Hilbert curve is a type of space-filling curves that folds one dimensional axis into a two dimensional space, but still keeps the locality. It has advantages to visualize data with long axis in following two aspects:
greatly improve resolution of the visualization fron \(n\) to \(\sqrt{n}\);
easy to visualize clusters because generally data points in the axis will also be close in the 2D space.
This package aims to provide an easy and flexible way to visualize data through Hilbert curve. The implementation and example figures are based on following sources:
Following plots show Hilbert curves with level 2, 3, 4, 5:
## Warning in matrix(c(cos(theta), sin(theta), -sin(theta), cos(theta)), 2, : data length differs from
## size of matrix: [60 != 2 x 2]
## Warning in matrix(c(cos(theta), sin(theta), -sin(theta), cos(theta)), 2, : data length differs from
## size of matrix: [252 != 2 x 2]
## Warning in matrix(c(cos(theta), sin(theta), -sin(theta), cos(theta)), 2, : data length differs from
## size of matrix: [1020 != 2 x 2]
As shown in the above plots, as level increases, the length of the curve becomes longer and the curve folds more densely. The number of segments (one segment is marked in red) on the Hilbert curve is 4^level - 1. If a Hilbert curve with level 11 is used to map to human chromosome 1, the resolution would be 249250621/4^11 (approximately 59bp per segment).
Locality
Hilbert curve folds one-dimensional axis into a two-dimensional space while still preserves the locality of the data points. We visualize this attribute with a Hilbert curve with level 5. In following animation, the point moves with its natural order on the axis.
Next, we calculate the pairwise distance between data points in the 2D space and visualize it as heatmap. The numbers on x-axis (top) and y-axis (left) illustrate the position of data points in the original 1D axis.
Basically, from the heatmap, data points which are close in the 1D axis are still close in the 2D space (illustrated in the areas around diagonals).
Also there are some data points in (e.g, 1 ~ 256) which are close to data points in (768 ~ 1024)(bottom left or top right area in the heatmap), but it is only due to that the curve folds back. This exception can be easily distinguished by adding assistant grid lines or if users are clear with the structure of the curve.
Basic settings
The HilbertCurve package provides a rather simple way to display the data in the form of Hilbert curve. It hides all the technical parts and users only need to think that they are adding graphics on a novel axis based on specifying positions.
Generally, customizing a Hilbert curve follows following steps:
HilbertCurve() is a constructor function and initializes the Hilbert curve. Following example means initializing a Hilbert curve with level 4 which maps data ranging from 1 to 100. The function returns a HilbertCurve class instance and it can be used to add more graphics later.
reference argument here is only used to show the structure of the curve.
By default, the Hilbert curve starts from the bottom left corner. The start position can be specifyed by start_from option and the orientation of the first segment can be set by first_seg:
## Warning in matrix(c(cos(theta), sin(theta), -sin(theta), cos(theta)), 2, : data length differs from
## size of matrix: [60 != 2 x 2]
## Warning in matrix(c(cos(theta), sin(theta), -sin(theta), cos(theta)), 2, : data length differs from
## size of matrix: [60 != 2 x 2]
## Warning in matrix(c(cos(theta), sin(theta), -sin(theta), cos(theta)), 2, : data length differs from
## size of matrix: [60 != 2 x 2]
## Warning in matrix(c(cos(theta), sin(theta), -sin(theta), cos(theta)), 2, : data length differs from
## size of matrix: [60 != 2 x 2]
## Warning in matrix(c(cos(theta), sin(theta), -sin(theta), cos(theta)), 2, : data length differs from
## size of matrix: [60 != 2 x 2]
## Warning in matrix(c(cos(theta), sin(theta), -sin(theta), cos(theta)), 2, : data length differs from
## size of matrix: [60 != 2 x 2]
## Warning in matrix(c(cos(theta), sin(theta), -sin(theta), cos(theta)), 2, : data length differs from
## size of matrix: [60 != 2 x 2]
## Warning in matrix(c(cos(theta), sin(theta), -sin(theta), cos(theta)), 2, : data length differs from
## size of matrix: [60 != 2 x 2]
The curve can be though as a folded axis. When the coordinate for this folded axis is initialized, low-level graphics can be added by specifying the positions.
There are several ways to specify the “positions” of the data points. The most common way is to construct a IRanges object:
Here ir contains intervals which are composed by integers. In later sections, you will see it is also possible to specify positions with numeric values and very large values.
Points
There are two modes for adding points. Normally, intervals are always long in the curve and can not be sufficiently represented by single points, thus, by default, a list of e.g. circles compose the intervals.
The number of circles used to represent the intervals can be controlled by np (number of points per segment). np controls number of tiny segments that split every Hilbert curve segment (e.g. the first horizontal segment in the left bottom in the curve). Following plot is under np = 3.
Graphic parameters can be set by gp. Note under this mode, the size of points can only be controlled by np argument. To make it more interesting, you can choose different shapes for the points. There are some pre-defined shapes that you can choose from: “circle”, “square”, “triangle”, “hexagon”, “star”.
In above figure, you may notice for some points, the color is more light in the ends. This is because the segments represented by these light circles are not fully covered by user’s input intervals, thus, averaging is applied here. E.g. if only the half of the segment represented by the circle is covered by user’s interval and the color for this circle is set to red, then after the averaging, the color would be semi-red (#FF8080, an average between red and white). Averaging is very important when you visualize the genome with zooming. You can find more detailed explanation in the Averaging section.
If np is set to 1 or NULL, there will be second mode for adding points that the points are plotted at the center of every interval in ir. In this case, size argument is used to control the size of the points. This mode is useful if you have a lot of small intervals.
hc_rect() fills intervals by a list of squares. The squares always locate at the turning points of the Hilbert curve and the width/height cannot be modified and it is automatically adjusted according to the level of the curve.
There is a difference from hc_points() with setting shape to square. The number of squares per segment is always 2 for hc_rect() while it can be any number for hc_points(). When the interval is long enough, hc_rect() can completely fill the surrounding areas.
As you can see some rectangles are not full red as well. It is also because of averaging. See Averaging sections for explanation.
Polygons
Basically polygons are quite similar to the rectangles. The major difference is 1. borders can be set for irregular polygons; 2. there is only single color in one polygon while for rectangles, the color can change in the ends of the interval if the corresponding segment is not completely covered by the input interval.
When the intervals are long or when there are polygons representing the intervals, it is better to put text in the visual center of the polygons. For this, centered_by argument can be set to polygon. Compare postions of a and b in following plot:
This is also useful to specify the positions with huge integers which can not be handled by the IRanges class (e.g. the summation of total length of human chromosomes).
The pixel mode can be thought as a high resolution version of hc_rect(). When the level is high (e.g. > 10), the whole 2D space will be almost completely filled by the curve and it is impossible to add or visualize e.g. points on the curve. In this case, the ‘pixel’ mode visualizes each tiny ‘segment’ as a pixel and maps values to colors. Internally, the whole plot is represented as an RGB matrix and every time a new layer is added to the plot, the RGB matrix will be updated according to the color overlay. When all the layers are added, normally a PNG figure is generated directly from the RGB matrix. So the Hilbert curve with level 11 will generate a PNG figure with 2048x2048 resolution. This is extremely useful for visualize genomic data. E.g. If we make a Hilbert curve for human chromosome 1 with level 11, then each pixel can represent 60bp (249250621/2048/2048) which is of very high resolution.
Under ‘pixel’ mode, every time a new layer is added, the image is added to the interactive device as a rastered image.
Since color is graphic representation to values, there is only one graphic setting col. You can add more than one layers, and just remember to set transparent colors for the additional layers.
To map values to colors, users can try colorRamp2() function in circlize package to generate a color mapping function. Another advantage of using colorRamp2() is it can be used to generate the corresponding color legend.
Grid lines can be added to the plot for better distinguishing blocks in the Hilbert curve. The 2D space will be split into 2^(grid_line-1) rows and 2^(grid_line-1) columns.
Setting border argument in hc_layer directly modifies the RGB matrix. It adds one pixel border for each polygon and sometimes the border dispears when compressing the image. It is more suggested to add borders by hc_polygon():
For the low-level graphic functions introduced before, only hc_polygon() and hc_text() can work for pixel mode. E.g. hc_polygon() draws partitions and hc_text() adds labels for the partitions.
The Hilbert curve can be save as a PNG figure by hc_png() with resolution 2^level x 2^level. Note only the curve itself not including the graphics by hc_polygon()/hc_text() is exported.
As discussed before, under “pixel” mode, the Hilbert curve is stored in an RGB matrix, every time a new layer is added, the RGB values that correspond to the new input intervals will be updated according to the color and transparency of the new layer.
By default, the color overlay for layers is applied based on the transparency of the new layer. Under “pixel” mode, each pixel is represented by red, green and blue colors for which the value is between 0 and 1. When a new layer with transparency color is added, the overlayed RGB is calculated as follows (r0, g0 and b0 correspond to the layer which is already added, r, g, b and alpha correspond to the layer which is going to be added). The method is implemented in default_overlay() function.
Insteading of globally controlling the color overlay, hc_layer() provides an overlay argument for which users can specificly control the color overlay in the regions that the input invervals cover. In following example, we change the red regions which are overlapped with the blue regions to green.
The self-defined overlay should accept seven arguments for which, r0, g0 and b0 correspond to RGB channles for the layers which are already added, r, g, b and alpha correspond to the layer which are going to be added. The elements of i.e. r0, g0, b0, r, g, b only correspond to the pixels that are covered by the intervals in the new layer, and overlay function is applied to every interval in the new layer iteratively.
For default_overlay(), if any of r0, g0, b0 has NA, the color from the new layer is only used. If any of r, g, b has NA, the color from the background is still used. The overlap of the two-layer colors are only calculated when there are no NA values in the colors.
The color overlay can also be applied to continuous colors. If the color mapping is generated by circlize::colorRamp2(), circlize::col2value() function can transform from colors back to the original values (it can be thought as the reversed function of colorRamp2()). Since the overlay function only provides color information, we can use col2value() to obtain back the original value and use colorRamp2() to generate a new set of colors.
In following plot, we change the color theme in the overlapped areas from “white-red” to “white-blue”. To do this, we first select the non-white pixels, transform back to the original values by col2value() and finally change to another color schema.
In above example, actually you can implement such overlay by first splitting the “red” regions into two parts by whether they overlap with the “grey” regions. Then add the not-overlapped regions, the “grey” regions, and overlapped regions as three independent layers.
hc_layer(red regions not ovelap with blue regions, use white-red color schema)
hc_layer(grey regions)
hc_layer(white-blue regions)
You can see the overlay option provides an easier and more straightforward way to do it.
In later section, we will demonstrate using self-defined overlay method can efficiently visualize the correspondance between H3K36me3 histone mark and gene body.
Add title and legends
Title can be set by the title argument in HilbertCurve(). Legend can be passed to legend argument in HilbertCurve() as a grob/ComplexHeatmap::Legends object or a list of these objects. Basically, colors are the main aesthetic mappings to values, thus, Legend() function in ComplexHeatmap package can be very helpful to create a legend. Also you can consider legendGrob() in grid package.
Under “pixel” mode, and when the curve is shown on the interactive device as raster image, title and legends can be added as well. But if the plot is exported by hc_png(), there will be not title and no legend.
Note: this “averaging” concept is very important. If you have unexpected plot, try other mean_mode option.
When using points or rectangles under “normal” mode, or using pixels under “pixel” mode, each point/rectangle/pixel corresponds to a small interval. When overlapping these intervals to the segments of Hilbert curve, and when the segment on the curve is not fully covered by the input intervals, proper averaging method should be applied to summarize the value in the corresponding segments.
Depending on different scenarios, HilbertCurve provides three metrics for averaging.
The overlapping model is illustrated in following plot. The red line in the bottom represents one segment on the Hilbert curve. Black lines on the top are the parts of input intervals that overlap with the segment. The thick lines indicate the intersected part between the input intervals and the segment.
For a segment on the curve, \(n\) is the number of input intervals which overlap with the segment (it is 3 in the above plot), \(w_i\) is the width of the intersected segments (black thick lines), \(x_i\) is the value associated with the original intervals.
The “absolute” method which is denoted as \(v_a\) and is simple calculated as the mean regardless of the width:
\[ v_a = \frac{\sum_i^n{x_i}}{n} \]
The “weighted” method which is denoted as \(v_w\) is the mean weighted by the width of intersections:
“Absolute” and “weighted” mode should be applied when background information is not of interest. E.g. when summarizing mean methylation in a small segment, since methylation is only related with CpG sites while not other bases, non-CpG background should be ignored.
The “w0” mode is the weighted mean between the intersected parts and un-intersected parts:
\[ v_{w0} = \frac{v_wW + v_bW'}{W+W'} \]
for which \(W\) is sum of width of the intersected parts and \(W'\) is the sum of width for the un-intersected parts. \(V_b\) is the value corresponding to the background. E.g when averaging colors which are represented as numeric RGB values, the background value is set to 255 which corresponds to white.
There is one more value for mean_mode which is max_freq. max_freq is mainly for discrete signals and in a segment, value with the highest frequency (or with the highest length) is selected for this segment.
To make it easy with genomic data which is always stored as a data frame or a GRanges object, the package provides specific class and functions to deal with. GenomicHilbertCurve() initializes the Hilbert curve by specifying chromosomes or specific background regions. After that, the data frame or GRanges object which contains genomic intervals can be mapped to the curve.
Following functions are depatched to the GenomicHilbertCurve class, and the second argument should be a data frame (which is in bed-format) or a GRanges object.
The conserved regions on human chromosome 1 between mouse and human or between zebrafish and human are downloaded from UCSC Table Browser. The parameters for downloading are:
clade: Mammal
genome: Human
assembly: Feb. 2009(GRCh37/hg19)
group: Comparative Genomics
= for mouse
track: Placental Chain/Net
table: Mouse Net (netMm10)
= for zebrafish
track: Vertebrate Chain/Net
table: Zebrafish Net(netDanRer7)
The “net” alignment which is used here shows the best alignment (the alignment algorithm allows longer gaps than traditional affine gap scoring systems) between each part in human genome and other genomes. For a detailed description of the alignment strategy, please go to UCSC Table Browser, and click describe table schema button.
In following example, the conservation data is mapped to a level 6 Hilbert curve under “normal” mode. Points are used as the graphic and each segment on the Hilbert curve is split by 3 points. In this case, each point on the curve represents approximately 30kb. Only chromosome 1 is visualized.
The point is fully red if the window which is represented by this point is completely covered by the conserved regions, and it is yellow if it is not covered by any region. Colors are interpolated linearly between red and yellow if it is partially covered by the conserved regions.
Compareing these two plots, we have following conclusions:
Although it is already known that mouse is far more orthologically close to human, the Hilber curve illustrates more clearly that almost all parts of human chromosome 1 have similar sequences on mouse genome.
The start and end of the human chromosome 1 is less conserved to mouse compared to other parts in the chromosome. Also there are several less conserved regions inside the human chromosome.
It illustrates how the regions which are conserved to zebrafish spread in the human chromosome and it gives an estimation of the positions and length of the conserved regions.
The idea of visualizing sequencing conservation through the Hilbert curve is from here.
GC percent
Following figure visualizes GC percent on chromosome 1, under “normal” mode. The GC content per 1KB window is calculated by hgGcPercent.bin binary which is downloaded from http://hgdownload.soe.ucsc.edu/admin/exe/linux.x86_64/. To speed up, we only used chromosome 1 for demonstration.
Background is not necessary to be the whole chromosome, it can also be self-defined regions. The background can be specified as a GRanges object and pass to background argument. In following example, only the first half part of chromosome 1 is used to initialize the curve.
Note: when using background, one chromosome can only exist in one row in background.
Different modifications (e.g. methylation, acetylation) of the histones affect the chromatin-DNA interaction, thus play roles in transcription regulation. Different types of modifications relates to different types of gene regulation. E.g. H3H4me3 modification is found in actively transcribed promoters. ChIP-sequencing (ChIPSeq) utilizes highthroughput DNA sequencing technology to identify status of chromatin states genome-wide, thus it provides a way to study gene regulation in a global aspect.
In this section, we visualize four types of histone modifications (H3K27ac, H3K36me3, H3K4me3, H3K9me3). The ChIPSeq data are downloaded from Roadmap (http://genboree.org/EdaccData/Release-9/sample-experiment/Lung/, The bed files are alignments of reads, you need to convert to the basepair or window-based coverage before using following R commands). Following files for four histone modifications are used to visualize:
In above code, intensities of histone modifications are mapped to a level 9 Hilbert curve so that each pixel approximately represents a window of 950bp. In each plot, there is an additional grey layer which represents the regions of genes (RefSeq genes) (g is from system.file("extdata", "refseq_chr1.RData", package = "HilbertCurve")).
H3K27ac is a mark for active enhancers which is narrow and sharp. The Hilbert curve shows this attributes and also shows enhancers may exist in form of clusters.
H3K36me3 is a mark for actively transcribed gene bodies. The Hilbert curve shows that basically most of H3K36me3 modification with high intensity exists as big blocks.
H3K4me3 is a mark for TSS of actively transcribed genes which is also narrow and sharp. The Hilbert curve confirms this and also shows the size of marks are quite consistent.
When there are more than one layers added and users want to see the correspondance between different genomic features, they can define a color overlay function to enhance the visual effects of the overlapping areas. In following example, when regions for H3K36me3 mark are overlapped with gene bodies, the color theme is changed to white-purple.
Next figure visualizs methylation on human chromosome 1. Blue corresponds to un-methylation and red corresponds to full-methylation. One thing which is very important here is that methylation is always measured at single CpG sites, so the mode for averaging should be set to absolute in order to sumarize mean methylation values only on CpG sites. Data are from http://genboree.org/EdaccData/Release-9/sample-experiment/Lung/.
The methylation plot shows quite clear patterns for un-methylated regions which are small and sharp, fully methylated regions and intermediate methylated regions which are represented as large blocks.
We demonstrated how to visualize one single chromosome through Hilbert curve, but sometimes we also want to put all chromosomes into one plot to get a full view of the genome. One solution is to construct a ‘huge fake’ chromosome which merges all real chromosomes. GenomicHilbertCurve() makes it easy. You can specify multiple chromosomes by sending a vector to chr argument or specify more than one background intervals in background argument.
In following examples, copy number loss and gain for 22 human chromosomes are visualized. Also color for the overlapping regions between layer for gain events and layer for loss events is changed to purple by defining overlay function. Data is from Database of Genomic Variants, CNV map.
If you want to compare several plots simutaneously, you can use grid.layout() to put them together. Remember to set newpage to FALSE in HilbertCurve() or GenomicHilbertCurve(). Also note we set labels in hc_map() to define the labels added to each chromosome.
