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Vicus luminescence dating reports will include plots of sample data. These plots are used to help us determine whether all grains/minerals in a sample have a similar De value, or if they show a range of De values. This helps us decide on what statistical model to apply to the data to calculate the De value for the sample. It also helps to identify various pre- and post-depositional processes that the sampled sedimentary deposit may have experienced.
Samples that show a range of De values may have experienced incomplete sun-exposure prior to burial (common in rivers, for example), partial sun-exposure due to sampling error, or mixing of grains of different ages by natural processes related to soil development or disturbance by insects or animals.
Let’s take a sample we’ll call VCS-1, for which we measure 500 grains, of which 117 have luminescence signals suitable for dating. We will generate a plot of these 117 De values to look at their distribution. One could argue that the simplest way to display this data is in a histogram. A histogram is a graph of relative numbers of grains falling into different De intervals (bins), and the data might look as follows:
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Figure 1. A histogram for De values from Sample VCS-1, where n=117.
Two disadvantages of histograms include the following:
They require the use of pre-determined bin intervals (rectangle widths) that may be arbitrarily chosen by the plotter, and
They do not show the precision of each individual De value measurement, which can vary substantially.
To overcome these limitations, luminescence dating specialists more commonly present De distributions in the form of Kernel density estimate (KDE) plots and/or radial plots.
Kernel density estimates
A KDE plot is like a histogram, but rather than delineating bin intervals with rectangles, the data is presented as a smooth curve (Fig. 2). You can imagine this curve being drawn simply by joining the tops of all rectangles in the histogram. The wider the chosen bin intervals (rectangle widths), the smoother the curve will be, and the lower the resolution in curve shape. In a very large sample, where very small bin intervals are chosen, the smooth curve will approximate the shape of what statisticians call the “probability density function, or f(x)”. You can think of the probability density function as the true De distribution of the material from which the measured De values have been sampled.
The vertical scale (or y-axis) of a KDE plot is a continuous ‘density’ proportional to the number of aliquots (or grains) per unit Gray.
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Figure 2. A KDE plot of sample VCS-1.
Vicus typically superimposes additional graphical elements onto the KDE plot to aid visualization of the data. These include a box and whisker plot and an empirical distribution function of the De values (Fig. 3).
The box and whisker plot covers the interquartile interval where 50% of the data is found, and the left and right sides of the box represent the upper and lower quartiles, respectively. The vertical line splitting the box into two is the median De value. The whiskers extend from the minimum (or maximum) De value to the lower (or upper) quartile limits of the box. Dots extending beyond the whiskers are outlying De values.
An empirical distribution function is simply a plot of the De values in rank order that allows visualization of individual De values (points) and their errors (error bars on the points). Often the steepest part of the empirical distribution function slope aligns with the KDE peak, and if this slope appears near vertical, it may signify a distinct bleaching event.
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Figure 3. The KDE plot of sample VCS-1 with an empirical distribution function superimposed on the KDE curve and a box plot.
Radial plots
A radial plot allows visualization of individual De estimates and their standard errors. These plots can be less intuitive than histograms or KDE plots, however they convey a lot of information. This includes:
individual De values,
individual De value errors and precision,
the spread of the data (otherwise known as “overdispersion”),
the presence or absence of discrete components (or modes), and
the statistical consistency of De values with a reference value or weighted mean.
De values are plotted as points on a radial plot and their value in Gray, can be read off the curved radial axis by imagining a line extending from the origin (the standardized estimate of 0), through the data point onto the radial axis (Fig. 4A). Figure 4B shows the De values from sample VCS-1 plotted in a radial plot. The radial axis is on a log scale.
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Figure 4. A) Two De values plotted on a radial plot. The black lines intersect the radial axis at the De value. From Galbraith & Roberts (2012). B) Sample VCS-1 De values plotted in a radial plot.
The radial plot can be centered on any arbitrary value. We’ve centered the radial plot for VCS-1 on the sample weighted mean, 5.38 (dashed line). The grey shading marks the 95% confidence interval around the weighted mean, and therefore all points that lie within the shaded region are considered statistically consistent with the weighted mean at 2 standard deviations (2 σ). Points that lie outside the shaded region are considered “overdispersed”. In practice, it’s common for De values to be overdispersed due to factors such as incomplete re-setting of the signal prior to burial, heterogeneities in the dose field of the burial environment, mixing of grains from sedimentary units of different age, and variability in internal luminescence characteristics.
The radial plot includes 3 axes:
The vertical (y) axis on the left (standardized estimate) is the scale that determines the width of 2 σ around the value the plot is centered on. In mathematical terms, it plots (yi-y0)/si where yi is the De value for the ith aliquot or grain, y0 is the chosen reference De value, and si is the De standard error.
The curved axis on the right provides the De value (i.e., the estimate of the burial dose in Gray) of each aliquot or grain.
The (x) axis at the bottom allows one to see the relative error and precision of each De value. More precise values plot to the right, and less precise values plot to the left.
Abanico plots
The abanico plot allows visualization of individual De values, relative errors and precision, as well as the De distribution shape and modality (Dietze et al., 2016). It modifies the more traditional radial plot by merging it with a kernel density estimate plot (or other univariate plot types of choice).
Below we illustrate the abanico plot (with a KDE plot) by plotting the De distribution for sample VCS-2 (Fig. 5). This sample has two statistically significant grain populations as identified by the Finite Mixture Model (discussed in another module), and the weighted mean of these populations are highlighted as purple lines.
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Figure 5. An abanico plot for the De distribution of sample VCS-2. The plot is centred around the mean De value (dashed line), and the light gray shading marks the standard deviation. Two statistically significant components have been identified by the Finite Mixture model (purple lines).
The light grey shaded region (i.e., the scatter polygon) allows a graphical display of measures of dispersion that characterise the De distribution and can be used to mark the upper and lower quartile range, the standard deviation of the mean, or the 5-95% range. The dark grey dispersion bar marks the agreement of individual De estimates to a specified reference value, as in the radial plot.
Use of graphical displays
KDE plots and radial plots allow us to observe the range, shape, modality and skewness of De distributions. These observations together with observations at the sampling site, help us infer:
how well a sample may have been bleached prior to burial,
the complexity of the dose rate field at the sample site, and
whether or not there is evidence of sediment mixing after burial.
This helps us decide which statistical model to apply to the data to calculate the De of the sample, and therefore the age.
For more insights on graphical displays of luminescence data, see Dietze et al. (2016), Galbraith & Roberts (2012), Galbraith (2010), Galbraith (1988), Galbraith (1990).