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    Time Series Forecasting

    -- Currency Exchange Rate Forecasting with ARIMA and STL

    This example shows time series forecasting of Euro-AUD exchange rates with the with the ARIMA and STL models. The data used are historical currency exchange rates from January 1999 to June 2014 provided by the European Central Bank.

    This example was produced with R Markdown. The Rmd and R source code files are provided at the bottom of this page.

    1. Downloading data from European Central Bank

    Download data from the European Central Bank at http://www.ecb.europa.eu/stats/exchange/eurofxref/html/index.en.html.

    url <- "http://www.ecb.europa.eu/stats/eurofxref/eurofxref-hist.zip" download.file(url, "eurofxref-hist.zip")

    2. Checking data

    rates <- read.csv(unz("eurofxref-hist.zip", "eurofxref-hist.csv"), 
                      header = T) rates[1:2, ]
    ##         Date   USD   JPY    BGN CYP   CZK   DKK EEK    GBP   HUF   LTL LVL
    ## 1 2014-07-01 1.369 139.0 1.9558 N/A 27.43 7.456 N/A 0.7981 310.4 3.453 N/A
    ## 2 2014-06-30 1.366 138.4 1.9558 N/A 27.45 7.456 N/A 0.8015 309.3 3.453 N/A
    ##   MTL   PLN ROL    RON   SEK SIT SKK   CHF ISK   NOK   HRK     RUB TRL
    ## 1 N/A 4.158 N/A 4.3881 9.160 N/A N/A 1.214 N/A 8.438  7.58  46.895 N/A
    ## 2 N/A 4.157 N/A  4.383 9.176 N/A N/A 1.216 N/A 8.403 7.576 46.3779 N/A
    ##      TRY   AUD    BRL   CAD    CNY   HKD      IDR     INR  KRW     MXN
    ## 1 2.9066 1.448 3.0349 1.459 8.4883 10.61 16251.94 82.2307 1385 17.7759
    ## 2 2.8969 1.454 3.0002 1.459 8.4722 10.59 16248.15 82.2023 1382 17.7124
    ##      MYR   NZD    PHP   SGD    THB   ZAR   ILS  X
    ## 1 4.3893 1.562 59.764 1.705 44.367 14.58 4.692 NA
    ## 2 4.3856 1.563 59.652 1.705 44.323 14.46 4.696 NA
    
    str(rates$Date)
    
    ##  Factor w/ 3968 levels "1999-01-04","1999-01-05",..: 3968 3967 3966 3965 3964 3963 3962 3961 3960 3959 ...
    
    ## convert into date format
    rates$Date <- as.Date(rates$Date, "%Y-%m-%d")
    str(rates$Date)
    
    ##  Date[1:3968], format: "2014-07-01" "2014-06-30" "2014-06-27" "2014-06-26" ...
    
    range(rates$Date)
    
    ## [1] "1999-01-04" "2014-07-01"
    
    rates <- rates[order(rates$Date), ]
    
    ## plot time series
    plot(rates$Date, rates$AUD, type = "l")
    original time series

    3. Forecasting with ARIMA

    The code below shows that there are no data for weekends or public holidays.

    head(rates$Date, 20)
    
    ##  [1] "1999-01-04" "1999-01-05" "1999-01-06" "1999-01-07" "1999-01-08"
    ##  [6] "1999-01-11" "1999-01-12" "1999-01-13" "1999-01-14" "1999-01-15"
    ## [11] "1999-01-18" "1999-01-19" "1999-01-20" "1999-01-21" "1999-01-22"
    ## [16] "1999-01-25" "1999-01-26" "1999-01-27" "1999-01-28" "1999-01-29"
    
    years <- format(rates$Date, "%Y")
    tab <- table(years)
    tab
    
    ## years
    ## 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 
    ##  259  255  254  255  255  259  257  255  255  256  256  258  257  256  255 
    ## 2014 
    ##  126
    
    ## number of days per year after removing 2014
    mean(tab[1:(length(tab) - 1)])
    
    ## [1] 256.1

    Based on above result, there are about 256 values per year, so the windows size is set to 256 in time series analysis in section 5. Another way is to fill weekends and public holidays with values in the previous populated days.

    source("forecast.R") ## see code file in section 5 result.arima <- forecastArima(rates, n.ahead = 90)
    forecasting with ARIMA


    source("plotForecastResult.R") ## see code file in section 5 plotForecastResult(result.arima, title = "Exchange rate forecasting with ARIMA")
    Forecasting with ARIMA - 2


    4. Forecasting with STL

    result.stl <- forecastStl(rates, n.ahead = 90)
    Forecasting with STL


    plotForecastResult(result.stl, title = "Exchange rate forecasting with STL")
    Forecasting with STL - 2


    ## exchange rate in 2014 result <- subset(result.stl, date >= "2014-01-01") plotForecastResult(result, title = "Exchange rate forecasting with STL (2014)")
    Forecasting with STL - 3

    5. Functions

    Below are two source files used in section 3 and 4.

    File forecast.R

    It provides functions for forecasting with ARIMA and STL.

    > 
    library(forecast) > forecastStl <- function(x, n.ahead = 30) { + myTs <- ts(x$AUD, start = 1, frequency = 256) + fit.stl <- stl(myTs, s.window = 256) + sts <- fit.stl$time.series + trend <- sts[, "trend"] + fore <- forecast(fit.stl, h = n.ahead, level = 95) + plot(fore) + pred <- fore$mean + upper <- fore$upper + lower <- fore$lower + output <- data.frame(actual = c(x$AUD, rep(NA, n.ahead)), + trend = c(trend, rep(NA, n.ahead)), pred = c(rep(NA, + nrow(x)), pred), lower = c(rep(NA, nrow(x)), lower), + upper = c(rep(NA, nrow(x)), upper), date = c(x$Date, + max(x$Date) + (1:n.ahead))) + return(output) + } > forecastArima <- function(x, n.ahead = 30) { + myTs <- ts(x$AUD, start = 1, frequency = 256) + fit.arima <- arima(myTs, order = c(0, 0, 1)) + fore <- forecast(fit.arima, h = n.ahead) + plot(fore) + upper <- fore$upper[, "95%"] + lower <- fore$lower[, "95%"] + trend <- as.numeric(fore$fitted) + pred <- as.numeric(fore$mean) + output <- data.frame(actual = c(x$AUD, rep(NA, n.ahead)), + trend = c(trend, rep(NA, n.ahead)), pred = c(rep(NA, + nrow(x)), pred), lower = c(rep(NA, nrow(x)), lower), + upper = c(rep(NA, nrow(x)), upper), date = c(x$Date, + max(x$Date) + (1:n.ahead))) + return(output) + }

    File plotForecastResult.R

    It provides a function for ploting time series forecasting result, incl. trend, forecast and bounds.

    > plotForecastResult <- function(x, title = NULL) { + x <- x[order(x$date), ] + max.val <- max(c(x$actual, x$upper), na.rm = T) + min.val <- min(c(x$actual, x$lower), na.rm = T) + plot(x$date, x$actual, type = "l", col = "grey", main = title, + xlab = "Time", ylab = "Exchange Rate", xlim = range(x$date), + ylim = c(min.val, max.val)) + grid() + lines(x$date, x$trend, col = "yellowgreen") + lines(x$date, x$pred, col = "green") + lines(x$date, x$lower, col = "blue") + lines(x$date, x$upper, col = "blue") + legend("bottomleft", col = c("grey", "yellowgreen", "green", + "blue"), lty = 1, c("Actual", "Trend", "Forecast", "Lower/Upper Bound")) + }

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