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February 12, 2021 15:28
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Medians vs Log-Means
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| #' @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? |
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