object's metadata to group cells by, e.g., a condition column — useful when the DEs were calculated between conditions (such as cell cycle phases) and you want to compare the expression of the markers across those conditions. A single value is passed directly to FeatureStatPlot: for heatmap and dot plots it is applied as the column annotation (ident), and it only takes effect when each includes a metadata column mapping; for violin, box, bar, and ridge plots it is passed as group_by. The "marker_column:metadata_column" syntax (see Metadata column mapping) restricts the object to only the cells involved in the comparisons: for example, if a comparison column in the markers data frame holds "G1:G2M", passing group_by = "comparison:Phase" keeps only G1 and G2M cells in the plot, with the Phase column re-factored to these two levels in the order they first appear in the comparison column. Without the restriction, e.g., group_by = "Phase", all phase cells (G1, G2M, and S) are included in the plot. Default: NULL.

each

A column name in markers indicating the grouping from which each marker was identified (e.g., the cluster column from FindAllMarkers()). Required for jitter and DE heatmap/dot plot types, where it defines the x-axis or column groups. For volcano plot types, it splits the plot by group (or facets it, with facet_each = TRUE). For expression plot types, each is used to select the markers within each group; a plain column name does not split the plot — use the "marker_column:metadata_column" syntax (see Metadata column mapping) to also split the plot by the mapped metadata column. Alternatively, pass ":metadata_column" with an empty marker part to split the expression plot by the metadata column directly, without selecting markers per group (markers are selected overall) and without merging metadata. Default: NULL.

facet_each

Logical. Only for volcano plot types: if TRUE, facet the volcano plot by the each groups instead of splitting it into separate subplots. Ignored for other plot types. Default: FALSE.

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 DE heatmap plots (heatmap_log2fc, heatmap_pct), controls which cells receive significance marks. For expression plot types with a numeric select, only markers with a p-value below cutoff are eligible for selection. Ignored by DE dot plots (dot_log2fc, dot_pct). 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 of one or more comma-separated expressions used to order the markers (evaluated with dplyr::arrange()). Can reference columns in markers as well as metadata columns merged in via a colon-form each (see Metadata column mapping). Only the first value of each merged metadata column is kept. Example: "desc(avg_log2FC)" or "desc(avg_log2FC), desc(pct.1)". The ordering determines which markers are selected when select is numeric. For jitter plots, it is also passed to plotthis::JitterPlot(). Default: "desc(abs(avg_log2FC))".

select

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

Numeric: Top N markers per each group, or overall when each is NULL (default: 5 for volcano/jitter types and for expression plot types when each selects markers per group, 10 otherwise).
Single expression: Filter condition for dplyr::filter().
Character vector of multiple expressions (DE heatmap/dot plot types only): expressions mentioning the each column name filter the overall data, others filter within the remaining data.

flatten_markers

Logical. Only for the expression heatmap and dot plot types. When each is used to select markers per group, the markers are by default provided to FeatureStatPlot as a named list (one entry per group), which splits the feature rows of the plot by group. With flatten_markers = TRUE, the selected markers are collapsed into a single vector so the plot shows one unsplit block of features — useful e.g. to mimic Seurat::DoHeatmap() on globally selected markers. Default: FALSE.

...

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, layer, cell_type. Note that group_by, ident, and columns_split_by are set by MarkersPlot() from the group_by and each arguments.

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

plot_type determines which underlying plotting function is called and also what to be plotted. volcano, volcano_log2fc, volcano_pct jitter, jitter_log2fc, jitter_pct, heatmap_log2fc, heatmap_pct, dot_log2fc, and dot_pct are DE summary plots that visualize the DE statistics themselves, while heatmap, violin, box, bar, ridge, and dot are expression-based plots that visualize the actual expression values of the selected marker genes in the context of the original Seurat object.
each is required for jitter plots ("jitter", "jitter_log2fc", "jitter_pct") and DE heatmap/dot plots ("heatmap_log2fc", "heatmap_pct", "dot_log2fc", "dot_pct"). Its role depends on the plot type:
- Volcano plot types: the plot is split by the each groups (faceted when facet_each = TRUE).
- Jitter plot types: the x-axis grouping.
- DE heatmap/dot plot types: the columns of the heatmap/dot plot.
- Expression plot types: used to select the markers within each group; it does not split or facet the plot. Pass "marker_column:metadata_column" (e.g., "cluster:seurat_clusters") to also split the plot by the mapped metadata column (via columns_split_by for heatmap/dot, or ident for violin/box/bar), or ":metadata_column" (e.g., ":seurat_clusters") to split the plot by the metadata column without per-group marker selection.
When each uses the "marker_column:metadata_column" form with a non-empty marker 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. The ":metadata_column" form (empty marker part) skips the join entirely.
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.

Metadata column mapping

Both each and group_by accept a "marker_column:metadata_column" syntax that links a column in the markers data frame to a column in the Seurat object's metadata.

The part before the colon must be a column in markers (e.g., cluster) or be empty; the part after the colon must be a column in object@meta.data (e.g., seurat_clusters).
This syntax requires object to be provided; otherwise an error is raised.
When the marker part is non-empty, every value in the marker column must exist in the metadata column, otherwise an error is raised.
For each with a non-empty marker part, the metadata is merged into the markers data frame (keeping the first row of each metadata group for non-key columns), so metadata columns become available for arguments like order_by. On name conflicts, the merged columns get a .meta suffix. With an empty marker part (":metadata_column"), no merging or per-group selection happens; the metadata column is used only to split/annotate the expression plot (columns_split_by for heatmap/dot, ident for violin/box/bar).
For group_by, the object is subset to the cells whose metadata values occur in the marker column, and the metadata column is re-factored with those values in the order they first appear in the marker column. Values separated by a colon (e.g., "G1:G2M") are split into individual groups.

Marker selection and filtering

How select picks the markers depends on the plot type and the value provided:

Numeric — Select the top N markers (ordered by order_by) within each group defined by each, or overall when each is NULL. Jitter plots label the top N markers per group (a numeric select is required). Volcano plots ignore select — labeling is controlled via ... (e.g., nlabel). For expression plot types, a numeric select only keeps markers with a p-value below cutoff (when set) before the top-N selection.
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) — Only for DE heatmap/dot plot types ("heatmap_log2fc", "heatmap_pct", "dot_log2fc", "dot_pct"). Each element is evaluated independently: expressions that mention the each 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, and for expression plot types when each is provided to select markers per group (a plain marker column or a non-empty "marker_column:metadata_column"); 10 otherwise (DE heatmap/dot types and expression plots without per-group selection, e.g., each = NULL or ":metadata_column").

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.

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
#> Warning: no non-missing arguments to min; returning Inf
#> Warning: no non-missing arguments to max; returning -Inf
#> 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
#> Warning: no non-missing arguments to min; returning Inf
#> Warning: no non-missing arguments to max; returning -Inf
#> Warning: no non-missing arguments to min; returning Inf
#> Warning: no non-missing arguments to max; returning -Inf

MarkersPlot(allmarkers, each = "cluster", ncol = 2, facet_each = 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
#> Warning: no non-missing arguments to min; returning Inf
#> Warning: no non-missing arguments to max; returning -Inf
#> Warning: no non-missing arguments to min; returning Inf
#> Warning: no non-missing arguments to max; returning -Inf


MarkersPlot(allmarkers, plot_type = "jitter", each = "cluster")
#> Warning: [JitterPlot] `raster` is ignored when `size_by` is mapped to a column; falling back to vector points.

MarkersPlot(allmarkers, plot_type = "jitter_pct", order_by = "desc(abs(pct.1 - pct.2))",
    each = "cluster", add_hline = 0, shape = 16)
#> Warning: [JitterPlot] `raster` is ignored when `size_by` is mapped to a column; falling back to vector points.


MarkersPlot(allmarkers, plot_type = "heatmap_log2fc", each = "cluster",
    order_by = "desc(avg_log2FC)", select = 3)

MarkersPlot(allmarkers, plot_type = "heatmap_log2fc", each = "cluster",
    label = scales::label_number(accuracy = 0.01), select = 3,
    cutoff = 0.05, show_labels = TRUE, sig_mark = '{}')

MarkersPlot(allmarkers, plot_type = "heatmap_pct", each = "cluster",
    cutoff = 0.05, select = 3)


MarkersPlot(allmarkers, plot_type = "dot_log2fc", each = "cluster",
    add_reticle = TRUE, select = 3)


topmarkers <- allmarkers[order(allmarkers$avg_log2FC, decreasing = TRUE), ]
# Mimic Seurat's DoHeatmap()
MarkersPlot(topmarkers[1:20, ], object = pancreas_sub, plot_type = "heatmap",
   layer = "data", cell_type = "bars", flatten_markers = TRUE, cluster_rows = FALSE,
   show_column_names = "inplace", each = "cluster:seurat_clusters")
#> Warning: Layer counts isn't present in the assay object; returning NULL


# Select top 3 markers per cluster
MarkersPlot(allmarkers, object = pancreas_sub, plot_type = "heatmap",
   order_by = "desc(avg_log2FC)", select = 3,
   layer = "data", cell_type = "bars",
   show_column_names = "inplace", each = "cluster:seurat_clusters")
#> Warning: Layer counts isn't present in the assay object; returning NULL

# Suppose we did a DE between G2M and G1 phases in each cluster and
# stored the results in a new column "comparison"
allmarkers$comparison <- "G1:G2M"
MarkersPlot(allmarkers, object = pancreas_sub, plot_type = "heatmap",
   group_by = "comparison:Phase", each = "cluster:seurat_clusters",
   order_by = "desc(avg_log2FC)", select = 3, layer = "data")
#> Warning: Layer counts isn't present in the assay object; returning NULL


MarkersPlot(allmarkers, object = pancreas_sub, plot_type = "dot", select = 2,
   flatten_markers = TRUE, order_by = "desc(avg_log2FC)",
   group_by = "Phase", each = "cluster:seurat_clusters", layer = "data")
#> Warning: Layer counts isn't present in the assay object; returning NULL


MarkersPlot(allmarkers, object = pancreas_sub, plot_type = "violin", select = 2,
   position_dodge_preserve = "single", add_bg = TRUE, add_box = TRUE,
   group_by = "comparison:Phase", each = "cluster:seurat_clusters", layer = "data")
#> 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",
  select = c('cluster %in% c("1", "2", "0") & pct.2 - pct.1 > 0.6'),
  each = "cluster:seurat_clusters", cutoff = 0.05, layer = "data")
#> Warning: Layer counts isn't present in the assay object; returning NULL


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


MarkersPlot(allmarkers, object = pancreas_sub, plot_type = "ridge", select = 2,
   group_by = "Phase", each = "cluster:seurat_clusters", layer = "data",
   ncol = 4)
#> Warning: Layer counts isn't present in the assay object; returning NULL
#> Picking joint bandwidth of 0.441
#> Picking joint bandwidth of 0.226
#> Picking joint bandwidth of 0.157
#> Picking joint bandwidth of 0.144
#> Picking joint bandwidth of 0.232
#> Picking joint bandwidth of 0.441
#> Picking joint bandwidth of 0.293
#> Picking joint bandwidth of 0.334
#> Picking joint bandwidth of 0.229
#> Picking joint bandwidth of 0.345
#> Picking joint bandwidth of 0.33
#> Picking joint bandwidth of 0.433
#> Picking joint bandwidth of 0.881
#> Picking joint bandwidth of 0.375
#> Picking joint bandwidth of 0.202
#> Picking joint bandwidth of 0.319
#> Picking joint bandwidth of 0.586
#> Picking joint bandwidth of 0.0593
#> Picking joint bandwidth of 0.0355
#> Picking joint bandwidth of 0.102
#> Picking joint bandwidth of 0.283
#> Picking joint bandwidth of 0.337
#> Picking joint bandwidth of 0.242
#> Picking joint bandwidth of 0.129
#> Picking joint bandwidth of 0.27
#> Picking joint bandwidth of 0.498
#> Picking joint bandwidth of 0.358
#> Picking joint bandwidth of 0.498
#> Picking joint bandwidth of 0.223
#> Picking joint bandwidth of 0.498
#> Picking joint bandwidth of 0.363
#> Picking joint bandwidth of 0.426
#> Picking joint bandwidth of 0.125
#> Picking joint bandwidth of 0.131
#> Picking joint bandwidth of 0.187
#> Picking joint bandwidth of 0.327
#> Picking joint bandwidth of 0.497
#> Picking joint bandwidth of 0.0635
#> Picking joint bandwidth of 0.327
#> Picking joint bandwidth of 0.327
#> Picking joint bandwidth of 0.0624
#> Picking joint bandwidth of 0.255
#> Picking joint bandwidth of 0.102
#> Picking joint bandwidth of 0.0994
#> Picking joint bandwidth of 0.0465
#> Picking joint bandwidth of 0.393
#> Picking joint bandwidth of 0.126
#> Picking joint bandwidth of 0.216
#> Picking joint bandwidth of 0.203
#> Picking joint bandwidth of 0.294
#> Picking joint bandwidth of 0.167
#> Picking joint bandwidth of 0.325
#> Picking joint bandwidth of 0.241
#> Picking joint bandwidth of 0.173
#> Picking joint bandwidth of 0.468
#> Picking joint bandwidth of 0.358
#> Picking joint bandwidth of 0.203
#> Picking joint bandwidth of 0.499
#> Picking joint bandwidth of 0.0658
#> Picking joint bandwidth of 0.0904
#> Picking joint bandwidth of 0.525
#> Picking joint bandwidth of 0.16
#> Picking joint bandwidth of 0.195
#> Picking joint bandwidth of 0.465
#> Picking joint bandwidth of 0.413
#> Picking joint bandwidth of 0.457
#> Picking joint bandwidth of 0.481
#> Picking joint bandwidth of 0.574
#> Picking joint bandwidth of 0.413
#> Picking joint bandwidth of 0.206
#> Picking joint bandwidth of 0.441
#> Picking joint bandwidth of 0.226
#> Picking joint bandwidth of 0.157
#> Picking joint bandwidth of 0.144
#> Picking joint bandwidth of 0.232
#> Picking joint bandwidth of 0.441
#> Picking joint bandwidth of 0.293
#> Picking joint bandwidth of 0.334
#> Picking joint bandwidth of 0.229
#> Picking joint bandwidth of 0.345
#> Picking joint bandwidth of 0.33
#> Picking joint bandwidth of 0.433
#> Picking joint bandwidth of 0.881
#> Picking joint bandwidth of 0.375
#> Picking joint bandwidth of 0.202
#> Picking joint bandwidth of 0.319
#> Picking joint bandwidth of 0.586
#> Picking joint bandwidth of 0.0593
#> Picking joint bandwidth of 0.0355
#> Picking joint bandwidth of 0.102
#> Picking joint bandwidth of 0.283
#> Picking joint bandwidth of 0.337
#> Picking joint bandwidth of 0.242
#> Picking joint bandwidth of 0.129
#> Picking joint bandwidth of 0.27
#> Picking joint bandwidth of 0.498
#> Picking joint bandwidth of 0.358
#> Picking joint bandwidth of 0.498
#> Picking joint bandwidth of 0.223
#> Picking joint bandwidth of 0.498
#> Picking joint bandwidth of 0.363
#> Picking joint bandwidth of 0.426
#> Picking joint bandwidth of 0.125
#> Picking joint bandwidth of 0.131
#> Picking joint bandwidth of 0.187
#> Picking joint bandwidth of 0.327
#> Picking joint bandwidth of 0.497
#> Picking joint bandwidth of 0.0635
#> Picking joint bandwidth of 0.327
#> Picking joint bandwidth of 0.327
#> Picking joint bandwidth of 0.0624
#> Picking joint bandwidth of 0.255
#> Picking joint bandwidth of 0.102
#> Picking joint bandwidth of 0.0994
#> Picking joint bandwidth of 0.0465
#> Picking joint bandwidth of 0.393
#> Picking joint bandwidth of 0.126
#> Picking joint bandwidth of 0.216
#> Picking joint bandwidth of 0.203
#> Picking joint bandwidth of 0.294
#> Picking joint bandwidth of 0.167
#> Picking joint bandwidth of 0.325
#> Picking joint bandwidth of 0.241
#> Picking joint bandwidth of 0.173
#> Picking joint bandwidth of 0.468
#> Picking joint bandwidth of 0.358
#> Picking joint bandwidth of 0.203
#> Picking joint bandwidth of 0.499
#> Picking joint bandwidth of 0.0658
#> Picking joint bandwidth of 0.0904
#> Picking joint bandwidth of 0.525
#> Picking joint bandwidth of 0.16
#> Picking joint bandwidth of 0.195
#> Picking joint bandwidth of 0.465
#> Picking joint bandwidth of 0.413
#> Picking joint bandwidth of 0.457
#> Picking joint bandwidth of 0.481
#> Picking joint bandwidth of 0.574
#> Picking joint bandwidth of 0.413
#> Picking joint bandwidth of 0.206

# }