Author: Luc Steinbuch, Anna-Maartje de Boer, Wageningen University & Research

# Assessing crosstalk in single-grain luminescence detection

#### Luc Steinbuch, Anna-Maartje de Boer, Wageningen University & Research

#### February 2025

# Crosstalk

This vignette explores the crosstalk-related functions in the “Luminescence” package, related to research paper “A novel tool to assess crosstalk in single-grain luminescence detection”, by Anna-Maartje de Boer, Luc Steinbuch, Gerard Heuvelink and Jakob Wallinga. Crosstalk in single-grain luminescence imaging (EMCCD) is the overlapping of luminescence signals from adjacent grains on a single-grain disc. The actual signal on one grain location influences the observed signal on a neighboring grain location which happens to be on the same measurement disc (one “position” in the reader). In this research and the shown functions, we define “neighboring” as rook-wise (horizontally and vertically) only, and we assume that all measurement discs have a regular grid of 10x10 grain locations.

    library(Luminescence)
    #> Welcome to the R package Luminescence version 1.1.0 [Built: 2025-06-11 16:03:06 UTC]
    #> A Windows user: 'An apple a day keeps the doctor away.'

# Explore imaged luminescence signals from a single-grain disc (one “position” in the reader)

All grain location observations on one single-grain measurement disc are represented by a vector of 100 numbers. Here we simulate such a disc by randomly selecting from two normal distributions. The prefix “vn_” means: a vector of numbers:

    vn_simulated <- sample(x = c(rnorm(n = 30, mean = 2000, sd = 500),
                                 rnorm(n = 70, mean = 20, sd = 1)),
                           size = 100)

    vn_simulated <- round(vn_simulated) # Because photons are discrete

    head(vn_simulated, n = 25)
    #>  [1]   19   21   21   21   20   20   19   20   20   19   20 2036   21 2837   20
    #> [16] 1495   20 1821 1798   21   19   19   19 1652   22

    hist(vn_simulated,
         main = "Simulated signal (histogram)",
         xlab = "Photon counts",
         ylab = "Frequency",
         breaks = 30
         )

## plot_SingleGrainDisc

Let’s visualize the disc/position:

    par(mar = c(1, 4, 6, 4))

    plot_SingleGrainDisc(object = vn_simulated,
              main = "Simulated signal (measurement disc)"
              )

Let’s calculate Moran’s I, and the associated pseudo-p, of our randomly ordered simulated disc:

    calc_MoransI(object = vn_simulated)
    #> [1] -0.08614305

    calc_MoransI(object = vn_simulated, compute_pseudo_p = TRUE)
    #> [1] 0.85

what changes if we add a serious amount, say 10%, of crosstalk?

    vn_simulated_with_crosstalk <- apply_Crosstalk(object = vn_simulated,
                                                  n_crosstalk = 0.10)

    vn_simulated_with_crosstalk <- round(vn_simulated_with_crosstalk)

    hist(vn_simulated_with_crosstalk,
         main = "Simulated signal with crosstalk (histogram)",
         xlab = "Photon counts",
         ylab = "Frequency",
         breaks = 30
         )

    plot_SingleGrainDisc(object = vn_simulated_with_crosstalk,
              main = "Simulated signal with crosstalk (measurement disc)")

    calc_MoransI(object = vn_simulated_with_crosstalk)
    #> [1] 0.1349534

    calc_MoransI(object = vn_simulated_with_crosstalk, compute_pseudo_p = TRUE)
    #> [1] 0.03

Let’s try several amounts of simulated crosstalk:

    df_MoransI <- data.frame(crosstalk = seq(from = 0,
                        to = 0.30,
                        length.out=50))

    df_MoransI$MoransI <- NA
    df_MoransI$pseudo_p <- NA

    old.opts <- options(warn = -1) # silence warnings from compute_pseudo_p
    for (i in 1:nrow(df_MoransI))
    {

      vn_simulated_with_crosstalk <- apply_Crosstalk(object = vn_simulated,
                                                  n_crosstalk = df_MoransI$crosstalk[i])

      df_MoransI$MoransI[i]  <- calc_MoransI(object = vn_simulated_with_crosstalk)
      df_MoransI$pseudo_p[i] <-
        calc_MoransI(object = vn_simulated_with_crosstalk, compute_pseudo_p = TRUE)
    }
    options(old.opts) # restore the default options

    n_expected_I_no_spatial_autocorr <- calc_MoransI(1:100,
                                                     spatial_autocorrelation = FALSE)

    ##

    plot(x = df_MoransI$crosstalk,
         y = df_MoransI$MoransI,
         ylim = range(
              pretty(x = c(df_MoransI$MoransI, n_expected_I_no_spatial_autocorr))
              ),
         ## Set ylim manually to make sure the value for I for no crosstalk is visible
         xlab = "Amount of added crosstalk",
         ylab = "Calculated Moran's I"
         )
    graphics::grid()
    abline(h = n_expected_I_no_spatial_autocorr,
           col = "purple")

    legend(x = "topleft",
           legend = "Expected I if no spatial autocorrelation",
           lty = "solid",
           col = "purple",
           cex = 0.8)

    plot(x = df_MoransI$crosstalk,
         y = df_MoransI$pseudo_p,
         xlab = "Amount of added crosstalk",
         ylab = "Generated pseudo-p of related Moran's I")
    graphics::grid()

Please note that above two plots are subject to randomness; for a good assessment many simulations have to be performed.

# Moran scatterplot

A way to visualise spatial auto-correlation is the Moran scatterplot:

    plot_MoranScatterplot(object = vn_simulated,
               main = "Moran scatterplot, simulated signal without crosstalk")

    vn_simulated_with_crosstalk <- apply_Crosstalk(object = vn_simulated,
                                                  n_crosstalk = 0.25)
    vn_simulated_with_crosstalk <- round(vn_simulated_with_crosstalk)

    plot_MoranScatterplot(object = vn_simulated_with_crosstalk,
               main = "Moran scatterplot, simulated signal with added crosstalk")

The plot area is divided into four quadrants using the mean in each dimension; the South-west and North-east quadrant represent a contribution to a positive spatial autocorrelation, while the North-west and South-east quadrants indicate a negative spatial correlation. Between the point, a least square line (which slopes indicates, but not exactly represents, Moran’s I) is added, as well as an 1:1 line (which indicates a Moran’s I of around 1, suggesting a perfect positive spatial correlation).

The internal function `.get_Neighbours()` was until now used on the background, but we can explicitly call it to generate a dataframe with all positions which are rook connected to each other (note that we need to use the `Luminescence:::` prefix as this function is not exported by the package):

        vn_simulated_with_holes <- c(rnorm(30, mean = 10, sd = 5),
                                  rnorm(30, mean = 500, sd = 50),
                                  rep(NA, times = 40)
                                  )

    df_Neighbours <- Luminescence:::.get_Neighbours(object = vn_simulated_with_holes)

    head(df_Neighbours)
    #>    location neighbour weight
    #> 11       11         1      1
    #> 12       12         2      1
    #> 13       13         3      1
    #> 14       14         4      1
    #> 15       15         5      1
    #> 16       16         6      1

And we can plot the first disc, while indicating which borders are taken into account. The corresponding `adjacent_grain_locations` is calculated in the background:

        plot_SingleGrainDisc(object = vn_simulated_with_holes,
                  show_neighbours = TRUE,
                  show_location_ids = TRUE)

Also other functions, such as `calc_MoransI()`, work as long as the remaining grid does not become too sparse. Note that in this context all observations, even “islands” who do not border any other grains, are used for the calculation of Moran’s I.

# Add borders with certain weight

One can manually change `df_Neighbours`, for example add the diagonal borders and attain a certain weight to it, and add it as argument to almost all functions. Assume that we want to add a few diagonal borders with weight 1/sqrt(2) to a full disc (note that the standard relative weight for rook borders is set to one):

    df_Neighbours_with_diag <- Luminescence:::.get_Neighbours(object = vn_simulated)

    for (i in c(1:9, 11:19, 21:29) )
    {
      df_Neighbours_with_diag <- rbind(df_Neighbours_with_diag,
                                                c(i, i+11, 1/sqrt(2))
                                              )
    }

    tail(df_Neighbours_with_diag)
    #>     location neighbour    weight
    #> 202       24        35 0.7071068
    #> 203       25        36 0.7071068
    #> 204       26        37 0.7071068
    #> 205       27        38 0.7071068
    #> 206       28        39 0.7071068
    #> 207       29        40 0.7071068

    plot_SingleGrainDisc(object = vn_simulated,
              df_neighbours = df_Neighbours_with_diag,
              show_neighbours = TRUE,
              show_location_ids = TRUE)

To exclude all border effects, we can use the `ignore_borders` option available in `calc_MoransI()` and `plot_SingleGrainDisc()`, which can take out all border rows and columns when computing the data frame of neighbours:

    vn_values_to_show <-
          sample(x = c(rnorm(n = 30, mean = 2000, sd = 500),
                       rnorm(n = 70, mean = 20, sd = 1)),
                 size = 100)

    ## Set the outer rows to NA before adding crosstalk
    vn_disc_border_locations <- c(1:10,
                                  91:100,
                                  seq(from = 11, to = 81, by = 10),
                                  seq(from = 20, to = 90, by = 10)
                                 )
    vn_values_to_show[vn_disc_border_locations] <- NA

    vn_values_to_show <- apply_Crosstalk(object = vn_values_to_show,
                                         n_crosstalk = 0.15)

    calc_MoransI(object = vn_values_to_show)
    #> [1] 0.2165462

    plot_SingleGrainDisc(object = vn_values_to_show,
                         show_neighbours = TRUE,
                         ignore_borders = TRUE)

    calc_MoransI(object = vn_values_to_show,
                 ignore_borders = TRUE)
    #> [1] 0.2165462

# Plot disc options

    plot_SingleGrainDisc(object = vn_simulated,
              main = "",
              legend = TRUE,
              show_coordinates = TRUE,
              show_location_ids = TRUE,
              show_positioning_holes = FALSE)

When there is a wide range in values, it can be helpful to apply a logarithmic scale in plotting (note that the default is “square root”):

    plot_SingleGrainDisc(object = vn_simulated,
              main = "Linear scale",
              legend = TRUE,
              show_coordinates = FALSE,
              show_location_ids = FALSE,
              show_positioning_holes = TRUE,
              str_transform = "lin")

    plot_SingleGrainDisc(object = vn_simulated,
              main = "Logarithmic scale",
              legend = TRUE,
              show_coordinates = FALSE,
              show_location_ids = FALSE,
              show_positioning_holes = TRUE,
              str_transform = "log")

# Moran scatterplot options

    vn_simulated <- c(rnorm(75, mean = 10, sd = 5),
                      rnorm(25, mean = 500, sd = 50)  )
    vn_simulated <- sample(size = 100, vn_simulated)

    vn_simulated_with_crosstalk <- apply_Crosstalk(object = vn_simulated,
                                                   n_crosstalk = 0.15)

    ## Base use
    plot_MoranScatterplot(object = vn_simulated,
                          main = "Without crosstalk")

    plot_MoranScatterplot(object = vn_simulated_with_crosstalk,
                          main = "With crosstalk")

    ## Layout options
    plot_MoranScatterplot(object = vn_simulated_with_crosstalk,
               pch = "show_location_ids",
               legend = FALSE,
               log = "xy",
               main = "With location ID's; and with log scales"
               )

    plot_MoranScatterplot(object = vn_simulated_with_crosstalk,
               pch = "show_n_neighbours",
               legend = FALSE,
               str_y_def = "weighted_sum",
               main = "With number of neighbours, and other y calculation"
               )

# Moran’s I function options

The function `calc_MoransI()` can return many intermediate calculation numbers:

    calc_MoransI(object = 1:100,
                 return_intermediate_values = TRUE)
    #> $n
    #> [1] 100
    #>
    #> $n_mean
    #> [1] 50.5
    #>
    #> $n_population_variance
    #> [1] 833.25
    #>
    #> $n_sum_similarities
    #> [1] 133320
    #>
    #> $n_sum_weights
    #> [1] 180
    #>
    #> $n_average_auto_correlation
    #> [1] 740.6667
    #>
    #> $n_moransI
    #> [1] 0.8888889

If the weights (and thus the spatial pattern under investigation) and the number of observations remain the same, this can be useful to understand what is happening. For example, if we add crosstalk, we can see that the population variance in most cases slightly increases (the values are indeed spatially smoothed, but the average increases) but the spatial autocorrelation strongly increases:

    vn_simulated <- sample(x = c(rnorm(n = 30, mean = 2000, sd = 500),
                                 rnorm(n = 70, mean = 20, sd = 1)),
                           size = 100)
    vn_simulated <- round(vn_simulated)

    vn_simulated_with_crosstalk <- apply_Crosstalk(object = vn_simulated,
                                                   n_crosstalk = 0.20)
    vn_simulated_with_crosstalk <- round(vn_simulated_with_crosstalk)

    df_compare <-
      data.frame(Case = c("Without crosstalk", "With crosstalk"),
                 MoransI =  c(calc_MoransI(object = vn_simulated),
                              calc_MoransI(object = vn_simulated_with_crosstalk)
                              ),
                 PopulationVar =
                   c(calc_MoransI(object = vn_simulated,
                     return_intermediate_values = TRUE)$n_population_variance,
                     calc_MoransI(object = vn_simulated_with_crosstalk,
                     return_intermediate_values = TRUE)$n_population_variance),
                 SpatialAutoCor =
                   c(calc_MoransI(object = vn_simulated,
                     return_intermediate_values = TRUE)$n_average_auto_correlation,
                     calc_MoransI(object = vn_simulated_with_crosstalk,
                     return_intermediate_values = TRUE)$n_average_auto_correlation)
          )

    df_compare[,2] <-  round(df_compare[,2],2)

    (df_compare)
    #>                Case MoransI PopulationVar SpatialAutoCor
    #> 1 Without crosstalk    0.06      782235.7       48993.16
    #> 2    With crosstalk    0.29      946747.1      274315.87