Directional Statistics in PCVR

pcvr v0.1.0

Josh Sumner, DDPSC Data Science Core Facility

2024-04-22

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.simulated {
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}
Code 
library(pcvr)
library(brms) # for rvon_mises
library(ggplot2)
library(patchwork) # for easy ggplot manipulation/combination

What are directional statistics?

Directional (or circular/spherical) statistics is a subset of statistics which focuses on directions and rotations. The reason directional statistics are separated from general statistics is that normally we think about numbers being on a line such that the mean of 4 and 356 would be 180, but if they are degrees in a circle then the mean would be at 0 (360) degrees.

Most distributions that we talk about in statistics are defined on a line. The Beta distribution is defined on the interval [0,1], the normal can exist on a line in [-Inf, Inf], gamma on [0, Inf], etc. Directional statistics allows us to wrap those distributions around a circle but that can sometimes add difficulty to interpretation or extra error prone steps. Currently, pcvr does not support “wrapping” distributions in this way and instead uses to Von-Mises distribution to handle circular data.

The Von-Mises distribution is a mathematically tractable circular distribution that can range from the circular uniform to roughly the circular normal depending on the precision parameter \(\kappa\), with the uniform corresponding to \(\kappa = 0\).

Why are they in pcvr?

This is relevant to pcvr mainly for the color use case. PlantCV returns some single and multi value traits that are circular, hue_circular_mean/median and hue_frequencies. Luckily for simplified plant phenotyping, the Hue circle has red at 0/360 degrees (0/\(2\pi\) in radians) and much of the time we will not have to worry about the circular nature of the data since values are confined to the more green part of the hue circle. Still, for cases where color does wrap around the circle it may be important to your research to take that into account. Those special cases are where the Von-Mises distribution can help you.

conjugate

The simplest way to use the Von-Mises distribution in pcvr is through the conjugate function, where “vonmises” and “vonmises2” are valid methods. As with other conjugate methods these are implemented for single or multi value traits, but unlike other methods these are only necessarily supported in comparing to other samples from the same distribution. If more distributions become tenable to add as circular or wrapped functions then this may be revisted.

“vonmises” method

The “vonmises” method uses the fact that the conjugate prior for the direction parameter (\(\mu\)) is itself a Von-Mises distribution. Utilizing this conjugacy requires that we assume a known \(\kappa\) for the complete distribution so that updating the \(\mu\) parameter is straightforward. Conceptually it may be helpful to consider this similarly to the “T” method for comparing the means of guassians.

Priors for this method should specify a list containing “mu”, “kappa”, “boundary”, “known_kappa”, and “n” elements. In that prior “mu” is the direction of the circular distribution, “kappa” is the precision of the mean, “boundary” is a vector including the two values that are the where the circular data “wraps” around the circle, “known_kappa” is the fixed value of precision for the total distribution, and “n” is the number of prior observations. If the prior is not specified then the default is list(mu = 0, kappa = 1, boundary = c(-pi, pi), known_kappa = 1, n = 1). As per other methods for the conjugate function, the “posterior” part of the output is of the same form as the prior.

Example

First we’ll simulate some multi value data

Code 
mv_gauss <- mvSim(
  dists = list(
    rnorm = list(mean = 50, sd = 10),
    rnorm = list(mean = 60, sd = 12)
  ),
  n_samples = c(30, 40)
)

Next we’ll run conjugate specifying that our data is on a circle defined over [0, 180] with an expected direction around 45 (90 degrees on the full [0,360] or \(\pi/2\) radians) and low precision.

Code 
vm_ex1 <- conjugate(
  s1 = mv_gauss[1:30, -1],
  s2 = mv_gauss[31:70, -1],
  method = "vonmises",
  priors = list(mu = 45, kappa = 1, boundary = c(0, 180), known_kappa = 1, n = 1),
  plot = TRUE, rope_range = c(-5, 5), rope_ci = 0.89,
  cred.int.level = 0.89, hypothesis = "equal"
)

The summary shows normal conjugate output, here showing a posterior probability of ~91% that our samples have equal means (remember the difference in our simulated data is now on a circle).

Code 
vm_ex1$summary
##      HDE_1 HDI_1_low HDI_1_high    HDE_2 HDI_2_low HDI_2_high   hyp post.prob
## 1 49.46678  32.66822   64.46699 59.52812  41.66264   79.59501 equal 0.6375772
##    HDE_rope HDI_rope_low HDI_rope_high rope_prob
## 1 -10.49173    -34.87026      14.28739 0.2237951

Displaying plots of these data can be slower than for other conjugate methods due to the density of the support. To explain, the Von-Mises distribution is defined in on the unit circle [\(-\pi\), \(\pi\)] so in order to have support that works to project that data into whatever space the boundary in the prior specifies the support has to be very dense.

Note also that our rope_range is specified in the boundary units space, which is not necessarily the unit circle.

Code 
vm_ex1$plot # not printed due to being a very dense ggplot

We get very similar results using roughly analogous single value traits.

Code 
vm_ex1_1 <- conjugate(
  s1 = rnorm(30, 50, 10),
  s2 = rnorm(40, 60, 12),
  method = "vonmises",
  priors = list(mu = 0, kappa = 1, known_kappa = 1, boundary = c(0, 180), n = 1),
  plot = FALSE, rope_range = c(-0.1, 0.1), rope_ci = 0.89,
  cred.int.level = 0.89, hypothesis = "equal"
)
do.call(rbind, vm_ex1_1$posterior)
##      mu       kappa    known_kappa n  boundary 
## [1,] 50.04425 7.002869 1           31 numeric,2
## [2,] 57.94215 8.385856 1           41 numeric,2

Single value traits work in the same way. Note that if we omit parts of the prior then they will be filled in with the default prior values.

Code 
set.seed(42)
vm_ex2 <- conjugate(
  s1 = brms::rvon_mises(100, -3.1, 2),
  s2 = brms::rvon_mises(100, 3.1, 2),
  method = "vonmises",
  priors = list(mu = 0, kappa = 1, known_kappa = 2),
  plot = TRUE, rope_range = c(-0.1, 0.1), rope_ci = 0.89,
  cred.int.level = 0.89, hypothesis = "equal"
)

We check our summary and see around 75% chance that these are equal

Code 
vm_ex2$summary
##       HDE_1 HDI_1_low HDI_1_high    HDE_2 HDI_2_low HDI_2_high   hyp post.prob
## 1 -3.071143  2.393966  -2.224163 3.140433  2.381274  -2.297627 equal  0.940651
##     HDE_rope HDI_rope_low HDI_rope_high rope_prob
## 1 0.07243397    -1.081833      1.232842 0.1241434
Code 
do.call(rbind, vm_ex2$posterior)
##      mu        kappa    known_kappa n   boundary 
## [1,] -3.071143 4.204577 2           101 numeric,2
## [2,] 3.140433  4.499838 2           101 numeric,2

Here our plot is much faster to make since the support is a roughly a thirtieth the size of the previous example.

Code 
vm_ex2$plot # not printed due to being a very dense ggplot

Sometimes it may be helpful to use polar coordinates to consider this data, although limitations in plotting area style geometries in the polar coordinates can be frustrating.

Code 
p <- vm_ex2$plot
p[[1]] <- p[[1]] +
  ggplot2::coord_polar() +
  ggplot2::scale_y_continuous(limits = c(-pi, pi))

“vonmises2” method

The “vonmises2” method updates \(\mu\) and \(\kappa\) of the complete Von-Mises distribution. This is done by first taking a weighted average of the prior \(\kappa\) and the MLE of \(\kappa\) based on the sample data then updating \(\mu\) as above.

Priors for this method should specify “mu”, “kappa”, “boundary”, and “n”. Where “mu” is still mean direction, “kappa” is the precision, and boundary/n are as above.

Example

Using the same test data as above we can run the “vonmises2” method.

Code 
vm2_ex1 <- conjugate(
  s1 = mv_gauss[1:30, -1],
  s2 = mv_gauss[31:70, -1],
  method = "vonmises2",
  priors = list(mu = 45, kappa = 1, boundary = c(0, 180), n = 1),
  plot = TRUE, rope_range = c(-5, 5), rope_ci = 0.89,
  cred.int.level = 0.89, hypothesis = "equal"
)
Code 
do.call(rbind, vm2_ex1$posterior)
##      mu       kappa    n  boundary 
## [1,] 49.46678 8.371329 31 numeric,2
## [2,] 59.52812 6.07     41 numeric,2
Code 
set.seed(42)
vm2_ex2 <- conjugate(
  s1 = brms::rvon_mises(100, -3.1, 2),
  s2 = brms::rvon_mises(100, 3.1, 2),
  method = "vonmises2",
  priors = list(mu = 0, kappa = 1),
  plot = TRUE, rope_range = c(-0.75, 0.75), rope_ci = 0.89,
  cred.int.level = 0.89, hypothesis = "equal"
)
Code 
vm2_ex2$plot # much lighter to print this since it is in radians

Plot of the posterior distribution using the vonmises2 method

Code 
do.call(rbind, vm2_ex2$posterior)
##      mu        kappa    n   boundary 
## [1,] -3.071143 2.091375 101 numeric,2
## [2,] 3.140433  2.237544 101 numeric,2

growthSS and brms

The “von_mises” family is an option in brms::brm() and can be used via growthSS by specifying it in the model using the form model = "von_mises: linear". While this will let you specify a Von-Mises model it does not necessarily mean the model will be as ready to go as the default student_t models or gaussian or count models. The Von-Mises family can be more difficult to fit, particularly with non-linear models. Von-Mises mixture models (which may be useful for modeling color changes due to disease or abiotic stress that affects only a part of the plant at a time) are very difficult to fit but can be at least hypothetically very useful.

Example of specifying a circular model in growthSS

Here we set up a model with growthSS only for example purposes

Code 
nReps <- 25
time <- 1:20
muTrend1 <- -2 + (0.25 * time)
muTrend2 <- -1 + (0.2 * time)
kappaTrend1 <- (0.5 * time)
kappaTrend2 <- (0.3 * time)
set.seed(123)
vm2 <- do.call(rbind, lapply(1:nReps, function(rep) {
  do.call(rbind, lapply(time, function(ti) {
    v1 <- brms::rvon_mises(1, muTrend1[ti], kappaTrend1[ti])
    v2 <- brms::rvon_mises(1, muTrend2[ti], kappaTrend2[ti])
    data.frame(y = c(v1, v2), x = ti, group = c("a", "b"), rep = rep)
  }))
}))

ss <- growthSS(
  model = "von_mises: int_linear", form = y ~ x | rep / group, sigma = "int", df = vm2,
  start = NULL, type = "brms"
)
ss$prior # default priors
##   prior class   coef group resp  dpar nlpar lb ub       source
##  (flat)     b                             A            default
##  (flat)     b groupa                      A       (vectorized)
##  (flat)     b groupb                      A       (vectorized)
##  (flat)     b                             I            default
##  (flat)     b groupa                      I       (vectorized)
##  (flat)     b groupb                      I       (vectorized)
##  (flat)     b                   kappa                  default
##  (flat)     b groupa            kappa             (vectorized)
##  (flat)     b groupb            kappa             (vectorized)
Code 
ss$formula # formula specifies kappa based on sigma argument
## y ~ I + A * x 
## autocor ~ tructure(list(), class = "formula", .Environment = <environment>)
## kappa ~ 0 + group
## I ~ 0 + group
## A ~ 0 + group

Example using brms directly

Single Timepoint Model

Code 
set.seed(123)
n <- 1000
vm1 <- data.frame(
  x = c(brms::rvon_mises(n, 1.5, 3), brms::rvon_mises(n, 3, 2)),
  y = rep(c("a", "b"), each = n)
)

basePlot <- ggplot(vm1, aes(x = x, fill = y)) +
  geom_histogram(binwidth = 0.1, alpha = 0.75, position = "identity") +
  labs(fill = "Group") +
  guides(fill = guide_legend(override.aes = list(alpha = 1))) +
  scale_fill_viridis_d() +
  theme_minimal() +
  theme(legend.position = "bottom")

basePlot +
  coord_polar() +
  scale_x_continuous(breaks = c(-2, -1, 0, 1, 2, 3.1415), labels = c(-2, -1, 0, 1, 2, "Pi"))
basePlot + scale_x_continuous(breaks = c(-round(pi, 2), -1.5, 0, 1.5, round(pi, 2)))

prior1 <- set_prior("student_t(3,0,2.5)", coef = "ya") +
  set_prior("student_t(3,0,2.5)", coef = "yb") +
  set_prior("normal(5.0, 0.8)", coef = "ya", dpar = "kappa") +
  set_prior("normal(5.0, 0.8)", coef = "yb", dpar = "kappa")

fit1 <- brm(bf(x ~ 0 + y, kappa ~ 0 + y),
  family = von_mises,
  prior = prior1,
  data = vm1,
  iter = 1000, cores = 2, chains = 2, backend = "cmdstanr", silent = 0, init = 0,
  control = list(adapt_delta = 0.999, max_treedepth = 20)
)
fit1

x <- brmsfamily("von_mises")
pars <- colMeans(as.data.frame(fit))
mus <- pars[grepl("b_y", names(pars))]
x$linkinv(mus) # inverse half tangent function
# should be around 1.5, 3
kappas <- pars[grepl("kappa", names(pars))]
exp(kappas) # kappa is log linked
# should be around 3, 2
pred_draws <- as.data.frame(predict(fit1, newdata = data.frame(y = c("a", "b")), summary = FALSE))
preds <- data.frame(
  draw = c(pred_draws[, 1], pred_draws[, 2]),
  y = rep(c("a", "b"), each = nrow(pred_draws))
)
predPlot <- ggplot(preds, aes(x = draw, fill = y)) +
  geom_histogram(binwidth = 0.1, alpha = 0.75, position = "identity") +
  labs(fill = "Group", y = "Predicted Draws") +
  guides(fill = guide_legend(override.aes = list(alpha = 1))) +
  scale_fill_viridis_d() +
  theme_minimal() +
  theme(legend.position = "bottom")
predPlot + scale_x_continuous(breaks = c(-round(pi, 2), -1.5, 0, 1.5, round(pi, 2)))
predPlot +
  coord_polar() +
  scale_x_continuous(breaks = c(-2, -1, 0, 1, 2, 3.1415), labels = c(-2, -1, 0, 1, 2, "Pi"))

Longitudinal Model

Code 
nReps <- 25
time <- 1:20
muTrend1 <- -2 + (0.25 * time)
muTrend2 <- -1 + (0.2 * time)
kappaTrend1 <- (0.5 * time)
kappaTrend2 <- (0.3 * time)
set.seed(123)
vm2 <- do.call(rbind, lapply(1:nReps, function(rep) {
  do.call(rbind, lapply(time, function(ti) {
    v1 <- rvon_mises(1, muTrend1[ti], kappaTrend1[ti])
    v2 <- rvon_mises(1, muTrend2[ti], kappaTrend2[ti])
    data.frame(y = c(v1, v2), x = ti, group = c("a", "b"), rep = rep)
  }))
}))

ggplot(vm2, aes(x = x, y = y, color = group, group = interaction(group, rep))) +
  geom_line() +
  labs(y = "Y (Von Mises)") +
  theme_minimal()

ggplot(vm2, aes(y = x, x = y, color = group, group = interaction(group, rep), alpha = x)) +
  geom_line() +
  labs(y = "Time", x = "Von Mises") +
  theme_minimal() +
  guides(alpha = "none") +
  coord_polar() +
  scale_x_continuous(
    breaks = c(-2, -1, 0, 1, 2, 3.1415),
    limits = c(-pi, pi),
    labels = c(-2, -1, 0, 1, 2, "Pi")
  )

prior2 <- set_prior("normal(5,0.8)", nlpar = "K") +
  set_prior("student_t(3, 0, 2.5)", nlpar = "I") +
  set_prior("student_t(3, 0, 2.5)", nlpar = "M")

fit2 <- brm(
  bf(y ~ I + M * x,
    nlf(kappa ~ K * x),
    I + M ~ 0 + group,
    K ~ 0 + group,
    autocor = ~ arma(x | rep:group, 1, 1),
    nl = TRUE
  ),
  family = von_mises,
  prior = prior2,
  data = vm2,
  iter = 2000, cores = 4, chains = 4, backend = "cmdstanr", silent = 0, init = 0,
  control = list(adapt_delta = 0.999, max_treedepth = 20)
)
fit2
pars <- colMeans(as.data.frame(fit2))
pars[grepl("^b_", names(pars))]

outline <- data.frame(
  group = rep(c("a", "b"), each = 20),
  x = rep(1:20, 2)
)
probs <- seq(0.01, 0.99, 0.02)
preds <- cbind(outline, predict(fit2, newdata = outline, probs = probs))

pal <- viridis::plasma(n = length(probs))
p2 <- ggplot(preds, aes(y = x)) +
  facet_wrap(~group) +
  lapply(seq(1, 49, 2), function(lower) {
    geom_ribbon(aes(xmin = .data[[paste0("Q", lower)]], xmax = .data[[paste0("Q", 100 - lower)]]),
      fill = pal[lower]
    )
  }) +
  theme_minimal() +
  coord_polar() +
  scale_x_continuous(
    breaks = c(-2, -1, 0, 1, 2, 3.1415),
    limits = c(-pi, pi),
    labels = c(-2, -1, 0, 1, 2, "Pi")
  )

These models can be difficult to fit but they may be useful for your situation in which case the stan forums and pcvr github issues are reasonable places to get help.

Posterior Predictive intervals from a non-linear VM Model