DTU02417 TSA: Assignment 2, ARIMA and Seasonal Model

library(HistogramTools)
library(MASS)

Question 1, Stability

Simulate 10 realisations with 200 observations from the process and plot in the same plot.

lists_ts_ar2ma1 <- vector("list", 10)
set.seed(1)
for (i in seq(10)){
    lists_ts_ar2ma1[[i]] <- arima.sim(model = list(order = c(2, 0, 1), ar = c(0.8, 0.1), ma = 0.8), n = 200)
}
par(mfrow = c(10, 1), mar = c(0.5, 0.5, 0.5, 0.5), oma = c(2, 2, 2, 2))
for (i in seq(10)){
    plot.ts(lists_ts_ar2ma1[[i]], xaxt = "n", xlab = NULL, ylab = NULL, frame.plot = TRUE)
}
title(main = "10 Realizations from the ARIMA(2,0,1) Process", cex.main = 2, outer = TRUE)

axis(1, at = seq(0, 200, by = 10), las = 0, outer = TRUE)

vec_acf <- c(0.5896, 0.5596, 0.5066, rep(0, 18))
for (i in seq(4, 21)){
    vec_acf[i] <- 0.8 * vec_acf[i-1] + 0.1 * vec_acf[i-2]
}
vec_acf <- vec_acf / vec_acf[1]
par(mfrow = c(10, 1), mar = c(0.5, 0.5, 0.5, 0.5), oma = c(2, 2, 2, 2))
for (i in seq(10)){
    acf(lists_ts_ar2ma1[[i]], lag.max = 20, xaxt = "n", xlab = NULL, ylab = NULL, frame.plot = TRUE)
    lines(seq(0, 20), vec_acf, type = "l", col = "red", lty = 2, lwd = 2)
    if (i == 1){
        legend("topright", inset = .02, legend = c("Estimated ACF", "5% significance limits", "Calculated ACF"), 
            col = c("black", "blue", "red", "red", "red"), lty = c(1, 2, 2), lwd = c(1, 1, 2))
    }
}
title(main = "ACF of 10 Realizations from the ARIMA(2,0,1) Process", cex.main = 2, outer = TRUE)

axis(1, at = seq(0, 20), las = 0, outer = TRUE)

mat_acf <- matrix(nrow = 10, ncol = 41)

There were 25 warnings (use warnings() to see them)

for (i in seq(10)){
    mat_acf[i,] <- acf(lists_ts_ar2ma1[[i]], lag.max = 40, plot = F)$acf[,1,1]
}
# Manual Calculation of AC
vec_acf <- c(0.5896, 0.5596, 0.5066, rep(0, 38))
for (i in seq(4, 41)){
    vec_acf[i] <- 0.8 * vec_acf[i-1] + 0.1 * vec_acf[i-2]
}
vec_acf <- vec_acf / vec_acf[1]
# Functional Calculation of AC
vec_acf_2 <- ARMAacf(c(0.8, 0.1), 0.8, lag.max = 40, pacf = FALSE)
# Plot different ACFs
plot(seq(0, 40), vec_acf, ylim = c(min(mat_acf), 1),
     type = "l", col = "red", lty = 2, lwd = 3, xlab = "lags", ylab = "autocovariance function", 
     main = "Estimated ACF from Realizations and Manual Calculated ACF of ARIMA(2,0,1)")
lines(seq(0, 40), rep(0.05, 41), type = "l", col = "blue", lty = 2, lwd = 1)

lines(seq(0, 40), rep(0.05, 41), type = "l", col = "blue", lty = 2, lwd = 1)
lines(seq(0, 40), rep(-0.05, 41), type = "l", col = "blue", lty = 2, lwd = 1)

legend("topright", inset = .02, legend = c("Estimated ACF", "5% significance limits", "Manual Calculated ACF"), 
       col = c("black", "blue", "red"), lty = c(1, 2, 2), lwd = c(1, 1, 3))
for (i in seq(10)){
    lines(seq(0, 40), mat_acf[i,], type = "l", col = "black", lty = 1, lwd = 1)
}

par(mfrow = c(10, 1), mar = c(0.5, 0.5, 0.5, 0.5), oma = c(2, 2, 2, 2))
for (i in seq(10)){
    pacf(lists_ts_ar2ma1[[i]], lag.max = 20, xaxt = "n", xlab = NULL, ylab = NULL, frame.plot = TRUE)
}
title(main = "PACF of 10 Realizations from the ARIMA(2,0,1) Process", cex.main = 2, outer = TRUE)

axis(1, at = seq(0, 20), las = 0, outer = TRUE)

mat_pacf <- matrix(nrow = 10, ncol = 20)
for (i in seq(10)){
    mat_pacf[i,] <- pacf(lists_ts_ar2ma1[[i]], lag.max = 20, plot = F)$acf[,1,1]
}
# Functional Calculation of PACF
vec_pacf <- ARMAacf(c(0.8, 0.1), 0.8, lag.max = 20, pacf = TRUE)
# Plot different PACFs
plot(seq(1, 20), vec_pacf, ylim = c(min(mat_pacf), 1),
     type = "l", col = "red", lty = 2, lwd = 3, xaxt = "n", bty = "n", xlab = "lags", ylab = "autocovariance function", 
     main = "Estimated PACF from Realizations and Function-Calculated PACF of ARIMA(2,0,1)")
lines(seq(1, 20), rep(0.05, 20), type = "l", col = "blue", lty = 2, lwd = 1)

lines(seq(1, 20), rep(-0.05, 20), type = "l", col = "blue", lty = 2, lwd = 1)
legend("topright", inset = .02, legend = c("Estimated PACF", "5% significance limits", "Fun-Calculated PACF"), 
       col = c("black", "blue", "red"), lty = c(1, 2, 2), lwd = c(1, 1, 3))

for (i in seq(10)){
    lines(seq(1, 20), mat_pacf[i,], type = "l", col = "black", lty = 1, lwd = 1)
}
axis(1, at = seq(1, 20, by = 1), las = 0, outer = FALSE)

Question 2, Predicting the Carbon-Dioxide Level in an Office

list_co2.log.mean <- c(6.153, 6.308, 6.327, 6.344, 6.266, 6.121, 6.096, 6.081, 
                       6.178, 6.411, 6.442, 6.505, 6.195, 6.085, 6.086, 6.079)
list_co2.log <- list_co2.log.mean - 6.24
pred_co2.log_1.hat <- function(t, list_co2.log){
    return(0.547 * list_co2.log[t] + 0.86 * list_co2.log[t-7] - 0.47042 * list_co2.log[t-8])
}
pred_co2.log_2.hat <- function(t, list_co2.log){
    return(0.299209 * list_co2.log[t] + 0.86 * list_co2.log[t-5] - 0.47042 * list_co2.log[t-6] +
           0.47042 * list_co2.log[t-7] - 0.25731974 * list_co2.log[t-8])
}
co2.log_17.hat <- pred_co2.log_1.hat(16, list_co2.log)
co2.log_18.hat <- pred_co2.log_2.hat(16, list_co2.log)
# Prediction Interval
cal_vec_intervalPred <- function(y.hat, var, prob = 0.95){
    quantileNormDist <- qnorm(p = 0.95)
    boundUp <- y.hat + quantileNormDist * sqrt(var)
    boundLow <- y.hat - quantileNormDist * sqrt(var)
    return(list(boundUp = drop(boundUp), boundLow = drop(boundLow)))
}
vec_intervalPred_17.hat.mean <- cal_vec_intervalPred(y.hat = co2.log_17.hat + 6.24, var = 4.225e-3)
vec_intervalPred_18.hat.mean <- cal_vec_intervalPred(y.hat = co2.log_18.hat + 6.24, var = 6.536075E-3)

Plot the result

plot(seq(1, 16), list_co2.log.mean, 
     type = "b", col = "blue", lwd = 1, lty = 1, xaxt = "n", bty = "n", xlim = c(0, 19), ylim = c(6.0, 6.6),
     main = paste("Two-Step Prediction of Carbon-Dioxide Level in an Office by ARIMA(1,8,0) Model"), 
     xlab = "Series", ylab = "Degree")
points(c(17, 18), c(co2.log_17.hat + 6.24, co2.log_18.hat + 6.24), 
       type = "b", col = "blue", lty = 1, pch = 16)  # Predicted values

lines(c(16, 17), c(list_co2.log.mean[16], co2.log_17.hat + 6.24), 
      type = "c", col = "blue", lty = 1)  # Connect observations with one step prediction
legend("topleft", inset = .02, legend = c("Observations", "Prediction", "Pred Interval"), 
       col = c("blue", "blue", "red"), pch = c(1, 16, 15), 
       lty = c(1, 1, 3), lwd = c(1, 1, 3))

# Plot the prediction result
points(c(17, 17), vec_intervalPred_17.hat.mean, 
       type = "b", col = "red", pch = 15, lty = 3, lwd = 2)
points(c(18, 18), vec_intervalPred_18.hat.mean, 
       type = "b", col = "red", pch = 15, lty = 3, lwd = 2)

axis(1, at = seq(1, 18, by = 1), las = 0, outer = FALSE)

Question 3, Simulating Seasonal Processes

Define the function to plot the residuals for analysis.

# Plot the time series / residuals and its ACF and PACF. No Ljung-Box plot
plotTimeSeriesResidual <- function(dat, num_lag = 50, list_para, ...){
    # If the input dat is arima model, just take its residuals are the dat.
    if(class(dat) == "Arima")
        dat <- dat$residuals
    par(mfrow = c(3, 1), mgp = c(2, 0.7, 0), mar = c(1, 1, 1, 1), oma = c(2, 2, 4, 2), 
        cex.lab = 0.8, cex.axis = 0.8)
    plot(dat)
    title(main = paste("TS, ACF, PACF when phi_1(", list_para[1], "), phi_2(", list_para[2], 
                       "), phi.cap(", list_para[3], ") theta(", list_para[4], ") and theta.cap(", list_para[5], 
                       ")"), cex.main = 1.5, line = 1, outer = TRUE)
    acf(dat, lag.max = 20, xaxt = "n")
    axis(1, at = seq(0, 20, by = 1), las = 0, outer = FALSE)
    pacf(dat, lag.max = 20, xaxt = "n")
    axis(1, at = seq(1, 20, by = 1), las = 0, outer = FALSE)
    # Ljung-Box Plot
    # value_p <- sapply(1:num_lag, function(i) Box.test(dat, i, type = "Ljung-Box")$p.value)
    # plot(1L: num_lag, value_p, main = "p Values for Ljung-Box Statistic",
    #      xlab = "lag", ylab = "p value", ylim = c(0,1))
    # abline(h = 0.05, lty = 2, col = "blue")
}

jud_stationary_arima <- function(vec_phi){
    # Different definitions of ARMA models have different signs for the AR and/or MA coefficients.
    # How to decide the sign of phi is that phi and past observations are in left-hand-side
    # Edward J. Xu, 190402
    vec_phi <- c(1, vec_phi)
    vec.rev_phi <- rev(vec_phi)
    vec_root <- polyroot(vec.rev_phi)
    vec_root.abs <- abs(vec_root)
    vec_root.abs.unit <- abs(vec_root) <= rep(1, length(vec_root))
    return(all(vec_root.abs.unit))
}

There were 24 warnings (use warnings() to see them)

list_phi_1 <- c(0.8, 0, -0.7, -0.8, 0.6, 0)
list_phi_2 <- c(0, 0, 0, 0, 0.5, 0)
list_phi.cap_1 <- c(0, - 0.85, 0, 0.7, 0.9, 0)
list_theta_1 <- c(0, 0, 0, 0, 0, - 0.9)
list_theta.cap_1 <- c(0, 0, 0.8, 0, 0, - 0.7)
mat_phi <- matrix(0, nrow = 6, ncol = 14)
mat_theta <- matrix(0, nrow = 6, ncol = 13)
list_stationary <- logical(length = 6)
for (i in seq(6)){
    phi_1 <- list_phi_1[i]
    phi_2 <- list_phi_2[i]
    phi.cap_1 <- list_phi.cap_1[i]
    theta_1 <- list_theta_1[i]
    theta.cap_1 <- list_theta.cap_1[i]
    mat_phi[i,] <- c(phi_1, phi_2, rep(0, 9), phi.cap_1, phi_1 * phi.cap_1, phi_2 * phi.cap_1)
    mat_theta[i,] <- c(theta_1, rep(0, 10), theta.cap_1, theta_1 * theta.cap_1)
    list_stationary[i] <- jud_stationary_arima(mat_phi[i,])
}

list_ts_ar13ma13 <- vector("list", 6)
set.seed(99)
for(i in seq(6)){
    if(list_stationary[i]){
        list_para <- rep(0, 5)
        list_para[1] <- list_phi_1[i]
        list_para[2] <- list_phi_2[i]
        list_para[3] <- list_phi.cap_1[i]
        list_para[4] <- list_theta_1[i]
        list_para[5] <- list_theta.cap_1[i]
        list_ts_ar13ma13[[i]] <- arima.sim(model = list(ar = - mat_phi[i,], ma =  mat_theta[i,]), n = 200)
        plotTimeSeriesResidual(list_ts_ar13ma13[[i]], list_para = list_para)
    }
}

Simulate and Estimation

sim_ar1ma1 <- function(phi, theta, sigma, numtime_Sim){
    # Different definitions of ARMA models have different signs for the AR and/or MA coefficients.
    set.seed(99)
    lists_ts_ar1ma1 <- vector("list", 100)
    lists_mod_ar1ma1 <- vector("list", 100)
    vec_phi.est <- rep(0, 100)
    vec_theta.est <- rep(0, 100)
    # j <- 1
    # vec_series_seed <- rep(0, 100)
    for (i in seq(100)){
        # print(paste("i: ", i, "j:", j))
        # while(any(class(out <- tryCatch(arima(lists_ts_ar1ma1[[i]] <- arima.sim(model = list(ar = - phi_1, 
        #     ma = theta_1), sd = sigma, n = numtime_Sim), order = c(1, 0, 1), include.mean = FALSE), 
        #     error = function(e) e)) == "error")){
            # j <- j + 1
            # set.seed(j)
        # }
        lists_ts_ar1ma1[[i]] <- arima.sim(model = list(ar = - phi, ma = theta), sd = sigma, n = numtime_Sim)
        lists_mod_ar1ma1[[i]] <- arima(lists_ts_ar1ma1[[i]], order = c(1, 0, 1), method = "ML",
                                       include.mean = FALSE)
        vec_phi.est[i] <- lists_mod_ar1ma1[[i]]$coef[1]
        vec_theta.est[i] <- lists_mod_ar1ma1[[i]]$coef[2]
        # vec_series_seed[i] <- j
    }
    quan95 <- ApproxQuantile(hist(vec_phi.est, breaks = 50, plot = FALSE), 0.95)
    list_result <- list(quan95 = quan95, vec_phi.est = vec_phi.est, vec_theta.est = vec_theta.est)
    return(list_result)
}
histPhi_sim_ar1ma1 <- function(vec_phi.est, phi, theta, sigma, numtime_Sim){
    den_phi.est <- density(vec_phi.est, adjust = 0.2)
    hist_phi.est <- hist(vec_phi.est, breaks = 50, plot = FALSE)
    hist(vec_phi.est, breaks = 50, freq = FALSE, ylim = c(0, max(hist_phi.est$density, den_phi.est$y)), 
         xlab = "phi_1", main = NULL)  # cex.lab = 0.8, cex.axis = 0.8
    lines(den_phi.est, col = "blue", lwd = 2)
    title(main = paste("Hist of phi when times(",numtime_Sim,"), phi(", 
                       phi, "), theta(", theta, ") and sd(", sigma, ")"))  # cex.main = 0.8
}
histPhi.2_sim_ar1ma1 <- function(vec_phi.est_1, phi_1, theta_1, sigma_1, numtime_Sim_1, 
                                 vec_phi.est_2, phi_2, theta_2, sigma_2, numtime_Sim_2){
    par(mfrow = c(1, 2), mar = c(0.5, 1.5, 0.5, 0.5), oma = rep(2, 4), 
        cex.lab = 0.8, cex.axis = 0.8, cex.main = 0.8)
    histPhi_sim_ar1ma1(vec_phi.est_1, phi_1, theta_1, sigma_1, numtime_Sim_1)
    histPhi_sim_ar1ma1(vec_phi.est_2, phi_2, theta_2, sigma_2, numtime_Sim_2)
}
histPhi.4_sim_ar1ma1 <- function(mat_phi.est, vec_phi, vec_theta, vec_sigma, vec_numtime_Sim){
    par(mfrow = c(2, 2), mar = c(1.5, 1.5, 1.5, 1.5), oma = rep(1, 4), 
        cex.lab = 0.8, cex.axis = 0.8, cex.main = 0.8)
    for (i in seq(1, 4)){
        histPhi_sim_ar1ma1(mat_phi.est[i,], vec_phi[i], vec_theta[i], vec_sigma[i], vec_numtime_Sim[i])
    }
}
plotContour_sim_ar1ma1 <- function(vec_theta.est, vec_phi.est, phi, theta, sigma, numtime_Sim){
    kde_ar1ma1_1 <- kde2d(vec_theta.est, vec_phi.est, n = 20)
    contour(kde_ar1ma1_1, xlab = "theta", ylab = "phi", bty = "n", main = NULL) 
    points(theta, - phi, pch = 4, cex = 2, lwd = 2, col = "red")
    title(main = paste("Contour of theta(", theta, ") and phi(", phi, ") when times(", numtime_Sim ,
        "), sd(", sigma, ")"))
}
plotContour.2_sim_ar1ma1 <- function(vec_theta.est_1, vec_phi.est_1, phi_1, theta_1, sigma_1, numtime_Sim_1, 
                                     vec_theta.est_2, vec_phi.est_2, phi_2, theta_2, sigma_2, numtime_Sim_2){
    par(mfrow = c(1, 2), mar = c(0.5, 1.5, 0.5, 0.5), oma = rep(2, 4), 
        cex.lab = 0.8, cex.axis = 0.8, cex.main = 0.8)
    plotContour_sim_ar1ma1(vec_theta.est_1, vec_phi.est_1, phi_1, theta_1, sigma_1, numtime_Sim_1)
    plotContour_sim_ar1ma1(vec_theta.est_2, vec_phi.est_2, phi_2, theta_2, sigma_2, numtime_Sim_2)
}
plotContour.4_sim_ar1ma1 <- function(mat_theta.est, mat_phi.est, vec_phi, vec_theta, vec_sigma, vec_numtime_Sim){
    par(mfrow = c(2, 2), mar = c(1.5, 1.5, 1.5, 1.5), oma = rep(1, 4), 
        cex.lab = 0.8, cex.axis = 0.8, cex.main = 0.8)
    for (i in seq(1, 4)){
        plotContour_sim_ar1ma1(mat_theta.est[i,], mat_phi.est[i,], vec_phi[i], 
                               vec_theta[i], vec_sigma[i], vec_numtime_Sim[i])
    }
}
plotQQ_sim_ar1ma1 <- function(vec_phi.est, phi, theta, sigma, numtime_Sim){
    qqnorm(vec_phi.est, main = NULL, bty = "n")
    qqline(vec_phi.est)
    title(main = paste("Norm Q-Q Plot of phi when times(", numtime_Sim ,
                       "), phi(", phi, "), theta(", theta, ") and sd(", sigma, ")"))
}
plotQQ.2_sim_ar1ma1 <- function(vec_phi.est_1, phi_1, theta_1, sigma_1, numtime_Sim_1, 
                                vec_phi.est_2, phi_2, theta_2, sigma_2, numtime_Sim_2){
    par(mfrow = c(1, 2), mar = c(0.5, 1.5, 0.5, 0.5), oma = rep(2, 4), 
        cex.lab = 0.8, cex.axis = 0.8, cex.main = 0.8)
    plotQQ_sim_ar1ma1(vec_phi.est_1, phi_1, theta_1, sigma_1, numtime_Sim_1)
    plotQQ_sim_ar1ma1(vec_phi.est_2, phi_2, theta_2, sigma_2, numtime_Sim_2)
}
plotQQ.4_sim_ar1ma1 <- function(mat_phi.est, vec_phi, vec_theta, vec_sigma, vec_numtime_Sim){
    par(mfrow = c(2, 2), mar = c(1.5, 1.5, 1.5, 1.5), oma = rep(1, 4), 
        cex.lab = 0.8, cex.axis = 0.8, cex.main = 0.8)
    for (i in seq(1, 4)){
        plotQQ.2_sim_ar1ma1(mat_phi.est[i,], vec_phi[i], vec_theta[i], vec_sigma[i], vec_numtime_Sim[i])
    }
}

Judge whether the given process are stationary.

jud_stationary_arima(- 0.9)

[1] TRUE

jud_stationary_arima(- 0.995)

[1] TRUE

list_result_1 <- sim_ar1ma1(- 0.9, 0.4, 0.1, 300)
list_result_2 <- sim_ar1ma1(- 0.9, 0.4, 5, 300)
list_result_3 <- sim_ar1ma1(- 0.995, 0.4, 0.1, 300)
list_result_4 <- sim_ar1ma1(- 0.995, 0.4, 5, 300)
mat_phi.est <- matrix(nrow = 4, ncol = 100)
mat_phi.est[1,] <- list_result_1$vec_phi.est
mat_phi.est[2,] <- list_result_2$vec_phi.est
mat_phi.est[3,] <- list_result_3$vec_phi.est
mat_phi.est[4,] <- list_result_4$vec_phi.est
mat_theta.est <- matrix(nrow = 4, ncol = 100)
mat_theta.est[1,] <- list_result_1$vec_theta.est
mat_theta.est[2,] <- list_result_2$vec_theta.est
mat_theta.est[3,] <- list_result_3$vec_theta.est
mat_theta.est[4,] <- list_result_4$vec_theta.est

mat_phi.est <- matrix(nrow = 4, ncol = 100)
mat_phi.est[1,] <- list_result_1$vec_phi.est
mat_phi.est[2,] <- list_result_2$vec_phi.est
mat_phi.est[3,] <- list_result_3$vec_phi.est
mat_phi.est[4,] <- list_result_4$vec_phi.est
histPhi.4_sim_ar1ma1(mat_phi.est, c(- 0.9, - 0.9, - 0.995, - 0.995), rep(0.4, 4), c(0.1, 5, 0.1, 5), rep(300, 4))

plotContour.4_sim_ar1ma1(mat_theta.est, mat_phi.est, c(- 0.9, - 0.9, - 0.995, - 0.995), rep(0.4, 4), c(0.1, 5, 0.1, 5), rep(300, 4))

plotQQ.4_sim_ar1ma1(mat_phi.est, c(- 0.9, - 0.9, - 0.995, - 0.995), rep(0.4, 4), c(0.1, 5, 0.1, 5), rep(300, 4))

phi_1	sigma	95% quantile
-0.900	0.1	0.9325
-0.900	5.0	0.9325
-0.995	0.1	1.0005
-0.995	5.0	1.0005

The Effect of Number of Simulated Observations

list_result_5 <- sim_ar1ma1(- 0.9, 0.4, 1, 100)

Quantiles computed for histograms with fewer than 100 buckets may be inaccurate. Consider using histograms with more granular buckets.

list_result_6 <- sim_ar1ma1(- 0.9, 0.4, 1, 1000)

Quantiles computed for histograms with fewer than 100 buckets may be inaccurate. Consider using histograms with more granular buckets.

histPhi.2_sim_ar1ma1(list_result_5$vec_phi.est, - 0.9, 0.4, 1, 100,
                     list_result_6$vec_phi.est, - 0.9, 0.4, 1, 1000)

plotQQ.2_sim_ar1ma1(list_result_5$vec_phi.est, - 0.9, 0.4, 1, 100,
                    list_result_6$vec_phi.est, - 0.9, 0.4, 1, 1000)

plotContour.2_sim_ar1ma1(list_result_5$vec_theta.est, list_result_5$vec_phi.est, -0.9, 0.4, 1, 300,
                         list_result_6$vec_theta.est, list_result_6$vec_phi.est, -0.9, 0.4, 1, 1000)