Clustering of Currency Exchange Rates Using Hierarchical Methods Based on Dynamic Time Warping

Clustering the daily spot exchange rates of the 30 currencies.

In this study, we collect and analyze historical exchange rate data for the 30 currencies listed in the table below, considering the US dollar as reference. Reuters is the source of the data and you can access it by downloading post29.xlsx file here.

country	currency
Argentine	USDARS
Brazil	USDBRL
Bulgaria	USDBGN
Chile	USDCLP
Colombia	USDCOP
Czech Republic	USDCZK
Egypt	USDEGP
Hungary	USDHUF
India	USDINR
Indonesia	USDIDR
Kazakhstan	USDKZT
Kenya	USDKES
Malaysia	USDMYR
Mexico	USDMXN
Nigeria	USDNGN
Pakistan	USDPKR
Peru	USDPEN
Philippines	USDPHP
Poland	USDPLN
Romania	USDRON
Russia	USDRUB
Singapore	USDSGD
South Africa	USDZAR
South Korea	USDKRW
Sri Lanka	USDLKR
Taiwan	USDTWD
Thailand	USDTHB
Turkey	USDTRY
Ukranie	USDUAH
Vietnam	USDVND

library(tidyverse)
library(dtwclust)

By calculating the average, we can convert the daily frequency data to monthly data.

df <- readxl::read_excel("data.xlsx") %>% 
  na.omit() %>% 
  pivot_longer(!DATE, names_to = "Vars", values_to = "Vals") %>% 
  mutate(
    Months = format(DATE,"%m"),
    Years = format(DATE,"%Y")
  ) %>% 
  group_by(Vars,Months,Years) %>% 
  summarise(
    MeanVals = mean(Vals)
  ) %>% 
  ungroup() %>% 
  arrange(Vars,Years) %>% 
  mutate(
    DATE = as.Date(paste0(Years,"-",Months,"-",1))
  ) %>% 
  select(DATE,Vars,MeanVals) %>% 
  pivot_wider(names_from = "Vars", values_from = "MeanVals")

It would be better if we standardize the data. The following formula can be used to standardize the values in a dataset.

\(z = \frac{(x - \mu)}{\sigma}\)

\(z\): Standardised value or Z-score

\(x\): Original value

\(\mu\): Sample mean

\(\sigma\): Sample standard deviation

standardize <- function(x){
  
  (x - mean(x)) / sd(x)
  
}

df_standardize <- df %>% 
  mutate_at(
    vars(-DATE), function(x) standardize(x)
  )

df_standardize %>% 
  pivot_longer(!DATE, names_to = "Vars", values_to = "Vals") %>% 
  ggplot(aes(x = DATE, y = Vals)) +
  geom_line() +
  facet_wrap(~Vars, ncol = 5) +
  theme_minimal() +
  theme(axis.title = element_blank(),
        axis.text.x = element_text(size = 15),
        axis.text.y = element_blank(),
        strip.text = element_text(size = 20))

What happens when the data is standardised? An example is provided below. I’d like to draw your attention to the y-axis.

Type: h or Hierarchical

An algorithm called hierarchical clustering divides objects into clusters based on how similar they are. The result is a collection of clusters, each of which differs from the others while having things that are generally similar to one another.

Distance: dtw or Dynamic Time Warping

We can explain DTW by comparing it to the Euclidean distance. The distance between two points in Euclidean space is known as the Euclidean distance.

Figure 1: https://rtavenar.github.io/blog/dtw.html

Dynamic Time Warping uses temporal distortions to create the best possible alignment, whereas Euclidean only allows one-to-one point comparison.

As a result of various trials, I decided to create 3 different clusters.

k <- 3L

data_cluster <- tsclust(
  t(df_standardize[,-1]), # data
  type = "h", # What type of clustering method to use
  k = k, # Number of desired clusters
  distance = "dtw" # Dynamic time warping
)
cluster <- as.data.frame(cutree(data_cluster, k=k)) %>% 
  rownames_to_column(., var = "Vars") %>% 
  rename("Cluster"=2)

Vars	Cluster
ARS	1
EGP	1
HUF	1
IDR	1
INR	1
KES	1
KZT	1
LKR	1
MXN	1
NGN	1
PKR	1
RON	1
RUB	1
TRY	1
UAH	1
VND	1
ZAR	1
BGN	2
BRL	2
CLP	2
COP	2
CZK	2
KRW	2
MYR	2
PEN	2
PHP	2
PLN	2
SGD	3
THB	3
TWD	3

df2 <- df_standardize %>% 
  pivot_longer(!DATE, names_to = "Vars", values_to = "Vals") %>% 
  arrange(Vars) %>% 
  left_join(cluster, by = "Vars")

for(i in 1:length(unique(cluster$Cluster))){
  
  g <- ggplot(df2 %>% filter(Cluster == i), aes(x = DATE, y = Vals)) +
    geom_line(data = df2 %>% filter(Cluster == i) %>% rename(Vars2 = Vars), aes(group = Vars2), color = "gray", size = 1) +
    geom_line(color = "dark blue", size = 2) +
    ggthemes::theme_fivethirtyeight() +
    theme(strip.text = element_text(size = 20),
          axis.text = element_blank()) +
    facet_wrap(~Vars)
  
  plot(g)
  
}