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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
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<title></title>
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<body>
<div class="include-before">
</div>
<div class="frontmatter">
<div class="title"><h1></h1></div>
<div class="author"><h2></h2></div>
<div class="date"><h3></h3></div>
</div>
<div class="body">
<p>Replication output for article, “hermiter: R package for Sequential
Nonparametric Estimation”. The output below corresponds to the output
produced by running knitr::spin() on the article_replication.R script,
with short_run set to FALSE.</p>
<p>Load libraries.</p>
<pre><code class="language-r">options(dplyr.summarise.inform = F)
if(!require(benchden)) devtools::install_github("thmild/benchden")
</code></pre>
<pre><code>## Loading required package: benchden
</code></pre>
<pre><code class="language-r">library(hermiter)
</code></pre>
<pre><code>##
## Attaching package: 'hermiter'
</code></pre>
<pre><code>## The following objects are masked from 'package:stats':
##
## cor, IQR
</code></pre>
<pre><code class="language-r">library(tdigest)
library(microbenchmark)
library(dplyr)
</code></pre>
<pre><code>##
## Attaching package: 'dplyr'
</code></pre>
<pre><code>## The following objects are masked from 'package:stats':
##
## filter, lag
</code></pre>
<pre><code>## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union
</code></pre>
<pre><code class="language-r">library(randtoolbox)
</code></pre>
<pre><code>## Loading required package: rngWELL
</code></pre>
<pre><code>## This is randtoolbox. For an overview, type 'help("randtoolbox")'.
</code></pre>
<pre><code class="language-r">library(ggplot2)
library(patchwork)
library(arrow)
</code></pre>
<pre><code>##
## Attaching package: 'arrow'
</code></pre>
<pre><code>## The following object is masked from 'package:utils':
##
## timestamp
</code></pre>
<pre><code class="language-r">library(colorspace)
</code></pre>
<p>Choose whether to run quick version of reproduction script i.e. under
an hour of processing time on hardware specified in manuscript versus
full version which would take several hours of run-time.</p>
<pre><code class="language-r">short_run <- FALSE
if (short_run == TRUE){
total_number_of_runs <- 10
} else {
total_number_of_runs <- 100
}
</code></pre>
<p>Set random seed for reproducibility.</p>
<pre><code class="language-r">set.seed(10)
</code></pre>
<p>Reproduce the output of the merge code blocks in section 8.5
First block with standardize = FALSE</p>
<pre><code class="language-r">observations_1 <- rlogis(n=1000)
observations_2 <- rlogis(n=1000)
hermite_est_1 <- hermite_estimator(N=50,standardize=FALSE,
observations = observations_1)
hermite_est_2 <- hermite_estimator(N=50,standardize=FALSE,
observations = observations_2)
hermite_est_merged <- merge_hermite(list(hermite_est_1,
hermite_est_2))
hermite_est_full <- hermite_estimator(N=50,standardize=FALSE,observations =
c(observations_1,observations_2))
all.equal(hermite_est_merged,hermite_est_full)
</code></pre>
<pre><code>## [1] TRUE
</code></pre>
<p>Second block with standardize = TRUE</p>
<pre><code class="language-r">observations_1 <- rlogis(n=1000)
observations_2 <- rlogis(n=1000)
hermite_est_1 <- hermite_estimator(N=50,standardize=TRUE,observations=observations_1)
hermite_est_2 <- hermite_estimator(N=50,standardize=TRUE,observations=observations_2)
hermite_est_merged <- merge_hermite(list(hermite_est_1,hermite_est_2))
hermite_est_full <- hermite_estimator(N=50,standardize=TRUE,observations =
c(observations_1,observations_2))
all.equal(hermite_est_merged,hermite_est_full)
</code></pre>
<pre><code>## [1] "Component \"coeff_vec\": Mean relative difference: 0.006074491"
</code></pre>
<p>Reproduce univariate PDF, CDF and Q-Q plot figures i.e. <strong>Figures 1, 2, 3</strong>
in the text respectively.</p>
<pre><code class="language-r">observations <- rlogis(n=5000)
hermite_est <- hermite_estimator(observations = observations)
x <- seq(-15,15,0.1)
pdf_est <- dens(hermite_est,x)
cdf_est <- cum_prob(hermite_est,x)
p <- seq(0.05,0.95,0.05)
quantile_est <- quant(hermite_est,p)
actual_pdf <- dlogis(x)
actual_cdf <- plogis(x)
df_pdf_cdf <- data.frame(x,pdf_est,cdf_est,actual_pdf,actual_cdf)
actual_quantiles <- qlogis(p)
df_quant <- data.frame(p,quantile_est,actual_quantiles)
pdf_comp_plot <- ggplot(df_pdf_cdf,aes(x=x)) +
geom_line(aes(y=pdf_est, colour="Estimated")) +
geom_line(aes(y=actual_pdf, colour="Actual")) +
scale_colour_manual("",
breaks = c("Estimated", "Actual"),
values = c("blue", "black")) + ylab("Probability Density")
pdf_comp_plot / (ggplot(df_pdf_cdf,aes(x=x,y=pdf_est-actual_pdf)) +
geom_line(color="red")+ ylab("Estimated - Actual"))
</code></pre>
<p><img src="data:image/png;base64,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" alt="plot of chunk reproduce_univar" /></p>
<pre><code class="language-r">cdf_comp_plot <- ggplot(df_pdf_cdf,aes(x=x)) +
geom_line(aes(y=cdf_est, colour="Estimated")) +
geom_line(aes(y=actual_cdf, colour="Actual")) +
scale_colour_manual("",
breaks = c("Estimated", "Actual"),
values = c("blue", "black")) +
ylab("Cumulative Probability")
cdf_comp_plot / (ggplot(df_pdf_cdf,aes(x=x,y=cdf_est-actual_cdf)) +
geom_line(color="red")+ ylab("Estimated - Actual"))
</code></pre>
<p><img src="data:image/png;base64,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" alt="plot of chunk reproduce_univar" /></p>
<pre><code class="language-r">quantile_comp_plot <-ggplot(df_quant,aes(x=actual_quantiles)) +
geom_point(aes(y=quantile_est), color="blue") +
geom_abline(slope=1,intercept = 0) +xlab("Theoretical Quantiles") +
ylab("Estimated Quantiles")
quantile_comp_plot / (ggplot(df_quant,aes(x=p,y=quantile_est-
actual_quantiles)) + geom_point(color="red") +
ylab("Estimated - Actual"))
</code></pre>
<p><img src="data:image/png;base64,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" alt="plot of chunk reproduce_univar" /></p>
<p>Reproduce bivariate PDF, CDF figures, namely <strong>Figure 4</strong> and <strong>Figure 5</strong> in
the text respectively.</p>
<pre><code class="language-r">sig_x <- 1
sig_y <- 1
num_obs <- 5000
rho <- 0.5
observations_mat <- mvtnorm::rmvnorm(n=num_obs,mean=rep(0,2),
sigma = matrix(c(sig_x^2,rho*sig_x*sig_y,
rho*sig_x*sig_y,sig_y^2),
nrow=2,ncol=2, byrow = TRUE))
hermite_est <- hermite_estimator(est_type = "bivariate",
observations = observations_mat)
vals <- seq(-5,5,by=0.25)
x_grid <- as.matrix(expand.grid(X=vals, Y=vals))
pdf_est <- dens(hermite_est,x_grid, clipped = TRUE)
cdf_est <- cum_prob(hermite_est,x_grid,clipped = TRUE)
spear_est <- spearmans(hermite_est)
kendall_est <- kendall(hermite_est)
actual_pdf <-mvtnorm::dmvnorm(x_grid,mean=rep(0,2),
sigma = matrix(c(sig_x^2,rho*sig_x*sig_y,
rho*sig_x*sig_y,sig_y^2),
nrow=2,ncol=2, byrow = TRUE))
actual_cdf <- rep(NA,nrow(x_grid))
for (row_idx in seq_len(nrow(x_grid))) {
actual_cdf[row_idx] <- mvtnorm::pmvnorm(lower = c(-Inf,-Inf),
upper=as.numeric(x_grid[row_idx,]),
mean=rep(0,2), sigma =
matrix(c(sig_x^2, rho*sig_x*sig_y,
rho*sig_x*sig_y,sig_y^2), nrow=2,
ncol=2, byrow = TRUE))
}
actual_spearmans <- cor(observations_mat,method = "spearman")[1,2]
actual_kendall <- cor(observations_mat,method = "kendall")[1,2]
df_pdf_cdf <- data.frame(x_grid,pdf_est,cdf_est,actual_pdf,actual_cdf)
p1 <- ggplot(df_pdf_cdf) + geom_tile(aes(X, Y, fill= actual_pdf)) +
scale_fill_continuous_sequential(palette="Oslo",
breaks=seq(0,.2,by=.05),
limits=c(0,.2))
p2 <- ggplot(df_pdf_cdf) + geom_tile(aes(X, Y, fill= pdf_est)) +
scale_fill_continuous_sequential(palette="Oslo",
breaks=seq(0,.2,by=.05),
limits=c(0,.2))
pdf_diff <- ggplot(df_pdf_cdf) + geom_tile(aes(X, Y, fill
= pdf_est - actual_pdf)) +
scale_fill_continuous_sequential(palette="Oslo",
breaks=seq(-.04,.04,by=.04),
limits=c(-.04,.04))
(p1+ ggtitle("Actual PDF")+ theme(legend.title = element_blank()) + p2 +
ggtitle("Estimated PDF") +theme(legend.title = element_blank()) +
plot_layout(guides = 'collect')) / (pdf_diff +
ggtitle("Estimated PDF - Actual PDF") +
theme(legend.title = element_blank()))
</code></pre>
<p><img 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" alt="plot of chunk reproduce_bivar_fig" /></p>
<pre><code class="language-r">p1 <- ggplot(df_pdf_cdf) + geom_tile(aes(X, Y, fill= actual_cdf)) +
scale_fill_continuous_sequential(palette="Oslo",
breaks=seq(0,1,by=.2),
limits=c(0,1))
p2 <- ggplot(df_pdf_cdf) + geom_tile(aes(X, Y, fill= cdf_est)) +
scale_fill_continuous_sequential(palette="Oslo",
breaks=seq(0,1,by=.2),
limits=c(0,1))
cdf_diff <- ggplot(df_pdf_cdf) + geom_tile(aes(X, Y, fill
= cdf_est - actual_cdf)) +
scale_fill_continuous_sequential(palette="Oslo",
breaks=seq(-.02,.02,by=.02),
limits=c(-.02,.02))
(p1+ ggtitle("Actual CDF") + theme(legend.title = element_blank()) + p2 +
ggtitle("Estimated CDF") + theme(legend.title = element_blank())+
plot_layout(guides = 'collect')) / (cdf_diff +
ggtitle("Estimated CDF - Actual CDF") +
theme(legend.title = element_blank()))
</code></pre>
<p><img 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" alt="plot of chunk reproduce_bivar_fig" /></p>
<p>Reproduce actual and estimated Spearman and Kendall correlation coefficient
results for <strong>Table 3</strong> in the text.</p>
<p>Actual Spearmans</p>
<pre><code class="language-r">print(round(actual_spearmans,3))
</code></pre>
<pre><code>## [1] 0.479
</code></pre>
<p>Estimated Spearmans</p>
<pre><code class="language-r">print(round(spear_est,3))
</code></pre>
<pre><code>## [1] 0.475
</code></pre>
<p>Actual Kendall</p>
<pre><code class="language-r">print(round(actual_kendall,3))
</code></pre>
<pre><code>## [1] 0.331
</code></pre>
<p>Estimated Kendall</p>
<pre><code class="language-r">print(round(kendall_est,3))
</code></pre>
<pre><code>## [1] 0.33
</code></pre>
<p>Reproduce quantile estimate results on EUR/USD and GBP/USD as presented in
<strong>Table 4</strong> in the text.</p>
<pre><code class="language-r">spread_data <-
arrow::read_parquet("./eurusd_gbpusd_spread_2021_10.parquet")
percs <- c(0.01,0.1,0.25,0.5,0.75,0.9,0.99)
hermite_ests <- by(spread_data, list(spread_data$hr,spread_data$currency_pair),
function(x){hermite_estimator(observations = log(x$spread_bps+1e-8))})
quantiles_ests <- t(sapply(hermite_ests,FUN=function(x){exp(quant(x,percs))}))
eur_usd_merged <- merge_hermite(as.list(hermite_ests)[1:24])
gbp_usd_merged <- merge_hermite(as.list(hermite_ests)[25:48])
gbp_all_hours <- exp(quant(gbp_usd_merged,percs))
eur_all_hours <- exp(quant(eur_usd_merged,percs))
dim(gbp_all_hours) <- c(1,length(percs))
dim(eur_all_hours) <- c(1,length(percs))
result <- data.frame(currency_pair = rep(attr(hermite_ests,"dimnames")[[2]],
each=24), hour_utc = rep(attr(hermite_ests,"dimnames")[[1]],2),
quantiles_ests)
result <- rbind(result,
data.frame(currency_pair = "EUR/USD", hour_utc = "All", eur_all_hours))
result <- rbind(result,
data.frame(currency_pair = "GBP/USD", hour_utc = "All", gbp_all_hours))
colnames(result) <- c("currency_pair", "hour_utc", paste0("p_",percs*100,"%"))
print(result,digits=1)
</code></pre>
<pre><code>## currency_pair hour_utc p_1% p_10% p_25% p_50% p_75% p_90% p_99%
## 1 EUR/USD 0 0.2 0.3 0.3 0.4 0.4 0.5 0.7
## 2 EUR/USD 1 0.2 0.3 0.3 0.4 0.4 0.4 0.5
## 3 EUR/USD 2 0.2 0.3 0.3 0.4 0.4 0.5 0.6
## 4 EUR/USD 3 0.2 0.3 0.3 0.4 0.4 0.5 0.6
## 5 EUR/USD 4 0.2 0.3 0.3 0.4 0.4 0.5 0.6
## 6 EUR/USD 5 0.2 0.3 0.3 0.4 0.4 0.5 0.6
## 7 EUR/USD 6 0.2 0.3 0.3 0.4 0.4 0.4 0.5
## 8 EUR/USD 7 0.2 0.3 0.3 0.4 0.4 0.5 0.5
## 9 EUR/USD 8 0.2 0.3 0.3 0.4 0.4 0.4 0.5
## 10 EUR/USD 9 0.2 0.3 0.3 0.4 0.4 0.4 0.5
## 11 EUR/USD 10 0.1 0.2 0.3 0.4 0.4 0.5 0.6
## 12 EUR/USD 11 0.2 0.3 0.3 0.4 0.4 0.5 0.6
## 13 EUR/USD 12 0.2 0.3 0.3 0.3 0.4 0.4 0.5
## 14 EUR/USD 13 0.2 0.3 0.3 0.4 0.4 0.4 0.5
## 15 EUR/USD 14 0.2 0.3 0.3 0.4 0.4 0.5 0.5
## 16 EUR/USD 15 0.2 0.3 0.3 0.4 0.4 0.5 0.5
## 17 EUR/USD 16 0.2 0.3 0.3 0.4 0.4 0.5 0.5
## 18 EUR/USD 17 0.2 0.3 0.3 0.4 0.4 0.5 0.6
## 19 EUR/USD 18 0.2 0.3 0.3 0.4 0.4 0.5 0.6
## 20 EUR/USD 19 0.2 0.3 0.3 0.4 0.4 0.5 0.6
## 21 EUR/USD 20 0.3 0.3 0.4 0.5 0.7 0.8 1.0
## 22 EUR/USD 21 1.8 2.6 3.0 4.0 6.0 8.5 15.9
## 23 EUR/USD 22 0.4 0.5 0.6 0.7 0.7 0.9 1.2
## 24 EUR/USD 23 0.3 0.4 0.5 0.5 0.6 0.7 0.7
## 25 GBP/USD 0 0.4 0.5 0.6 0.7 0.9 1.0 1.2
## 26 GBP/USD 1 0.4 0.5 0.6 0.8 0.9 1.0 1.2
## 27 GBP/USD 2 0.4 0.5 0.5 0.7 0.9 1.0 1.2
## 28 GBP/USD 3 0.4 0.5 0.6 0.8 0.9 1.0 1.1
## 29 GBP/USD 4 0.4 0.5 0.6 0.7 0.9 1.0 1.1
## 30 GBP/USD 5 0.4 0.5 0.6 0.7 0.8 0.9 1.2
## 31 GBP/USD 6 0.4 0.5 0.6 0.7 0.8 0.8 1.0
## 32 GBP/USD 7 0.3 0.5 0.6 0.7 0.8 0.9 1.0
## 33 GBP/USD 8 0.3 0.4 0.5 0.6 0.7 0.8 1.0
## 34 GBP/USD 9 0.3 0.4 0.5 0.6 0.7 0.8 1.0
## 35 GBP/USD 10 0.3 0.4 0.5 0.6 0.7 0.8 0.9
## 36 GBP/USD 11 0.4 0.5 0.6 0.7 0.7 0.8 0.9
## 37 GBP/USD 12 0.3 0.4 0.5 0.6 0.7 0.8 1.0
## 38 GBP/USD 13 0.3 0.4 0.5 0.6 0.7 0.8 0.9
## 39 GBP/USD 14 0.3 0.4 0.5 0.6 0.7 0.8 0.9
## 40 GBP/USD 15 0.3 0.4 0.5 0.6 0.7 0.7 0.9
## 41 GBP/USD 16 0.3 0.4 0.5 0.6 0.7 0.8 0.9
## 42 GBP/USD 17 0.3 0.4 0.5 0.6 0.7 0.8 0.9
## 43 GBP/USD 18 0.4 0.5 0.6 0.7 0.8 0.8 0.9
## 44 GBP/USD 19 0.3 0.5 0.6 0.7 0.8 0.8 1.0
## 45 GBP/USD 20 0.5 0.7 0.8 1.0 1.1 1.3 2.3
## 46 GBP/USD 21 2.1 2.5 3.3 5.0 8.9 12.0 24.7
## 47 GBP/USD 22 0.9 1.0 1.1 1.3 1.6 1.9 2.5
## 48 GBP/USD 23 0.7 0.8 0.9 1.0 1.2 1.4 1.6
## 49 EUR/USD All 0.2 0.3 0.3 0.4 0.5 0.6 6.4
## 50 GBP/USD All 0.4 0.5 0.6 0.7 0.9 1.1 9.0
</code></pre>
<p>Reproduce sequential quantile estimate results on EUR/USD and GBP/USD as
presented in <strong>Figure 6</strong> in the text along with sequential Spearman
and Kendall correlation estimates presented in <strong>Figure 7</strong> in the text.</p>
<pre><code class="language-r">eur_data <- spread_data[which(spread_data$currency_pair == "EUR/USD" &
spread_data$time_stamp >= as.POSIXct(as.Date("2021-10-07")) &
spread_data$time_stamp < as.POSIXct(as.Date("2021-10-08"))),]
gbp_data <-spread_data[which(spread_data$currency_pair == "GBP/USD" &
spread_data$time_stamp >= as.POSIXct(as.Date("2021-10-07")) &
spread_data$time_stamp < as.POSIXct(as.Date("2021-10-08"))),]
eur_data <- eur_data[order(eur_data$time_stamp),]
gbp_data <- gbp_data[order(gbp_data$time_stamp),]
h_est_eur <- hermite_estimator(exp_weight_lambda = 0.05)
h_est_gbp <- hermite_estimator(exp_weight_lambda = 0.05)
h_est_bivariate <- hermite_estimator(est_type = "bivariate",
exp_weight_lambda = 0.05)
output_eur <- rep(NA,nrow(eur_data))
output_gbp <- rep(NA,nrow(gbp_data))
output_spearman<- rep(NA,nrow(gbp_data))
output_kendall<- rep(NA,nrow(gbp_data))
for (idx in seq_len(nrow(eur_data))) {
current_obs_eur <- eur_data[idx,]$spread_bps
current_obs_gbp <- gbp_data[idx,]$spread_bps
h_est_eur <- update_sequential(h_est_eur,log(current_obs_eur+1e-8))
h_est_gbp <- update_sequential(h_est_gbp,log(current_obs_gbp +1e-8))
h_est_bivariate <- update_sequential(h_est_bivariate,
c(log(current_obs_eur+1e-8),log(current_obs_gbp+1e-8)))
output_eur[idx] <- exp(quant(h_est_eur,p=0.5))
output_gbp[idx] <- exp(quant(h_est_gbp,p=0.5))
output_spearman[idx] <- spearmans(h_est_bivariate)
output_kendall[idx] <- kendall(h_est_bivariate)
}
output_res_df_eur <- data.frame(time_stamp = eur_data$time_stamp,
median_spread = output_eur, currency_pair="EUR/USD")
output_res_df_gbp <- data.frame(time_stamp = gbp_data$time_stamp,
median_spread = output_gbp, currency_pair="GBP/USD")
output_res_df <- rbind(output_res_df_eur,output_res_df_gbp)
output_spearman_df <- data.frame(time_stamp = eur_data$time_stamp,
correlation=output_spearman, type="spearman" )
output_spearman_df <- output_spearman_df[seq(1,nrow(output_spearman_df),by=10),]
output_kendall_df <- data.frame(time_stamp = eur_data$time_stamp,
correlation=output_kendall , type="kendall" )
output_kendall_df <- output_kendall_df[seq(1,nrow(output_kendall_df),by=10),]
output_correl_df <- rbind(output_spearman_df, output_kendall_df)
output_correl_df <- output_correl_df[order(output_correl_df$time_stamp),]
output_res_df_plot <- output_res_df[seq(1,nrow(output_res_df),by=10),]
ggplot(output_res_df_plot,mapping=aes(x=time_stamp ,y=median_spread,color =
currency_pair)) + geom_line() + xlab("Timestamp") +
ylab("Median Spread (bps)")
</code></pre>
<pre><code>## Warning: Removed 1 row containing missing values (`geom_line()`).
</code></pre>
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAfgAAAH4CAMAAACR9g9NAAABEVBMVEUAAAAAADoAAGYAOmYAOpAAZmYAZrYAv8QzMzM6AAA6ADo6AGY6OmY6OpA6ZmY6ZrY6kLY6kNtNTU1NTW5NTY5NbqtNjshmAABmADpmAGZmOgBmOjpmOpBmZrZmkNtmtttmtv9uTU1uTY5ubqtujshuq+SOTU2OTW6OTY6ObquOjk2OjsiOq+SOyP+QOgCQOjqQOmaQZpCQkLaQtpCQtv+Q2/+rbk2r5P+2ZgC2Zjq2Zma225C2/7a2///Ijk3Ijm7Ijo7IyP/I///bkDrbkGbbtmbb25Db/7bb/9vb///kq27kq47k/8jk///r6+vy8vL4dm3/tmb/yI7/25D/5Kv/5OT//7b//8j//9v//+T///+QKkbgAAAACXBIWXMAAAsSAAALEgHS3X78AAAU2UlEQVR4nO3dC2PbyHHAcdl1tIrv3AuVxPalL6WJ+5avaao7NY2cxEqdWtdjQcuVdfj+H6QLYLFYgABFCQPsLPnfO0t8DGcA/IDFEgCpg5y2l+0g9gTQ4jTg97QBv6cN+D1tW8Avu239kaG2fWTspOrqAz9PUnX1gZ8nqbr6wM+TVF194OdJqq4+8PMkVVcf+HmSqqsP/DxJ1dUHfp6k6uoDP09SdfWBnyepuvrAz5NUXX3g50mqrj7w8yRVVx/4eZKqqw/8PEnV1Qd+nqTq6gM/T1J19YGfJ6m6+sDPk1RdfeDnSaquPvDzJJ2uvnlYVuDnSQo88MJJgZcKBR74FOoDLxUKPPAp1AdeKjQteAO8VCjwwCdQH3ixUOCBT6A+8GKhwAOfQH3gxUKBBz6B+sCLhQIPfAL1gRcLBR74BOoDLxaaFLzh7JxYKPDA668PvFwo8MDrrw+8XCjwwOuvD7xcKPDA668PvFwo8MDrrw+8XCjwwOuvD7xcKPDA668PvFwo8MDrrw+8XCjwM8HTRjRj/1PZ2OKnrZ/yFj9izuMv+Nj1gZcLBR54/fWBlwsFHnj99YGXCwUeeP31gZcLBR54/fWBlwsFHnj99c22X2YLfJykwAMvmhR4uVDggddfH3i5UOCB118feLlQ4IHXXx94uVDggddfH3i5UOCB118feLlQ4IHXXx94uVDggddfH3i5UOCB118feLlQ4IHXXx94uVDggddfH3i5UOCB118feLlQ4IFXXz8DXi4UeODV1wdeMBR44NXXB14wFHjg1dcHXjAUeODV1wdeMBR44NXXB14wFHjg1dcHXjAUeODV1wdeMBR44NXXB14wFHjg1dcHXjAUeODV1wdeMBR44NXXB14wFHjg1dcHXjAUeODV1wdeMBR44NXXB14wFHjg1dcHXjA0JXgDvFwo8MCrrw+8YCjwwKuvD7xgKPDAq69fwW9DrwP+6jT//uvj58CPTZoY/Jvj0/zDq/zNKfAjk6YF/+kPdov/9pvCPj86Opp6Ena5GWP/2f/0tcGu/ncVfM4WPyZpWlt8Cf8t8AJJC/ilSQqefbxE0gThGdVLJE0NvtVGzHn0BR+7PvCCocADr74+8IKhwAOvvn5JDrxMKPDAq68PvGAo8MCrrw+8YCjwwKuvD7xgKPDAq68PvGBoUvDZEnipUOCBV18feMFQ4IFXX7+ET+hiS+CFkgIvGAo88OrrAy8YCjzw6usDLxgKPPDq6wMvGAo88OrrAy8YCjzw6usDLxgKPPDq6wMvGAo88OrrAy8YCjzw6usDLxgKPPDq6wMvGAo88OrrAy8YCrws/OWBbQvgRZPqh19V5pcHJ8ALJlUPf/NX9QO/uQBeLql6+E1txJxHX/Cx66cAvzq8edHT0QM/JmkC8LdfnV0eXj97D7xk0gTgb768OF/YH8BLJk0A/vZff/vyjC1eOGkC8PYN3ZM/vTzrCRkx59EXfOz6KcAPthFzHn3Bx66fArwd0x886enpgR+RNAH429f2rdyqT37EnEdf8LHrJwBfDugZ1QsnTQA+P18UB3F6QkbMefQFH7t+AvDFLr5ojzlWL5g0AfjhNmLOoy/42PVTgGdUP0HSBOAZ1U+RNAF4RvVTJE0APj8/YVQvnlQ9/PCYHvgxSdXDb2oj5jz6go9dXz0819ztKTxX2e4rPNfVT5I0BfjBNmLOoy/42PWBFwwFHnj19YEXDAX+AW3tMloO4ExaXw38WuOQ7aT1JeGvf2i3S7vl3n51dv35zx7/p/13sTo4OMyv//yF/Vk9H15OY6OKxy/LmGfvivgeeE7STJFUEP7m5VleftapgH96lpf/bAd+fnL92YWFc89/dlGeZy3hn57Z2zbGPmvhn7aunee07KT1BeGrvbSDt7eLf6vy2EvzWMnod+fFjctF0RM8ss++a+/lt7gQg/bwZrLyp0QqD3/zsoE/dM94ePvQ5SJ4xeVi9fiifMUg/GAbscpH3+Ji1xfu6ldP3n12YXvwGt535ha+et6uFp/Xu+vqWbtyrDZu8ZeM6uWTig/u8vODP/sLD59Xg7sS3j1v9/P+BeXgzvbkP3hxMgxvV5jzE0b1wknnfztXrAI1fN9HYNfhv7y45GPS0klnh1+5N3a2Pfr1VvB2TVkdAi+cNIUDOKsn//2a07LCSVOAH2wj5jz6go9dPwV4O/r7m6/4YgTZpILw/9ttQvD2Hd/5CV+FIpw0Afjiy49OGNwJJ00AvtziOVYvnDQB+PJYfd+BO+BHJE0AvreXB35kUuAFQ4GX7erP+77HFvhxSSeHL8/In1x/YW/+x1l5qHZhx2v/5h7Iz8uTOPbhR9036p2vQuHsnGzS6eHLY601/BfVSZrrf3cPFCfdzsvVYq1D58jdpPUl4bNW2wT/x7P6gerijeLOZadD56tQJq0/S1f/+KLd1d/+6n3d1a8K0/LOqnMahmvuJq0/e1dfXqLx1/UDxXOXi81bPFfZTpF0JvjCrbiU9ovy2uo/ntUPFFfjVPAb9vF8weEESecZ1S/KXyfVdn7+o19V12QVn3g/L7v6LUb1fMGhbNIU3scPthFzHn3Bx66vH/7mxYntFzhWL5xUP/z5STGkZ1QvnFQ9fDHou1wwqpdOmgJ8ccgHeOGk6uGLD+J8dhF8CgN4kaTq4YszOCe3rzlkK5xUP/yGNmLOoy/42PWBFwwFHnj19aeHL47HVods7bvx6sjrNtdhAD9t/cnhi0/FFx97XJTXXSzKszbbXIfB5+OnrS8Jb1qtdXau/PWbCwe/zXUYrc/Hs8WLJ518i7egtoNfVJ189XOr6zC4ynba+vOcj18til+XC3dVxjbXYXCV7aT1TTbHPj4/r+BPKvitrsPgKttJ608PX7r5UX0Bf7vVdRiM6ietPwP8Qxuj+inrpwDPt15NkDQJeL71Sj5pCvB869UESSXhhRvfejVl/STgB9uIOY+94GPXd/DbyEeCv/nyt7yPl0+qH35TGzHnsRd87Pr64fmbNHsKn/M3aaZImgI8n5adIGkK8Hw+foKkKcDzjRgTJE0CfrCNmPPYCz52/STg+fZq+aQpwPPt1RMkTQGeb6+eIGkK8Hx79QRJU4Dn26snSJoE/GAbMeexF3zs+hZ+2/OywMdJuq/wt685STNBUpMtlcMX19jygQrxpMUHaQp5xfB5efU9gzvZpGnA27aiqxdNWoCrhy9282zxskkTgL/s+XgV8GOT6odnVL+n8JvaiDmPveBj1wdeMhR44LXXB14yFHjgtdcHXjIUeOC11wdeMhR44LXXTxH++6+Pf/wN8OOSpgj/6Z/rWyPmPPaCj10/RfiPPz/+6Xd5fnR0NPUk7HAz9l/mfitrg/AfXuUf/6W8NWKVj73Fxa6f4hZf2QM/KmmK8Fen+YdT4MclTRHejuqf58CPS5oifNNGzHnsBR+7PvCSocADr70+8JKhicEX8sBLhAIPvPb6wEuGAg+89vrAS4YCD7z2+sBLhgIPvPb6wEuGAg+89vrAS4YCD7z2+sBLhgIPvPb6wEuGAg+89vrAS4YCD7z2+sBLhiYLv5Ef+DhJgQdeMCnwkqHAA6+9PvCSocADr70+8JKhqcIb4MeFJgpvgB8ZCjzw2usDLxkKPPDa6wMvGQo88NrrAy8ZmiZ8ZjiAMzIUeOC11wdeMhR44LXXB14yFHjgtddv4A3wo0PThDecjx8bCjzw2usDLxmaJvxdn6cBPk5S4IEXTAq8ZGii8BnwI0OBB157feAlQ4EHXnt94CVDU4Q3S+BHhwIPvPb6wEuGAg+89vrAS4YCD7z2+g4+A14iNEX4JfDjQ4EHXnt94CVDU4O35hnwAqFJwhf/AT8uNEX4YpMHfmQo8MBrrw+8ZGii8OXYfvuswM+TFHjgBZM6+OJw/fIOeYXwtAc3435n1U8zHDl7Y4ufsn69iae5xY+Y89gLPnb9tjTw40KBB157feAlQ4EHXnt94CVDgQdee33gJUOBB157feAlQ4EHXnt94CVDgQdee33gJUOBB157feAlQ4EHXnt94CVD04E3Wesu8ONCgQdeeX3gRUOThd/4URrg4yQFHni5yB74QXvg4ySdBX7TnyADPk5S4IGXiwReNDRleAO8sqTzwG/483PAx0kKPPBykcCLhqYNX4zv+uiBj5N0TvjeIR7wcZLOBF/9fWHg9SQFHni5SNMJLdcDY3rlgY+TdE74JfB6kgIPvFik6Yev7Lvv6YCPk3RW+CXwapJGgnd9vkkDfuN3sj5sGe0RfP1k1eebeiUAPk7SSeCzzfCmHugZ4KMlFaxfLx6zEX7p+nlTHdIBPlJSaXizBXyziiwTg3e91PoT2y6jB4Uqh6/OvJZv1beFXyYGb9pHH836yea8NXZdblpJdgu+2iLugt+UVTV8617f6YfcuLXB+QfP7i68G7Avdw8+b23AndWg9WBeDVz9fi/s2+44gLH9Ipo79E5438PtKHxbPIB3G77r7xy8qd6zBqtL0FXcvTg3LKKeVqeOBe9u7iK8GYBvDkv4jt/t9OqXmDCsicrNcst2N5GJDF+3XYfvDuCcZ+sZr16vDT7Ydfr7Bt9dCVKFD4d4fgG0D1nW23cQH3T+wwO/oWHg4IcTmvHDjPCd/q9sOwnfHqmFd4JZ73tH0N7Td+DD7qA9XAxTLN0atJa9Ozl5+JKB4E7o3S33I5m1qeqMWXcOvntCuQ2/YW78smltH8GQ3wSpzUb49jLuGrj1ydXvpmj1Li349rC1ZzKryHrYujZV94DvyKcD31rsA5vUwGZk2kvU7RX8GMCvA2u+zWvCJ1obv3tx68zHMHwjVddfNuMTn9b0w7emrD7wHu7iu/PfWLs/Phm0RODbc31v+I5ECN884pd9uzcxeR0YugSUfiyxDm88bNOdOPvcj0bD2qb9MuNmynTWhealrQ1+N+GDdl/4nuFbH3zN196nNPDhWMCH+WMI3a7eBE/W50TrB00A7/ffzfrR7nza8P53OKHuDwyuw2fNrSThO+3+8N3XO/hw59pwBocDaviAvOmfTfsh44w8XH3DXfxgmgdN81TYd/jcfg/iZipcI1q7JdeA73947fWmfOfXe0jXb2v1NroG79eL1vvE8kfe9MDNSQPTPGDCcq6L7+xYms7fxblVtNlVLNfgK9RsDX7pO/hsbVifJvyA+/bw1VVK4UJvpawH++Xi7Yd3d037VX6saELzsAcPQrsJws7CQwfnYJZ1F2H64Isf6/Pv14kKvvmxe/CbjlPWr1/eAe+Fq2tVwlF5K3Dtnh+7+XeJwQ6hA98S9UGtXn5pOmFNfxJOc+maLftX/Kwe2Dn94l7POpIE/GDLs+Xa8LWn9cH3hVV4G+Hb8a33baGO6YZ3RV3LukmNGwYGAR7eP7YRvtm7lzv6rLqxS/CZW5mb7qx39t3dMmlvz+EjHUsLftmhDqoXS9OElTs7hlY9+3DPVxNlWTuqbHnrfVn5zTatl5RbdNbbf6+lb4Z76cB3xyf1pp018+/Us/V3rcEA14Xn9bCn3giauGoDyuqvgs3rPehyzaQF7wZXa9c7hZMZzFTWhve9cWuaq804mJ+AbtmQ10/ftYkEL04Bvpq9eq1uwJaBSBYMbprtvonP6t1bHdy56bNWd5qHXGT/DsT43NUk5vWdYC2tV6Ol72XrR+vVxs9Wa932HXPerJmuTph42Tw76iSRRvg8XLOzegxbj1PCxeb7hjrCb/1+IWV+kTZrQu1Wd4VZw5B5M186XC+yQCMMc+tq1kxoON2ugqm38qxZV7K6fpMzr2egmXq/kWfBvx2EX4YQS78V1vv0piMMX9v0/SG8h/QbVLN2BMc7siBDHqxCHtSvJvVPv5PxNZst0oc0rdor+Mnx61JTZdlMw7KZAR+0rB/dpJk+/GBrz/n6AQwX09kyqtAN4//wmbz9mH9v1PQQndAgb2c3kWWd0KCU3wkEyGHS7l6grwHffjbrbhl3JN0Y6ffR7aFEGNqt1qc0dKxlMLI78NtiUje11OG3i+xZpkKXPm1e8R6adfqkewI/KnSeBR+7PvDzJFVXH/h5kqqrD/w8SdXVB36epOrqR4P//uvj58DHqx8N/sOr/M0p8NHqR4P/9pvCPj86Opp6Emgx2iD87yr4nC0+Tv1o8N8CH7V+NHj28XHrR4NnVB+3fjT4po2YnYQXfOz6wM+TVF194OdJqq4+8PMkVVcf+HmSqqsP/DxJ1dUHfp6k6uoDP09SdfUVwK+1SU7b7HnSibION+B1JAV+T5MmAU/bgQb8njbg97QBv6dtGP7jz49//I27/X/f5e4M/dXx8fGr+k59zr51Z2PbLulPv8vlkub51Wn+6e+Of/JWOun9p3Qga2+ie2R9SBuE//i3b/NPf19N5Kd/LCarvibn92/rO/UjrTub2jZJ7RK9kkyavzkuM169kk5qH7nflA5l7U20fdYHtUF4NyV2lfvJ26tytXNX4VVX4pV3yh92Bpo7d1TbJunv/8mu/oJJP/2hDMg/nEonLb3uk3Qoaw3/sIX6sHYXvC374Xm1crrrbqs1vrxTX4nburOpbZP0zalsUhcQPCOW9Krq9rdOOpi1L9H2WR/U7oIPFlq19n38Zfl0a32873a0MaldsqJJq4BP//B2bbLHJrU7j6t7TelQ1t5EsbZ4tzsq5vUX/xPs5Nx0jNrHb0xa9KGSScvFXcSsTfbYpMWUPr9P0qGsvYli7ePrAegb+/P7r4vJqkaY1Q4zGHPaNffeY+U7kpb7eKmk5Y03xWhcOmn5VuFeSQeyrr/nuF/WhzTex+9pA35PG/B72oDf0wb8njbg97TtAPz1Dw9se/TrZ+97nrz5y75HabsAb9vlYuiZ697VgbZD8NfP3n3+s4MfvT5Y5KuDg8P89vXBwcL+ePL+srh77Z4sfh/Gnl4Fbafgn55d2/+fvbNb+fnJ6jC/+fLCbvH2/+JG/eTTs9vXJ7EnOH7bKfgS+frZfxX7/IXd9R9WXb299ejs2j1ZrBXDe4b9aTsJ77pyC35iH1g9vrh5CXyn7SJ81ZtfWv7zRQF/mK+CLZ6uvmy7CP++HNzl5wd2YHfz4smfXhz84MWJh2dwV7bdgL9H4/1d1YDf07Z38LSqAb+nDfg9bcDvaQN+T9v/AwPpWK/DPT5kAAAAAElFTkSuQmCC" alt="plot of chunk unnamed-chunk-7" /></p>
<pre><code class="language-r">ggplot(output_correl_df,mapping=aes(x=time_stamp, y=correlation,color=type)) +
geom_line() +geom_smooth() + xlab("Timestamp") + ylab("Correlation")
</code></pre>
<pre><code>## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'
</code></pre>
<pre><code>## Warning: Removed 2 rows containing non-finite values (`stat_smooth()`).
</code></pre>
<pre><code>## Warning: Removed 2 rows containing missing values (`geom_line()`).
</code></pre>
<p><img 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" alt="plot of chunk unnamed-chunk-7" /></p>
<p>Benchmark hermiter vs tdigest (parallel computation enabled), updating
with 1e6 observations. Reproduces <strong>Figure 8</strong> in the text.</p>
<pre><code class="language-r">obs <- rnorm(1e6)
bench_res <- microbenchmark::microbenchmark(
t_digest = tdigest(obs),
hermite_N_10 = hermite_estimator(N = 10, observations = obs),
hermite_N_20 = hermite_estimator(N = 20, observations = obs),
hermite_N_30 = hermite_estimator(N = 30, observations = obs),
hermite_N_50 = hermite_estimator(N = 50, observations = obs),
times = 20
)
autoplot(bench_res, log = TRUE)
</code></pre>
<pre><code>## Coordinate system already present. Adding new coordinate system, which will replace the existing one.
</code></pre>
<p><img src="data:image/png;base64,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" alt="plot of chunk benchmark_updating" /></p>
<pre><code class="language-r">print(bench_res)
</code></pre>
<pre><code>## Unit: milliseconds
## expr min lq mean median uq max neval
## t_digest 95.7025 96.59245 97.38696 97.25615 98.17650 99.1129 20
## hermite_N_10 23.6555 24.12250 26.94589 24.43730 31.68705 34.7963 20
## hermite_N_20 31.6230 32.28605 36.02094 32.88740 41.31345 42.1547 20
## hermite_N_30 40.2093 40.75345 45.21603 41.59195 50.28870 51.6656 20
## hermite_N_50 56.0286 57.00825 98.95538 57.51765 66.24465 834.4809 20
</code></pre>
<p>Benchmark hermiter vs tdigest (parallel computation disabled), updating
with 1e6 observations. Reproduces <strong>Figure 9</strong> in the text.</p>
<pre><code class="language-r">options(hermiter.parallel = FALSE)
bench_res <- microbenchmark::microbenchmark(
t_digest = tdigest(obs),
hermite_N_10 = hermite_estimator(N = 10, observations = obs),
hermite_N_20 = hermite_estimator(N = 20, observations = obs),
hermite_N_30 = hermite_estimator(N = 30, observations = obs),
hermite_N_50 = hermite_estimator(N = 50, observations = obs),
times = 20
)
autoplot(bench_res, log = TRUE)
</code></pre>
<pre><code>## Coordinate system already present. Adding new coordinate system, which will replace the existing one.
</code></pre>
<p><img src="data:image/png;base64,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" alt="plot of chunk benchmark_updating_serial" /></p>
<pre><code class="language-r">print(bench_res)
</code></pre>
<pre><code>## Unit: milliseconds
## expr min lq mean median uq max neval
## t_digest 95.9115 96.73475 97.19124 97.15615 97.7654 98.1648 20
## hermite_N_10 127.3002 129.02160 148.69904 132.63175 147.8537 376.5789 20
## hermite_N_20 251.0796 253.89215 290.41222 272.84590 275.1293 494.8028 20
## hermite_N_30 335.3084 345.46255 384.90503 363.59410 383.3717 596.9861 20
## hermite_N_50 557.2351 580.46980 613.15998 591.73035 605.2234 836.7987 20
</code></pre>
<pre><code class="language-r">options(hermiter.parallel = TRUE)
</code></pre>
<p>Benchmark hermiter vs tdigest, quantile estimation. Reproduces <strong>Figure 10</strong>
in the text.</p>
<pre><code class="language-r">obs <- rnorm(1e6)
td <- tdigest(obs)
h_est <-
hermite_estimator(observations = obs)
p_1 <- 0.5
p_100 <- seq(0.01, 1, 0.01)
p_10000 <- seq(0.0001, 1, 0.0001)
p_100000 <- seq(0.00001, 1, 0.00001)
bench_res <- microbenchmark::microbenchmark(
hermite_1_quantile = quant(h_est, p = p_1),
tdigest_1_quantile = quantile(td, probs = p_1),
hermite_100_quantiles = quant(h_est, p = p_100),
tdigest_100_quantiles = quantile(td, probs = p_100),
hermite_10_000_quantiles = quant(h_est, p = p_10000),
tdigest_10_000_quantiles = quantile(td, probs = p_10000),
hermite_100_000_quantiles = quant(h_est, p = p_100000),
tdigest_100_000_quantiles = quantile(td, probs = p_100000),
times = 1e2
)
autoplot(bench_res,
log = T,
xlab = "Algorithm",
ylab = "Time (millis)")
</code></pre>
<pre><code>## Coordinate system already present. Adding new coordinate system, which will replace the existing one.
</code></pre>
<p><img src="data:image/png;base64,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" alt="plot of chunk benchmark_quantile_est" /></p>
<pre><code class="language-r">print(bench_res)
</code></pre>
<pre><code>## Unit: microseconds
## expr min lq mean median uq max neval
## hermite_1_quantile 236.3 377.25 435.886 431.90 473.30 1038.5 100
## tdigest_1_quantile 10.1 12.65 23.262 18.85 31.50 99.7 100
## hermite_100_quantiles 281.7 387.05 440.205 426.50 493.20 634.8 100
## tdigest_100_quantiles 13.0 15.85 26.119 21.45 34.70 54.8 100
## hermite_10_000_quantiles 464.3 662.45 764.463 734.05 785.60 2467.7 100
## tdigest_10_000_quantiles 243.0 263.80 281.666 274.60 293.90 430.3 100
## hermite_100_000_quantiles 2955.1 3059.70 3585.874 3141.00 3224.05 44070.6 100
## tdigest_100_000_quantiles 2355.0 2457.55 2504.704 2489.05 2526.90 3501.5 100
</code></pre>
<p>Univariate simulation study comparing hermiter and tdigest for quantile
estimation.</p>
<pre><code class="language-r">calculate_miae_per_distro <- function(full_miae = FALSE) {
distros_index <- c(1:5, 7:8, 11, 13:17, 21:28)
numruns <- total_number_of_runs
num_obs_vec <- c(1e4, 1e5, 1e6, 1e7)
if (full_miae == TRUE) {
p <- randtoolbox::sobol(1000)
norm_factor <- 1
} else {
p <- randtoolbox::sobol(1000) * 0.98 + 0.01
norm_factor <- 0.98
}
distr_name_all <- c()
num_obs_all <- c()
mae_hermite_quant <- c()
mae_t_digest_quant <- c()
count <- 0
for (num_obs in num_obs_vec) {
for (current_distro_idx in seq_along(distros_index)) {
dnum <- distros_index[current_distro_idx]
distr_name <- benchden::berdev(dnum)$name
r_func <-
function(core_obs) {
benchden::rberdev(n = core_obs, dnum = dnum)
}
q_func <-
function(p_est) {
benchden::qberdev(p_est, dnum = dnum)
}
p_func <- function(x) {
benchden::pberdev(x, dnum = dnum)
}
res_hermite_quant <- rep(0, numruns)
res_t_digest_quant <- rep(0, numruns)
for (run in c(1:numruns)) {
obs <- r_func(num_obs)
h_est <- hermite_estimator(observations = obs )
td <- tdigest(obs)
q_est_hermite <- h_est %>% quant(p)
q_est_t_digest <- quantile(td, probs = p)
true_quant <- q_func(p)
res_hermite_quant[run] <- norm_factor *
mean(abs(q_est_hermite - true_quant))
res_t_digest_quant[run] <- norm_factor *
mean(abs(q_est_t_digest - true_quant))
}
mae_herm_quant <- mean(res_hermite_quant)
mae_t_dig_quant <- mean(res_t_digest_quant)
count <- count + 1
distr_name_all[count] <- distr_name
num_obs_all[count] <- num_obs
mae_hermite_quant[count] <- mae_herm_quant
mae_t_digest_quant[count] <- mae_t_dig_quant
}
}
result <- data.frame(
distribution_name = distr_name_all,
num_obs =
num_obs_all,
mae_hermite_quant,
mae_t_digest_quant
)
result <- result %>% mutate(hermite_better_quant =
ifelse(mae_hermite_quant < mae_t_digest_quant,
1, 0))
return(result)
}
</code></pre>
<pre><code class="language-r">calculate_miae <- function(miae_per_distro) {
univariate_quantile_results <- miae_per_distro %>%
group_by(num_obs) %>%
summarise(
num_herm_better = sum(hermite_better_quant),
total_distros = n(),
perc_herm_better =
sum(hermite_better_quant) / n()
)
return(univariate_quantile_results)
}
univar_quant_results_partial_per_distro <-
calculate_miae_per_distro(full_miae = FALSE)
univar_quant_results_full_per_distro <-
calculate_miae_per_distro(full_miae = TRUE)
univar_quant_results_partial <-
calculate_miae(univar_quant_results_partial_per_distro)
univar_quant_results_full <-
calculate_miae(univar_quant_results_full_per_distro)
</code></pre>
<p>Reproduces <strong>Table 5</strong> in the text:</p>
<pre><code class="language-r">print(univar_quant_results_full)
</code></pre>
<pre><code>## # A tibble: 4 × 4
## num_obs num_herm_better total_distros perc_herm_better
## <dbl> <dbl> <int> <dbl>
## 1 10000 12 21 0.571
## 2 100000 17 21 0.810
## 3 1000000 18 21 0.857
## 4 10000000 18 21 0.857
</code></pre>
<p>Reproduces <strong>Table 6</strong> in the text:</p>
<pre><code class="language-r">print(univar_quant_results_partial)
</code></pre>
<pre><code>## # A tibble: 4 × 4
## num_obs num_herm_better total_distros perc_herm_better
## <dbl> <dbl> <int> <dbl>
## 1 10000 13 21 0.619
## 2 100000 17 21 0.810
## 3 1000000 18 21 0.857
## 4 10000000 19 21 0.905
</code></pre>
<p>R implementation of count matrix algorithm of Xiao, Wei. “Novel online
algorithms for nonparametric correlations with application to analyze sensor
data.” 2019 IEEE International Conference on Big Data (Big Data). IEEE,
as an S3 class. The implementation below follows
<a href="https://github.com/wxiao0421/onlineNPCORR/">https://github.com/wxiao0421/onlineNPCORR/</a> reasonably closely in parts.</p>
<pre><code class="language-r">count_matrix_calculator <-
function(cut_points_inpt, normalize = FALSE) {
this <- list(
cut_points = cut_points_inpt,
count_matrix = matrix(
rep(0, cut_points_inpt ^ 2),
nrow = cut_points_inpt,
ncol = cut_points_inpt,
byrow = TRUE
),
n_row = rep(0, cut_points_inpt),
n_col = rep(0, cut_points_inpt),
x_breaks = qnorm(p = seq(0, 1, length.out =
(
cut_points_inpt + 1
)))[2:(cut_points_inpt)],
y_breaks = qnorm(p = seq(0, 1, length.out =
(
cut_points_inpt + 1
)))[2:(cut_points_inpt)],
num_obs = 0,
normalize_obs = normalize,
running_mean_x = 0,
running_mean_y = 0,
running_variation_x = 0,
running_variation_y = 0
)
class(this) <- append(class(this), "count_matrix_calculator")
return(this)
}
get_idx <- function(x, breaks) {
if (breaks[length(breaks)] < x) {
return(length(breaks) + 1)
}
return(which(breaks >= x)[1])
}
update_matrix <- function(this, x) {
UseMethod("update_matrix", this)
}
update_matrix.count_matrix_calculator <- function(this, x) {
this$num_obs <- this$num_obs + 1
if (this$normalize_obs == TRUE) {
prev_mean <- c(this$running_mean_x, this$running_mean_y)
upd_mean <- (prev_mean * (this$num_obs - 1) + x) / this$num_obs
this$running_mean_x <- upd_mean[1]
this$running_mean_y <- upd_mean[2]
if (this$num_obs < 2) {
return(this)
}
upd_var <- c(this$running_variation_x,
this$running_variation_y) + (x - prev_mean) *
(x - upd_mean)
this$running_variation_x <- upd_var[1]
this$running_variation_y <- upd_var[2]
x <- (x - upd_mean) / sqrt(upd_var / (this$num_obs))
}
idx_row <- get_idx(x[1], this$x_breaks)
idx_col <- get_idx(x[2], this$y_breaks)
this$count_matrix[idx_row, idx_col] <-
this$count_matrix[idx_row, idx_col] + 1
this$n_row[idx_row] <- this$n_row[idx_row] + 1
this$n_col[idx_col] <- this$n_col[idx_col] + 1
return(this)
}
get_spearmans <- function(this) {
UseMethod("get_spearmans", this)
}
get_spearmans.count_matrix_calculator <- function(this) {
len_x_breaks <- length(this$x_breaks)
len_y_breaks <- length(this$y_breaks)
r_row <- rep(0, len_x_breaks + 1)
r <- 0
for (k in c(1:(len_x_breaks + 1))) {
if (this$n_row[k] == 0) {
r_row[k] <- r
} else {
r_row[k] <- ((r + 1) + (r + this$n_row[k])) / 2
r <- r + this$n_row[k]
}
}
r_col <- rep(0, len_y_breaks + 1)
r <- 0
for (k in c(1:(len_y_breaks + 1))) {
if (this$n_col[k] == 0) {
r_col[k] <- r
} else {
r_col[k] <- ((r + 1) + (r + this$n_col[k])) / 2
r <- r + this$n_col[k]
}
}
r_row <- r_row - (this$num_obs + 1) / 2
r_col <- r_col - (this$num_obs + 1) / 2
r_row <- r_row / sqrt(sum(this$n_row * r_row ^ 2))
r_col <- r_col / sqrt(sum(this$n_col * r_col ^ 2))
corr <- t(r_row) %*% this$count_matrix %*% r_col
return(corr)
}
get_kendall <- function(this) {
UseMethod("get_kendall", this)
}
get_kendall.count_matrix_calculator <- function(this) {
len_n_row <- length(this$n_row)
len_n_col <- length(this$n_col)
count_mat_sum <-
matrix(rep(0, len_n_row * len_n_col), len_n_row, len_n_col, byrow = TRUE)
for (i in 2:len_n_row) {
count_mat_sum[i, 2:len_n_col] <-
cumsum(this$count_matrix[(i - 1), 1:(len_n_col - 1)])
}
for (i in 2:len_n_row) {
count_mat_sum[i, ] <- count_mat_sum[i, ] + count_mat_sum[(i - 1), ]
}
concord_pairs <- sum(this$count_matrix * count_mat_sum)
ties_in_x <- 0
for (i in 1:len_n_row) {
ties_in_x <- ties_in_x + (this$n_row[i] ^ 2 -
sum(this$count_matrix[i, ] ^ 2)) / 2
}
ties_in_y <- 0
for (j in 1:len_n_col) {
ties_in_y <- ties_in_y + (this$n_col[j] ^ 2 -
sum(this$count_matrix[, j] ^ 2)) / 2
}
ties_in_x_and_y <- sum(this$count_matrix * (this$count_matrix - 1)) / 2
discord_pairs <- this$num_obs * (this$num_obs - 1) / 2 - concord_pairs -
ties_in_x - ties_in_y - ties_in_x_and_y
corr <- (concord_pairs - discord_pairs) /
sqrt((concord_pairs + discord_pairs + ties_in_x) *
(concord_pairs + discord_pairs + ties_in_y))
return(corr)
}
</code></pre>
<p>Bivariate simulation study comparing hermiter and count matrix
algorithms for estimation of Spearman’s Rho and Kendall Tau coefficients.</p>
<pre><code class="language-r">rho_inpt <- c(-0.75, -0.5, -0.25, 0.25, 0.5, 0.75)
num_obs_inpt <- c(1e4, 5e4, 1e5)
runs <- total_number_of_runs
mae_matrix_kendall <- c()
mae_hermite_kendall <- c()
mae_matrix_spear <- c()
mae_hermite_spear <- c()
rho_vec <- c()
num_obs_vec <- c()
sig_x <- 1
sig_y <- 1
for (num_obs in num_obs_inpt) {
for (rho in rho_inpt) {
for (j in 1:runs) {
obs_mat <- mvtnorm::rmvnorm(
n = num_obs,
mean = rep(0, 2),
sigma = matrix(
c(sig_x ^ 2, rho * sig_x * sig_y,
rho * sig_x * sig_y, sig_y ^ 2),
nrow = 2,
ncol = 2,
byrow = T
)
)
matrix_est_c30 <-
count_matrix_calculator(cut_points_inpt = 30,
normalize = F)
matrix_est_c100 <-
count_matrix_calculator(cut_points_inpt = 100,
normalize = F)
hermite_est <-
hermite_estimator(est_type = "bivariate", standardize = F)
for (i in seq_len(nrow(obs_mat))) {
matrix_est_c100 <- matrix_est_c100 %>% update_matrix(obs_mat[i, ])
matrix_est_c30 <-
matrix_est_c30 %>% update_matrix(obs_mat[i, ])
hermite_est <-
hermite_est %>% update_sequential(obs_mat[i, ])
}
kendall_est_matrix <-
matrix_est_c100 %>% get_kendall()
kendall_est_hermite <- hermite_est %>% kendall()
spear_est_matrix <-
matrix_est_c30 %>% get_spearmans()
spear_est_hermite <- hermite_est %>% spearmans()
kendall_true <- 2 / pi * asin(rho)
spear_true <- cor(obs_mat, method = "spearman")[1, 2]
mae_matrix_kendall <- append(mae_matrix_kendall,
abs(kendall_est_matrix - kendall_true))
mae_hermite_kendall <- append(mae_hermite_kendall,
abs(kendall_est_hermite - kendall_true))
mae_matrix_spear <- append(mae_matrix_spear,
abs(spear_est_matrix - spear_true))
mae_hermite_spear <- append(mae_hermite_spear,
abs(spear_est_hermite - spear_true))
num_obs_vec <- append(num_obs_vec, num_obs)
rho_vec <- append(rho_vec, rho)
}
}
}
result <-
data.frame(
num_obs_vec,
rho_vec,
mae_matrix_kendall,
mae_hermite_kendall,
mae_matrix_spear,
mae_hermite_spear
)
summary_by_rho_and_num_obs <-
result %>%
group_by(num_obs_vec, rho_vec) %>%
summarise(
mae_matrix_kendall = mean(mae_matrix_kendall),
mae_hermite_kendall =
mean(mae_hermite_kendall),
mae_matrix_spear = mean(mae_matrix_spear),
mae_hermite_spear = mean(mae_hermite_spear)
)
summary_num_obs_kendall <- summary_by_rho_and_num_obs %>%
group_by(num_obs_vec) %>%
summarise(
mae_matrix_kendall_avg =
mean(mae_matrix_kendall) * 100,
mae_hermite_kendall_avg =
mean(mae_hermite_kendall) * 100,
sd_matrix_kendall =
sd(mae_matrix_kendall) * 100,
sd_hermite_kendall = sd(mae_hermite_kendall) * 100
)
summary_num_obs_spear <- summary_by_rho_and_num_obs %>%
group_by(num_obs_vec) %>%
summarise(
mae_matrix_spear_avg =
mean(mae_matrix_spear) * 100,
mae_hermite_spear_avg = mean(mae_hermite_spear) * 100,
sd_matrix_spear = sd(mae_matrix_spear) * 100,
sd_hermite_spear =
sd(mae_hermite_spear) * 100
)
</code></pre>
<p>Reproduces <strong>Table 7</strong> in the text.</p>
<pre><code class="language-r">print(summary_num_obs_spear)
</code></pre>
<pre><code>## # A tibble: 3 × 5
## num_obs_vec mae_matrix_spear_avg mae_hermite_spear_avg sd_matrix_spear sd_hermite_spear
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 10000 0.103 0.146 0.0288 0.0143
## 2 50000 0.0992 0.0646 0.0281 0.00474
## 3 100000 0.0995 0.0441 0.0288 0.00314
</code></pre>
<p>Reproduces <strong>Table 8</strong> in the text.</p>
<pre><code class="language-r">print(summary_num_obs_kendall)
</code></pre>