Medians vs Log-Means

medians_vs_logmeans.R

#' @title Log-means versus Medians
#' @author Daniel Egan
#' @description When data has a power law or extremely skewed distribution,
#' using a log-mean usually results in more stable and useful central estimates
#' compared to a mean or a median.
#' https://towardsdatascience.com/on-average-youre-using-the-wrong-average-geometric-harmonic-means-in-data-analysis-2a703e21ea0

library(dplyr)
library(tidyr)
library(ggplot2)

rdata<- 10^rnorm(30000, mean=3, sd=3)
n_samples <- 200
means_df <- data.frame(sample_id = 1:n_samples,
                       sample_median = NA_real_,
                       sample_mean = NA_real_,
                       sample_log_mean = NA_real_)

log10mean <- function(x){
  10 ^ (mean(log10(x)))
}
# log10mean(rdata[1:10])


for(i in 1:n_samples){
  this_sample <- sample(rdata, size = 200)
  means_df$sample_median[i] <- median(this_sample)
  means_df$sample_mean[i] <- mean(this_sample)
  means_df$sample_log_mean[i] <- log10mean(this_sample)
}


means_df %>%
  pivot_longer(c(sample_median, sample_mean, sample_log_mean), names_to = "type") %>%
  ggplot(aes(x = value, group = type)) +
  geom_density() +
  facet_wrap(. ~ type, scales = "free") +
  labs(title = "Sample means using different methods",
       subtitle = "Note fewer outliers/ more normal distribution for median & log-mean.")


#' Suppose we used each method and looked at out-of-sample
#' error, i.e. the RMSE  have compared to another.
rmse <- function(x){
  sqrt(mean(x))
}

median_ratio <- data.frame(type = "median", ratio = abs(as.vector(outer(X = means_df$sample_mean,Y = means_df$sample_median, FUN = "-"))))
mean_ratio <- data.frame(type = "mean", ratio = abs(as.vector(outer(X = means_df$sample_mean,Y = means_df$sample_mean, FUN = "-"))))
logmean_ratio <- data.frame(type = "logmean", ratio = abs(as.vector(outer(X = means_df$sample_mean,Y = means_df$sample_log_mean, FUN = "-"))))

ratios_df <- rbind(median_ratio, mean_ratio, logmean_ratio)
ratios_df %>%
  ggplot(aes(sample = ratio, group = type)) +
  geom_qq_line() +
  facet_wrap(. ~ type, scales = ) +
  labs(title = "Distribution of absolute errors",
       subtitle = "Note fewer outliers/ more normal distribution for median & log-mean.")

#' Log-means vs medians?
means_df %>%
  ggplot(aes(x = sample_median, y = sample_log_mean)) +
  geom_point() +
  geom_abline(slope = 1, intercept = 0) +
  coord_cartesian(xlim = c(0, 5000), ylim = c(0, 5000))
  labs(title = "Distribution of absolute errors",
       subtitle = "Note fewer outliers/ more normal distribution for median & log-mean.")



#' What about with data that aren't perfect base-10 generated dude?