Author: Josh Sumner, DDPSC Data Science Core Facility
Date: 2024-04-22

# Root Traits

### pcvr v0.1.0

#### Josh Sumner, DDPSC Data Science Core Facility

#### 2024-04-22

## Root imaging data

Root imaging is an emerging application of `PlantCV`. Due to the nature of available technologies for root imaging the output tends to be noisy and there are a different set of phenotypes that may be interesting for researchers. Fundamentally analysis should be very similar for most phenotypes, but in the interest of providing an example for root-focused researchers we will go over a few options for root data output from `PlantCV`. For this vignette we will work with simulated data based off of mini rhyzotron data collected from Fischer farms in the Fall of 2023.

## Simulating Minirhyzotron data

The data is simulated as a mixture between a Uniform background distribution and N gaussian distributions where N follows a uniform distribution and each gaussian is parameterized by mu and sigma. Mu also follows a uniform distribution and sigma follows a half-normal distribution. Pixels are assigned to the background or the gaussian mixture component according to theta. Pixels assigned to the gaussian mixture are randomly assigned to each of the N gaussian distributions. See the `rRhyzoDist` function below. We also define functions that will generate single value traits from the MV frequencies.

Code

    rRhyzoDist <- function(n, theta = 0.3, u1_max = 20, u2_max = 5500, sd = 200, abs_max = 5500) {
      #* split n_pixels based on theta into background and gaussians
      n_unif_pixels <- ceiling(n * theta)
      n_gauss_pixels <- floor(n * (1 - theta))
      #* background is uniform
      background <- runif(n_unif_pixels, 0, u2_max)
      #* simulate a number of gaussians randomly between 1 and u1_max
      n_gaussians <- runif(1, 1, u1_max)
      #* each gaussian has a mean that is uniform between 1 and u2_max
      mu_is <- lapply(seq_len(n_gaussians), function(i) {
        return(runif(1, 1, u2_max))
      })
      #* each gaussian has a sigma that is half-normal based on sd
      sd_is <- lapply(seq_len(n_gaussians), function(i) {
        return(extraDistr::rhnorm(1, sd))
      })
      #* assign pixels randomly to gaussians
      index <- sample(seq_len(n_gaussians), size = n_gauss_pixels, replace = TRUE)
      px_is <- lapply(seq_len(n_gaussians), function(i) {
        return(sum(index == i))
      })
      #* draws n_pixels time from each gaussian
      d <- unlist(lapply(seq_len(n_gaussians), function(i) {
        return(rnorm(px_is[[i]], mu_is[[i]], sd_is[[i]]))
      }))
      #* combine data
      d <- c(d, background)
      #* make sure no gaussians return data out of bounds
      d[d < 0] <- runif(sum(d < 0), 0, abs_max)
      d[d >= abs_max] <- runif(sum(d >= abs_max), 0, abs_max)
      return(d)
    }
    lastNonZeroBin <- function(d) {
      return(max(d[d$value > 0, "label"]))
    }
    tubeAngleToDepth <- function(x, theta) {
      return(sin(theta) * x)
    }
    mv_mean <- function(d) {
      return(weighted.mean(d$label, d$value))
    }
    mv_median <- function(d) {
      return(median(rep(d$label, d$value)))
    }
    mv_std <- function(d) {
      return(sd(rep(d$label, d$value)))
    }
    sv_from_mv <- function(df, theta) { # note this should also return mean/median/std
      metaCols <- colnames(df)[-which(grepl("value|label|trait", colnames(df)))]
      out <- aggregate(
        as.formula(paste0(
          "value ~ ",
          paste(metaCols, collapse = "+")
        )),
        data = df, sum, na.rm = TRUE
      )
      colnames(out)[ncol(out)] <- "area"
      out$max_pixel <- unlist(lapply(split(df, interaction(df[, metaCols])), lastNonZeroBin))
      out$height <- tubeAngleToDepth(out$max_pixel, 0.35)
      out$mean_x_frequencies <- unlist(lapply(split(df, interaction(df[, metaCols])), mv_mean))
      out$median_x_frequencies <- unlist(lapply(split(df, interaction(df[, metaCols])), mv_median))
      out$std_x_frequencies <- unlist(lapply(split(df, interaction(df[, metaCols])), mv_std))
      return(out)
    }

The simulated data looks realistic based on limited test data available at time of writing.

Code

    set.seed(123)
    ex <- do.call(rbind, lapply(1:20, function(rep) {
      n_total_pixels <- runif(1, 100, 3000)
      x <- rRhyzoDist(n = n_total_pixels)
      h <- hist(x, plot = FALSE, breaks = seq(0, 5500, 20))
      breaks <- h$breaks[-1]
      counts <- h$counts
      rep_df <- data.frame(
        rep = as.character(rep),
        value = counts, label = breaks,
        trait = "x_frequencies"
      )
      return(rep_df)
    }))
    pcv.joyplot(ex, "x_frequencies", group = c("rep"))

For the rest of this vignette we will use simulated data. The first simulated data `df` assumes that roots can appear and disappear from the minirhyzotron images over time. The second simulated data assumes that once a root is seen along the minirhyzotron that it will stay visible.

Code

    n_times <- 5
    parameters <- data.frame(
      time = c(1:n_times),
      n_min = rep(0, n_times),
      n_max = seq(2000, 5500, length.out = n_times),
      theta = rep(0.3, n_times),
      u1_max = seq(10, 20, length.out = n_times),
      u2_max = seq(2000, 5500, length.out = n_times),
      u2_max_noise = seq(400, 200, length.out = n_times),
      sd = seq(150, 250, length.out = n_times)
    )
    set.seed(123)
    df <- do.call(rbind, lapply(seq_len(nrow(parameters)), function(time) {
      pars <- parameters[parameters$time == time, ]
      time_df <- do.call(rbind, lapply(1:10, function(rep) {
        n_total_pixels <- runif(1, pars$n_min, pars$n_max)
        u2_max_iter <- ceiling(rnorm(1, pars$u2_max, pars$u2_max_noise))
        x <- rRhyzoDist(
          n = n_total_pixels, theta = pars$theta,
          u1_max = pars$u1_max,
          u2_max = u2_max_iter, sd = pars$sd
        )
        h <- hist(x, plot = FALSE, breaks = seq(0, 5500, 20))
        breaks <- h$breaks[-1]
        counts <- h$counts
        rep_df <- data.frame(
          rep = as.character(rep),
          time = as.character(time),
          value = counts, label = breaks,
          trait = "x_frequencies"
        )
        return(rep_df)
      }))
      return(time_df)
    }))
    df$rep <- factor(df$rep, levels = seq_along(unique(df$rep)), ordered = TRUE)
    sv <- sv_from_mv(df)

Code

    n_times <- 5
    parameters <- data.frame(
      time = c(1:n_times),
      n_min_new_px = rep(50, n_times),
      n_max_new_px = seq(1000, 2500, length.out = n_times),
      theta = rep(0.3, n_times),
      u1_max = seq(10, 20, length.out = n_times),
      mean_added_depth = seq(log(200), log(1200), length.out = n_times),
      added_depth_noise = rep(0.1, n_times),
      sd = seq(150, 250, length.out = n_times)
    )
    set.seed(567)
    df2 <- do.call(rbind, lapply(1:10, function(rep) {
      dList <- list()
      add_area <- 2000
      previous_max_depth <- 2000
      d <- numeric(0)
      for (time in seq_len(nrow(parameters))) {
        pars <- parameters[parameters$time == time, ]
        n_total_pixels <- add_area + runif(1, pars$n_min_new_px, pars$n_max_new_px)
        max_depth <- ceiling(previous_max_depth + rlnorm(1, pars$mean_added_depth, pars$added_depth_noise))
        x <- rRhyzoDist(
          n = n_total_pixels, theta = pars$theta,
          u1_max = pars$u1_max,
          u2_max = max_depth, sd = pars$sd
        )
        d <- append(d, x)
        h <- hist(d, plot = FALSE, breaks = seq(0, 5500, 20))
        breaks <- h$breaks[-1]
        counts <- h$counts
        dList[[time]] <- data.frame(
          rep = as.character(rep),
          time = as.character(time),
          value = counts, label = breaks,
          trait = "x_frequencies"
        )
        add_area <- 0
        previous_max_depth <- max_depth
      }
      return(do.call(rbind, dList))
    }))
    df2$rep <- factor(df2$rep, levels = seq_along(unique(df2$rep)), ordered = TRUE)
    sv2 <- sv_from_mv(df2)

## Outlier Removal

## Single Value Traits

If we assume that roots can only enter the minirhyzotron images then we would expect a positive trend over time for total root area.

Code

    ggplot(sv, aes(x = time, y = area, group = rep)) +
      geom_point() +
      geom_line() +
      labs(x = "Sampling Time", y = "Area (px)", title = "Assuming roots can leave the image")

Code

    ggplot(sv2, aes(x = time, y = area, group = rep)) +
      geom_point() +
      geom_line() +
      labs(x = "Sampling Time", y = "Area (px)", title = "Assuming roots can only enter the image")

It would also make sense that the mean of the distribution would move deeper over time as roots have time to grow. This is likely to be true regardless of whether roots can leave the image.

Code

    ggplot(sv, aes(x = time, y = mean_x_frequencies, group = rep)) +
      geom_point() +
      geom_line() +
      labs(x = "Sampling Time", y = "Mean Depth", title = "Assuming roots can leave the image")

Code

    ggplot(sv2, aes(x = time, y = mean_x_frequencies, group = rep)) +
      geom_point() +
      geom_line() +
      labs(x = "Sampling Time", y = "Mean Depth", title = "Assuming roots can only enter the image")

The same should be true for the median, although being more robust to outliers it may move more slowly.

Code

    ggplot(sv, aes(x = time, y = median_x_frequencies, group = rep)) +
      geom_point() +
      geom_line() +
      labs(x = "Sampling Time", y = "Median Depth", title = "Assuming roots can leave the image")

Code

    ggplot(sv2, aes(x = time, y = median_x_frequencies, group = rep)) +
      geom_point() +
      geom_line() +
      labs(x = "Sampling Time", y = "Median Depth", title = "Assuming roots can only enter the image")

Finally, depth (height as returned by `PlantCV`) should increase in both datasets over time but should be strictly monotone if roots can only enter the images.

Code

    ggplot(sv, aes(x = time, y = height, group = rep)) +
      geom_point() +
      geom_line() +
      labs(x = "Sampling Time", y = "Max Depth", title = "Assuming roots can leave the image")

Code

    ggplot(sv2, aes(x = time, y = height, group = rep)) +
      geom_point() +
      geom_line() +
      labs(x = "Sampling Time", y = "Max Depth", title = "Assuming roots can only enter the image")

### Statistical Analysis

These data are likely to be noisier than above ground phenotypes but the same general methods should be applicable. Here we will show a simple example of longitudinal modeling using `growthSS` and a pairwise comparison via `conjugate` but any methods in other vignettes may be broadly reasonable.

#### Longitudinal Modeling

For the purposes of this example we combine the two datasets as though each is from one genotype where the roots exhibit different behavior in being able to leave the image vs not being able to leave the image and model the root area over time.

Code

    sv$geno <- "a"
    sv2$geno <- "b"
    ex <- rbind(sv, sv2)
    ex$time <- as.numeric(ex$time)

We fit a (perhaps overly) simple linear model to this data using the `nls` backend. Other options include quantile modeling (`type = "nlrq"`), frequentist mixed effect modeling (`type = "nlme"`), General Additive modeling (`type = "mgcv"`), and Bayesian hierarchical modeling (`type = "brms"`).

Code

    ss <- growthSS("int_linear", area ~ time | rep / geno, df = ex, type = "nls")

    ## Individual is not used with type = 'nls'.

Code

    m1 <- fitGrowth(ss)

Any models fit by `fitGrowth` can be visualized using `growthPlot`.

Code

    growthPlot(m1, form = ss$pcvrForm, df = ss$df)

And non-`brms` models can be tested using `testGrowth`. Note that `brms::hypothesis` is a more flexible version of the third example of `testGrowth` below.

We might test that the intercept (amount of roots visible at the first timepoint) is different:

Code

    testGrowth(ss, m1, test = "I")$anova

    ## Analysis of Variance Table
    ##
    ## Model 1: area ~ I + A[geno] * time
    ## Model 2: area ~ I[geno] + A[geno] * time
    ##   Res.Df Res.Sum Sq Df  Sum Sq F value   Pr(>F)
    ## 1     97   81858044
    ## 2     96   76177689  1 5680355  7.1584 0.008773 **
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

That the effect of time is different:

Code

    testGrowth(ss, m1, test = "A")$anova

    ## Analysis of Variance Table
    ##
    ## Model 1: area ~ I[geno] + A * time
    ## Model 2: area ~ I[geno] + A[geno] * time
    ##   Res.Df Res.Sum Sq Df  Sum Sq F value  Pr(>F)
    ## 1     97   83399590
    ## 2     96   76177689  1 7221900  9.1011 0.00327 **
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Or more specific hypotheses on any coefficients such as the slope for the first group being 10% higher than that of the second group (to clarify groups you can always check the data returned by `growthSS`):

Code

    table(ss$df$geno, ss$df$geno_numericLabel)

    ##
    ##      1  2
    ##   a 50  0
    ##   b  0 50

Code

    testGrowth(ss, m1, test = "A1*1.1 - A2")

    ##          Form Estimate       SE  t-value    p-value
    ## 1 A1*1.1 - A2 -318.448 132.4264 2.404718 0.01810283

If you have a more nuanced hypothesis or want to model heteroskedasticity and autocorrelation then other backends as detailed in the intermediate growth modeling and advanced growth modeling tutorials may be of use.

#### Pairwise Comparisons

For non-longitudinal data/hypotheses any standard tests may be useful. Here we’ll only show `conjugate` since it is a departure from the norm.

The `conjugate` function makes pairwise Bayesian comparisons using distributions for which there are conjugate prior distributions that can be easily updated with observed data. This allows for more direct hypothesis testing and Region of Practical Equivalence (ROPE) testing.

Here we might use `conjugate` to compare to compare area on the last day between our two “genotypes”. In this example we’ll use the “T” distribution to run a Bayesian analog to a T-test.

Code

    s1 <- ex[ex$geno == "a" & ex$time == max(ex$time), "area"]
    s2 <- ex[ex$geno == "b" & ex$time == max(ex$time), "area"]

    conj_ex <- conjugate(
      s1, s2, # specify data, here two samples
      method = "t", # use the "T" distribution
      priors = list(mu = 3000, sd = 50), # prior distribution, here it is the same for both samples
      rope_range = c(-500, 500), # differences of <500 pixels deemed not meaningful
      rope_ci = 0.89, cred.int.level = 0.89, # default credible interval lengths
      hypothesis = "equal" # hypothesis to test
    )

The `conjugate` output includes a summary, the posterior distributions in the same format as the priors were supplied, and optionally a plot.

Print the object to view a summary which contains the HDE (highest density estimate) of each group’s posterior distribution, the HDI (highest density interval) of each group’s posterior distribution, the hypothesis that was tested, the posterior probability of that hypothesis, the HDE/HDI for the mean difference (if rope_range was specified), and the probability of the mean difference being within the rope_range.

Code

    conj_ex

    ## Normal distributed Mu parameter of T distributed data.
    ##
    ## Sample 1 Prior Normal(mu = 3000, sd = 50)
    ##  Posterior Normal(mu = 3010, sd = 49.629)
    ## Sample 2 Prior Normal(mu = 3000, sd = 50)
    ##  Posterior Normal(mu = 3082.806, sd = 49.4)
    ##
    ## Posterior probability that S1 is equal to S2 = 46.226%
    ##
    ## Probability of the difference between Mu parameters being within [-500:500] using a 89% Credible Interval is 100% with an average difference of -71.101
    ##
    ##
    ##   HDE_1 HDI_1_low HDI_1_high    HDE_2 HDI_2_low HDI_2_high   hyp post.prob
    ## 1  3010  2930.683   3089.316 3082.806  3003.856   3161.756 equal 0.4622612
    ##    HDE_rope HDI_rope_low HDI_rope_high rope_prob
    ## 1 -71.10135    -182.7525      40.26259         1

The posterior is returned as a list with the same elements as the prior. This allows for Bayesian updating if you wish to do so.

Code

    do.call(rbind, conj_ex$posterior)

    ##      mu       sd
    ## [1,] 3010     49.62892
    ## [2,] 3082.806 49.39955

The plot includes information from the summary graphically as a patchwork of 2 ggplots if rope_range was specified or is a single ggplot otherwise.

Code

    plot(conj_ex)

Here we would conclude that the distributions are different since the probability that they are the same is roughly 1% and that the mean difference is biologically meaningful since the HDI of the mean difference falls entirely outside of our rope_range.

## Multi Value Traits

The multi value traits returned by `analyze_distribution` will often be more difficult to analyze than standard multi value traits that describe spectral wavelengths or indices. The `conjugate` function can use MV traits by specifying matrices/data.frames for s1 and s2, but it will be very rare that a minirhyzotron image’s distribution will follow an easily parameterized pdf.

Code

    pcv.joyplot(df, "x_frequencies", group = c("rep", "time"))

Theoretically we could consider this as a mixture of uniform and gaussian distributions. A mixture of conjugate priors yields a mixture of conjugate posteriors, but this is out of the current scope of `conjugate`. Likewise, the complexity of a non-conjugate mixture model applied to this data does not seem warranted.

The other general option in `pcvr` to analyze multi-value traits is Earth-Mover’s Distance (EMD), which is a distance metric to classify how much “work” it would take to turn one histogram into another. This may be useful for minirhyzotron data depending on your hypothesis.

Here we will show an ad-hoc option to compare the number of “peaks” in the data and an example of using EMD.

### Numbers of Peaks

To compare the number of peaks we need a way to identify a peak. Here we do this with a quick function that finds intervals of counts above some cutoff for at least some duration.

Code

    getPeaks <- function(d = NULL, intensity = 20, duration = 3) {
      binwidth <- as.numeric(unique(diff(d$label)))
      if (length(binwidth) > 1) {
        stop("label column should have constant bin size")
      }
      labels <- sort(d[d$value >= intensity, "label"])
      r <- rle(diff(labels))
      peaks <- sum(r$lengths[r$values == binwidth] >= duration)
      return(peaks)
    }

Code

    d <- split(df, interaction(df[, c("rep", "time")]))
    peak_df <- data.frame(peaks = unlist(lapply(d, getPeaks)))
    rownames(peak_df) <- NULL
    peak_df$rep <- unlist(lapply(names(d), function(nm) {
      return(strsplit(nm, "[.]")[[1]][[1]])
    }))
    peak_df$time <- unlist(lapply(names(d), function(nm) {
      return(strsplit(nm, "[.]")[[1]][[2]])
    }))

Code

    s1 <- peak_df[peak_df$time == min(peak_df$time), "peaks"]
    s2 <- peak_df[peak_df$time == max(peak_df$time), "peaks"]

    conj_ex2 <- conjugate(
      s1, s2, # specify data, here two samples
      method = "poisson", # use the Poisson distribution
      priors = list(a = c(0.5, 0.5), b = c(0.5, 0.5)), # prior distributions for gamma on lambda
      rope_range = c(-1, 1), # differences of <500 pixels deemed not meaningful
      rope_ci = 0.89, cred.int.level = 0.89, # default credible interval lengths
      hypothesis = "equal" # hypothesis to test
    )

Code

    conj_ex2

    ## Gamma distributed Lambda parameter of Poisson distributed data.
    ##
    ## Sample 1 Prior Gamma(a = 0.5, b = 0.5, a = 0.5, b = 0.5)
    ##  Posterior Gamma(a = 16.5, b = 10.5)
    ## Sample 2 Prior Gamma(a = 0.5, b = 0.5, a = 0.5, b = 0.5)
    ##  Posterior Gamma(a = 31.5, b = 10.5)
    ##
    ## Posterior probability that S1 is equal to S2 = 11.874%
    ##
    ## Probability of the difference between Lambda parameters being within [-1:1] using a 89% Credible Interval is 23.896% with an average difference of -1.409
    ##
    ##
    ##     HDE_1 HDI_1_low HDI_1_high    HDE_2 HDI_2_low HDI_2_high   hyp post.prob
    ## 1 1.47619  1.006947   2.234277 2.904762  2.198332   3.900187 equal 0.1187409
    ##    HDE_rope HDI_rope_low HDI_rope_high rope_prob
    ## 1 -1.409191    -2.492106    -0.3709298 0.2389619

Code

    do.call(rbind, conj_ex2$posterior)

    ##      a    b
    ## [1,] 16.5 10.5
    ## [2,] 31.5 10.5

Code

    plot(conj_ex2)

### EMD

Earth Mover’s Distance measures how much work it takes to turn one histogram into another. Since multi- value traits are exported from `PlantCV` as histograms this can be useful for color or distribution analysis. In the following examples we make pairwise comparisons of all our rows and return a long dataframe of those distances. EMD can be computationally heavy with very large datasets since all the pairwise distances have to be calculated. The `mvAg` function may be useful if you need to summarize your data to make EMD faster. If you are only interested in a change of the mean then this is probably not the best way to use your data, but it is a reasonable option for comparing whether groups are more or less self-similar than other groups.

Here is a fast example of EMD. In this simulated data we have five generating distributions. Normal, Log Normal, Bimodal, Trimodal, and Uniform. We could use some gaussian mixtures to characterize the multi-modal histograms but that will get clunky for comparing to the unimodal or uniform distributions. The `conjugate` function would not work here since these distributions do not share a common parameterization. Instead, we can use EMD.

Code

    set.seed(123)

    simFreqs <- function(vec, group) {
      s1 <- hist(vec, breaks = seq(1, 181, 1), plot = FALSE)$counts
      s1d <- as.data.frame(cbind(data.frame(group), matrix(s1, nrow = 1)))
      colnames(s1d) <- c("group", paste0("sim_", 1:180))
      return(s1d)
    }

    sim_df <- rbind(
      do.call(rbind, lapply(1:10, function(i) {
        sf <- simFreqs(rnorm(200, 50, 10), group = "normal")
        return(sf)
      })),
      do.call(rbind, lapply(1:10, function(i) {
        sf <- simFreqs(rlnorm(200, log(30), 0.25), group = "lognormal")
        return(sf)
      })),
      do.call(rbind, lapply(1:10, function(i) {
        sf <- simFreqs(c(rlnorm(125, log(15), 0.25), rnorm(75, 75, 5)), group = "bimodal")
        return(sf)
      })),
      do.call(rbind, lapply(1:10, function(i) {
        sf <- simFreqs(c(rlnorm(100, log(15), 0.25), rnorm(50, 50, 5),
                         rnorm(50, 90, 5)), group = "trimodal")
        return(sf)
      })),
      do.call(rbind, lapply(1:10, function(i) {
        sf <- simFreqs(runif(200, 1, 180), group = "uniform")
        return(sf)
      }))
    )

    sim_df_long <- as.data.frame(data.table::melt(data.table::as.data.table(sim_df), id.vars = "group"))
    sim_df_long$bin <- as.numeric(sub("sim_", "", sim_df_long$variable))

    ggplot(sim_df_long, aes(x = bin, y = value, fill = group), alpha = 0.25) +
      geom_col(position = "identity", show.legend = FALSE) +
      pcv_theme() +
      facet_wrap(~group)

Our plots show very different distributions, so we get EMD between our images and see that we do have some trends shown in the resulting heatmap.

Code

    sim_emd <- pcv.emd(
      df = sim_df, cols = "sim_", reorder = c("group"),
      mat = FALSE, plot = TRUE, parallel = 1, raiseError = TRUE
    )

    ## Estimated time of calculation is roughly 3.1 seconds using 1 cores in parallel.

Code

    sim_emd$plot

Arranging these distances into a network of dissimilarities shows the different distributions clustering well.

Code

    n <- pcv.net(sim_emd$data, filter = "0.5")
    net.plot(n, fill = "group")

Using our simulated mini-rhyzotron data we can go through the same steps. Here we will show this with both simulated datasets (roots leaving the image and roots being stuck in the image once observed).

First we check our distributions via joyplot.

Code

    pcv.joyplot(df, "x_frequencies", group = c("rep", "time"))

We calculate EMD between our observations. Note here we have long input data as opposed to wide in the previous example.

Code

    df1_emd <- pcv.emd(
      df = df, cols = "x_frequencies", reorder = c("rep", "time"),
      id = c("rep", "time"),
      mat = FALSE, plot = TRUE, parallel = 1, raiseError = FALSE
    )

And we arrange the distances as a network of dissimilarities. Here we are filtering for only those edges that are above the 75th percentile in strength and we see a pretty clear temporal clustering.

Code

    n <- pcv.net(df1_emd$data, filter = "0.75")
    net.plot(n, fill = "time")

With our dataset that assumes roots cannot leave the image once observed we get similar results.

Code

    pcv.joyplot(df2, "x_frequencies", group = c("rep", "time"))

Code

    df2_emd <- pcv.emd(
      df = df2, cols = "x_frequencies", reorder = c("rep", "time"),
      id = c("rep", "time"),
      mat = FALSE, plot = TRUE, parallel = 1, raiseError = FALSE
    )
    n2 <- pcv.net(df2_emd$data, filter = "0.75")
    net.plot(n2, fill = "time")

## Conclusion

Root imaging raises several potentially interesting problems around having new phenotypes to consider and noisy data before and after segmentation. It is always important to consider the generating process for your data but that may be especially true where it comes to minirhyzotron image data. Hopefully this vignette helps provide some examples for how these data can be used, but if you have ideas or questions please raise them in the `pcvr` github issues or in the `help-datascience` slack channel for Danforth Center users.