Univariate models belong to a class of models that require only one variable to be formulated. They use information from the past values of a variable to forecast its future values.
Disadvantage: Univariate techniques cannot provide explanations for the possible causes of change in the variable of interest — they only describe what has happened, not why.
The methods covered in this chapter are:
| Method | Best For |
|---|---|
| Naive Forecast | No discernible pattern; benchmark |
| Naive with Trend | Series with slow, steady trend |
| Average Change | Short series with consistent changes |
| Average Percent Change | Constant percentage growth |
| Single Exponential Smoothing | Stable series, no trend |
| Double Exponential Smoothing | Linear trend, short series |
| Holt’s Method | Linear trend, more flexible |
| ARRES | Time-varying alpha needed |
| Holt-Winters | Trend + seasonality |
| Decomposition Forecasting | Clear trend + seasonal components |
The naive forecast strongly believes that what happens today will happen again tomorrow (or any other time in the future).
\[F_{t+m} = y_t\]
where \(m\) is the number of steps ahead.
When to use:
Compute the estimated values and forecast for 2024 using naive forecast. Then compute the MSE.
F_naive <- c(NA, price[-n_p])
e2_naive <- (price - F_naive)^2
mse_naive <- mean(e2_naive, na.rm = TRUE)
knitr::kable(
data.frame(Year = year, Price = price,
F_t = F_naive, e2 = e2_naive),
col.names = c("Year", "Price (RM)", "$F_t$", "$e_t^2$"),
align = "c", na_mark = "—"
)| Year | Price (RM) | \(F_t\) | \(e_t^2\) |
|---|---|---|---|
| 2018 | 6 | NA | NA |
| 2019 | 8 | 6 | 4 |
| 2020 | 9 | 8 | 1 |
| 2021 | 12 | 9 | 9 |
| 2022 | 15 | 12 | 9 |
| 2023 | 16 | 15 | 1 |
cat("Forecast for 2024 (Naive):", price[n_p], "\n")Forecast for 2024 (Naive): 16 cat("MSE:", round(mse_naive, 4), "\n")MSE: 4.8 ggplot(data.frame(Year = year, Actual = price, Forecast = F_naive),
aes(x = Year)) +
geom_line(aes(y = Actual, colour = "Actual"), linewidth = 0.9) +
geom_point(aes(y = Actual, colour = "Actual"), size = 2.5) +
geom_line(aes(y = Forecast, colour = "Naive Forecast"),
linewidth = 0.9, linetype = "dashed", na.rm = TRUE) +
geom_point(aes(y = Forecast, colour = "Naive Forecast"),
size = 2.5, na.rm = TRUE) +
scale_colour_manual(values = c("Actual" = "steelblue",
"Naive Forecast" = "#d73027")) +
labs(title = "Naive Forecast", x = "Year", y = "Price (RM)", colour = NULL) +
theme_ts()
Excel Tutorial: Naive Forecast in Excel
The naive method is modified to account for the trend component.
\[F_{t+1} = y_t \times \frac{y_t}{y_{t-1}}\]
| Condition | Interpretation |
|---|---|
| \(y_t > y_{t-1}\) | Upward trend |
| \(y_t < y_{t-1}\) | Downward trend |
This method is highly sensitive to changes in actual values.
# F_{t+1} = y_t * (y_t / y_{t-1}) -- needs TWO prior values, so first two rows are NA
growth <- c(NA, price[-1] / price[-n_p]) # growth rate at each t
F_trend <- c(NA, NA, sapply(3:n_p, function(i) # F(2019) = NA: needs y(2017) which is absent
price[i-1] * (price[i-1] / price[i-2])))
e2_trend <- (price - F_trend)^2
mse_trend <- mean(e2_trend, na.rm = TRUE)
knitr::kable(
data.frame(Year = year, Price = price,
Growth = round(c(NA, price[-1]/price[-n_p]), 4),
F_t = round(F_trend, 4), e2 = round(e2_trend, 4)),
col.names = c("Year", "Price (RM)", "$y_t/y_{t-1}$", "$F_t$", "$e_t^2$"),
align = "c", na_mark = "—"
)| Year | Price (RM) | \(y_t/y_{t-1}\) | \(F_t\) | \(e_t^2\) |
|---|---|---|---|---|
| 2018 | 6 | NA | NA | NA |
| 2019 | 8 | 1.3333 | NA | NA |
| 2020 | 9 | 1.1250 | 10.6667 | 2.7778 |
| 2021 | 12 | 1.3333 | 10.1250 | 3.5156 |
| 2022 | 15 | 1.2500 | 16.0000 | 1.0000 |
| 2023 | 16 | 1.0667 | 18.7500 | 7.5625 |
last_g <- price[n_p] / price[n_p - 1]
f2024_trend <- price[n_p] * last_g
cat("Forecast for 2024 (Naive with Trend):", round(f2024_trend, 4), "\n")Forecast for 2024 (Naive with Trend): 17.0667 cat("MSE:", round(mse_trend, 4), "\n")MSE: 3.714 Excel Tutorial: Naive with Trend in Excel
The forecast equals the actual value in the current period plus the average of the absolute changes.
\[F_{t+m} = y_t + \frac{(y_t - y_{t-1}) + (y_{t-1} - y_{t-2})}{2}\]
When to use:
chg <- c(NA, diff(price))
# F_{t+1} = y_t + [(y_t-y_{t-1}) + (y_{t-1}-y_{t-2})]/2
# Needs y_{t-2}: F(2019) needs y(2017), F(2020) needs y(2017) -> both NA
# First computable forecast is F(2021) at i=4
F_ac <- c(NA, NA, NA, sapply(4:n_p, function(i)
price[i-1] + (chg[i-1] + chg[i-2]) / 2))
e2_ac <- (price - F_ac)^2
mse_ac <- mean(e2_ac, na.rm = TRUE)
knitr::kable(
data.frame(Year = year, Price = price,
Change = round(chg, 4), F_t = round(F_ac, 4),
e2 = round(e2_ac, 4)),
col.names = c("Year", "Price (RM)", "$y_t - y_{t-1}$", "$F_t$", "$e_t^2$"),
align = "c", na_mark = "—"
)| Year | Price (RM) | \(y_t - y_{t-1}\) | \(F_t\) | \(e_t^2\) |
|---|---|---|---|---|
| 2018 | 6 | NA | NA | NA |
| 2019 | 8 | 2 | NA | NA |
| 2020 | 9 | 1 | NA | NA |
| 2021 | 12 | 3 | 10.5 | 2.25 |
| 2022 | 15 | 3 | 14.0 | 1.00 |
| 2023 | 16 | 1 | 18.0 | 4.00 |
f2024_ac <- price[n_p] + (chg[n_p] + chg[n_p - 1]) / 2
cat("Forecast for 2024 (Average Change):", round(f2024_ac, 4), "\n")Forecast for 2024 (Average Change): 18 cat("MSE:", round(mse_ac, 4), "\n")MSE: 2.4167 Excel Tutorial: Average Change in Excel
\[F_{t+m} = y_t + \frac{\left(\frac{y_t - y_{t-1}}{y_{t-1}}\right) + \left(\frac{y_{t-1} - y_{t-2}}{y_{t-2}}\right)}{2} \times y_t\]
May be unsuitable for forecasting beyond one or two periods due to compounding effects.
pct <- c(NA, diff(price) / price[-n_p])
# F_{t+1} = y_t * (1 + (p_t + p_{t-1})/2)
# p_{t-1} needs y_{t-2}: F(2019) and F(2020) both require y(2017) -> NA
# First computable forecast is F(2021) at i=4
F_apc <- c(NA, NA, NA, sapply(4:n_p, function(i)
price[i-1] * (1 + (pct[i-1] + pct[i-2]) / 2)))
e2_apc <- (price - F_apc)^2
mse_apc <- mean(e2_apc, na.rm = TRUE)
knitr::kable(
data.frame(Year = year, Price = price,
PctChg = round(pct * 100, 2), F_t = round(F_apc, 4),
e2 = round(e2_apc, 4)),
col.names = c("Year", "Price (RM)", "% Change", "$F_t$", "$e_t^2$"),
align = "c", na_mark = "—"
)| Year | Price (RM) | % Change | \(F_t\) | \(e_t^2\) |
|---|---|---|---|---|
| 2018 | 6 | NA | NA | NA |
| 2019 | 8 | 33.33 | NA | NA |
| 2020 | 9 | 12.50 | NA | NA |
| 2021 | 12 | 33.33 | 11.0625 | 0.8789 |
| 2022 | 15 | 25.00 | 14.7500 | 0.0625 |
| 2023 | 16 | 6.67 | 19.3750 | 11.3906 |
f2024_apc <- price[n_p] * (1 + (pct[n_p] + pct[n_p - 1]) / 2)
cat("Forecast for 2024 (Avg Percent Change):", round(f2024_apc, 4), "\n")Forecast for 2024 (Avg Percent Change): 18.5333 cat("MSE:", round(mse_apc, 4), "\n")MSE: 4.1107 Excel Tutorial: Average Percent Change in Excel
Also known as simple exponential smoothing (SES). Requires only one parameter \(\alpha\).
\[F_{t+1} = \alpha y_t + (1-\alpha)F_t, \quad 0 \leq \alpha \leq 1\]
Advantages over moving average:
Disadvantage: Cannot forecast more than one-step ahead.
Consider the one-step-ahead forecast:
\[F_{t+1} = \alpha y_t + (1-\alpha)F_t \tag{1}\]
Then,
\[F_t = \alpha y_{t-1} + (1-\alpha)F_{t-1} \tag{2}\]
Substitute Equation (2) into Equation (1):
\[F_{t+1} = \alpha y_t + (1-\alpha)\left[\alpha y_{t-1} + (1-\alpha)F_{t-1}\right]\]
\[F_{t+1} = \alpha y_t + \alpha(1-\alpha)y_{t-1} + (1-\alpha)^2 F_{t-1} \tag{3}\]
Let,
\[F_{t-1} = \alpha y_{t-2} + (1-\alpha)F_{t-2} \tag{4}\]
Substitute Equation (4) into Equation (3):
\[F_{t+1} = \alpha y_t + \alpha(1-\alpha)y_{t-1} + (1-\alpha)^2\left[\alpha y_{t-2} + (1-\alpha)F_{t-2}\right]\]
\[F_{t+1} = \alpha y_t + \alpha(1-\alpha)y_{t-1} + \alpha(1-\alpha)^2 y_{t-2} + (1-\alpha)^3 F_{t-2}\]
Continuing the recursive substitution for \(F_{t-2},\ F_{t-3}\) and so forth yields the final weighted average form:
\[F_{t+1} = \alpha y_t + \alpha(1-\alpha)y_{t-1} + \alpha(1-\alpha)^2 y_{t-2} + \cdots + \alpha(1-\alpha)^{t-1}y_1 + (1-\alpha)^t F_1\]
This shows that \(F_{t+1}\) is a weighted average of all past observations \(y_t, y_{t-1}, y_{t-2}, \ldots, y_1\) with respective weights:
\[\alpha,\quad \alpha(1-\alpha),\quad \alpha(1-\alpha)^2,\quad \alpha(1-\alpha)^3,\quad \ldots,\quad \alpha(1-\alpha)^{t-1},\quad (1-\alpha)^t\]
The weights decrease exponentially as we move further into the past — this is the damping effect. The most recent observation \(y_t\) provides the largest contribution; successive earlier observations provide smaller and smaller contributions.
Larger \(\alpha\) → faster damping → past observations lose influence quickly. Smaller \(\alpha\) → slower damping → past observations retain influence longer.
Example: \(\alpha = 0.8\)
The weights are \(0.8,\ 0.16,\ 0.032,\ 0.0064,\ 0.00128\) for \(y_t,\ y_{t-1},\ y_{t-2},\ y_{t-3},\ y_{t-4}\) respectively.
\[F_{t+1} = 0.8y_t + 0.16y_{t-1} + 0.032y_{t-2} + 0.0064y_{t-3} + 0.0012y_{t-4} + \cdots\]
Periods beyond \(t-3\) have almost minimal effect on the current estimate.
Example: \(\alpha = 0.2\)
The weights are \(0.2,\ 0.16,\ 0.128,\ 0.1024\) for \(y_t,\ y_{t-1},\ y_{t-2},\ y_{t-3}\) respectively.
\[F_{t+1} = 0.2y_t + 0.16y_{t-1} + 0.128y_{t-2} + 0.1024y_{t-3} + \cdots\]
The weights diminish slowly compared to a large value of \(\alpha\).
| \(\alpha\) | Behaviour |
|---|---|
| Near 1 | Reacts quickly; heavy weight on recent observation; fast damping |
| Near 0 | Smooth; weights diminish slowly; more historical influence |
Comparison with 3-period Moving Average:
\[\hat{y}_2 = \frac{y_1 + y_2 + y_3}{3} = \frac{1}{3}y_1 + \frac{1}{3}y_2 + \frac{1}{3}y_3\]
The MA uses constant equal weights of \(\frac{1}{3}\) — it does not distinguish between the most recent and older observations, unlike SES.
| Method | Description |
|---|---|
| Subjective | Near 1 for rapidly changing data; small for stable series |
| Objective | Minimise MSE/MAPE/MAE over the estimation set |
alpha <- 0.8
F_ses <- numeric(n_p); F_ses[1] <- price[1]
for (i in 2:n_p)
F_ses[i] <- alpha * price[i-1] + (1 - alpha) * F_ses[i-1]
e2_ses <- (price - F_ses)^2
mse_ses <- mean(e2_ses[-1])
knitr::kable(
data.frame(Year = year, Price = price,
F_t = round(F_ses, 4), e2 = round(e2_ses, 4)),
col.names = c("Year", "Price (RM)", "$F_t$", "$e_t^2$"),
align = "c"
)| Year | Price (RM) | \(F_t\) | \(e_t^2\) |
|---|---|---|---|
| 2018 | 6 | 6.0000 | 0.0000 |
| 2019 | 8 | 6.0000 | 4.0000 |
| 2020 | 9 | 7.6000 | 1.9600 |
| 2021 | 12 | 8.7200 | 10.7584 |
| 2022 | 15 | 11.3440 | 13.3663 |
| 2023 | 16 | 14.2688 | 2.9971 |
f2024_ses <- alpha * price[n_p] + (1 - alpha) * F_ses[n_p]
cat("Forecast for 2024 (SES, α = 0.8):", round(f2024_ses, 4), "\n")Forecast for 2024 (SES, <U+03B1> = 0.8): 15.6538 cat("MSE:", round(mse_ses, 4), "\n")MSE: 6.6164 SE_est <- ses(est_part)
summary(SE_est)
Forecast method: Simple exponential smoothing
Model Information:
Simple exponential smoothing
Call:
ses(y = est_part)
Smoothing parameters:
alpha = 0.9999
Initial states:
l = 21.3004
sigma: 2.0227
AIC AICc BIC
270.0030 270.5247 275.7391
Error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set 1.542145 1.981825 1.558155 2.974211 3.045736 0.9800978 0.4849583
Forecasts:
Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
2010 98.39994 95.80776 100.9921 94.43554 102.3643
2011 98.39994 94.73422 102.0657 92.79371 104.0062
2012 98.39994 93.91044 102.8894 91.53385 105.2660
2013 98.39994 93.21596 103.5839 90.47173 106.3282
2014 98.39994 92.60410 104.1958 89.53597 107.2639
2015 98.39994 92.05094 104.7489 88.68998 108.1099
2016 98.39994 91.54225 105.2576 87.91201 108.8879
2017 98.39994 91.06878 105.7311 87.18789 109.6120
2018 98.39994 90.62408 106.1758 86.50779 110.2921
2019 98.39994 90.20347 106.5964 85.86452 110.9354SE_eva <- ses(eva_part, alpha = 0.999)
summary(SE_eva)
Forecast method: Simple exponential smoothing
Model Information:
Simple exponential smoothing
Call:
ses(y = eva_part, alpha = 0.999)
Smoothing parameters:
alpha = 0.999
Initial states:
l = 100.002
sigma: 2.8618
AIC AICc BIC
62.50994 63.70994 63.63984
Error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set 2.093929 2.632431 2.309504 1.825748 2.005296 0.9238016 0.007182097
Forecasts:
Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
2023 127.1959 123.5284 130.8634 121.5870 132.8048
2024 127.1959 122.0119 132.3799 119.2676 135.1242
2025 127.1959 120.8479 133.5439 117.4874 136.9044
2026 127.1959 119.8664 134.5254 115.9864 138.4054
2027 127.1959 119.0017 135.3901 114.6640 139.7278
2028 127.1959 118.2199 136.1719 113.4683 140.9235
2029 127.1959 117.5010 136.8908 112.3688 142.0230
2030 127.1959 116.8318 137.5600 111.3453 143.0465
2031 127.1959 116.2032 138.1886 110.3840 144.0077
2032 127.1959 115.6087 138.7831 109.4749 144.9169autoplot(cpits) + ylab("CPI (2010 = 100)") + xlab("Year") +
autolayer(forecast(ses(cpits, alpha = 0.999), h = 5),
series = "SES Forecast", PI = TRUE) +
labs(title = "Malaysia CPI: Single Exponential Smoothing Forecast") +
theme_ts()
Useful for series exhibiting a linear trend. Generates multiple-step-ahead forecasts.
\[S_t = \alpha y_t + (1-\alpha)S_{t-1}\] \[S'_t = \alpha S_t + (1-\alpha)S'_{t-1}\] \[a_t = 2S_t - S'_t\] \[b_t = \frac{\alpha}{1-\alpha}(S_t - S'_t)\] \[F_{t+m} = a_t + b_t \times m\]
alpha <- 0.8
S <- numeric(n_p); Sp <- numeric(n_p)
S[1] <- price[1]; Sp[1] <- price[1]
for (i in 2:n_p) {
S[i] <- alpha * price[i] + (1 - alpha) * S[i-1]
Sp[i] <- alpha * S[i] + (1 - alpha) * Sp[i-1]
}
a_t <- 2 * S - Sp
b_t <- (alpha / (1 - alpha)) * (S - Sp)
F_des <- c(NA, a_t[-n_p] + b_t[-n_p] * 1)
e2_des <- (price - F_des)^2
mse_des <- mean(e2_des, na.rm = TRUE)
knitr::kable(
data.frame(Year = year, Price = price,
S = round(S, 4), Sp = round(Sp, 4),
a = round(a_t, 4), b = round(b_t, 4),
F_t = round(F_des, 4), e2 = round(e2_des, 4)),
col.names = c("Year", "Price", "$S_t$", "$S'_t$",
"$a_t$", "$b_t$", "$F_t$", "$e_t^2$"),
align = "c", na_mark = "—"
)| Year | Price | \(S_t\) | \(S'_t\) | \(a_t\) | \(b_t\) | \(F_t\) | \(e_t^2\) |
|---|---|---|---|---|---|---|---|
| 2018 | 6 | 6.0000 | 6.0000 | 6.0000 | 0.0000 | NA | NA |
| 2019 | 8 | 7.6000 | 7.2800 | 7.9200 | 1.2800 | 6.000 | 4.0000 |
| 2020 | 9 | 8.7200 | 8.4320 | 9.0080 | 1.1520 | 9.200 | 0.0400 |
| 2021 | 12 | 11.3440 | 10.7616 | 11.9264 | 2.3296 | 10.160 | 3.3856 |
| 2022 | 15 | 14.2688 | 13.5674 | 14.9702 | 2.8058 | 14.256 | 0.5535 |
| 2023 | 16 | 15.6538 | 15.2365 | 16.0710 | 1.6691 | 17.776 | 3.1542 |
f2024_des <- a_t[n_p] + b_t[n_p] * 1
cat("Forecast for 2024 (DES, α = 0.8, m = 1):", round(f2024_des, 4), "\n")Forecast for 2024 (DES, <U+03B1> = 0.8, m = 1): 17.7402 cat("MSE:", round(mse_des, 4), "\n")MSE: 2.2267 Excel Tutorial: Double Exponential Smoothing in Excel (α = 0.6)
Suitable for linear trend with more flexibility than DES — it uses two separate smoothing constants for level (\(\alpha\)) and slope (\(\beta\)).
Level: \(L_t = \alpha y_t + (1-\alpha)(L_{t-1} + T_{t-1})\)
Trend: \(T_t = \beta(L_t - L_{t-1}) + (1-\beta)T_{t-1}\)
Forecast: \(F_{t+m} = L_t + T_t \times m\)
where \(0 \leq \alpha \leq 1\), \(0 \leq \beta \leq 1\).
alpha_h <- 0.6; beta_h <- 0.2
L <- numeric(n_p); Tr <- numeric(n_p)
L[1] <- price[1]; Tr[1] <- price[2] - price[1]
for (i in 2:n_p) {
L[i] <- alpha_h * price[i] + (1 - alpha_h) * (L[i-1] + Tr[i-1])
Tr[i] <- beta_h * (L[i] - L[i-1]) + (1 - beta_h) * Tr[i-1]
}
F_holt <- c(NA, L[-n_p] + Tr[-n_p] * 1)
e2_h <- (price - F_holt)^2
mse_h <- mean(e2_h, na.rm = TRUE)
knitr::kable(
data.frame(Year = year, Price = price,
L = round(L, 4), Tr = round(Tr, 4),
F_t = round(F_holt, 4), e2 = round(e2_h, 4)),
col.names = c("Year", "Price (RM)", "$L_t$", "$T_t$", "$F_t$", "$e_t^2$"),
align = "c", na_mark = "—"
)| Year | Price (RM) | \(L_t\) | \(T_t\) | \(F_t\) | \(e_t^2\) |
|---|---|---|---|---|---|
| 2018 | 6 | 6.0000 | 2.0000 | NA | NA |
| 2019 | 8 | 8.0000 | 2.0000 | 8.0000 | 0.0000 |
| 2020 | 9 | 9.4000 | 1.8800 | 10.0000 | 1.0000 |
| 2021 | 12 | 11.7120 | 1.9664 | 11.2800 | 0.5184 |
| 2022 | 15 | 14.4714 | 2.1250 | 13.6784 | 1.7466 |
| 2023 | 16 | 16.2385 | 2.0534 | 16.5964 | 0.3556 |
f2024_holt <- L[n_p] + Tr[n_p] * 1
cat("Forecast for 2024 (Holt, α = 0.6, β = 0.2):", round(f2024_holt, 4), "\n")Forecast for 2024 (Holt, <U+03B1> = 0.6, <U+03B2> = 0.2): 18.292 cat("MSE:", round(mse_h, 4), "\n")MSE: 0.7241 Holt_est <- holt(est_part)
summary(Holt_est)
Forecast method: Holt's method
Model Information:
Holt's method
Call:
holt(y = est_part)
Smoothing parameters:
alpha = 0.9922
beta = 0.3708
Initial states:
l = 21.2128
b = -0.0636
sigma: 1.1718
AIC AICc BIC
217.2861 218.6498 226.8462
Error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set 0.1243695 1.12395 0.8068117 0.4127585 1.756522 0.5074939 0.1005745
Forecasts:
Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
2010 100.6626 99.16086 102.1643 98.36590 102.9593
2011 102.9050 100.36634 105.4436 99.02247 106.7875
2012 105.1474 101.51089 108.7839 99.58585 110.7089
2013 107.3898 102.57173 112.2078 100.02122 114.7583
2014 109.6322 103.54667 115.7176 100.32520 118.9391
2015 111.8745 104.43792 119.3112 100.50120 123.2479
2016 114.1169 105.24876 122.9851 100.55423 127.6796
2017 116.3593 105.98259 126.7361 100.48948 132.2292
2018 118.6017 106.64262 130.5608 100.31186 136.8916
2019 120.8441 107.23180 134.4564 100.02588 141.6624Holt_eva <- holt(eva_part, alpha = 0.9922, beta = 0.3708)
summary(Holt_eva)
Forecast method: Holt's method
Model Information:
Holt's method
Call:
holt(y = eva_part, alpha = 0.9922, beta = 0.3708)
Smoothing parameters:
alpha = 0.9922
beta = 0.3708
Initial states:
l = 97.4232
b = 2.5846
sigma: 2.0339
AIC AICc BIC
53.02336 55.69003 54.71821
Error measures:
ME RMSE MAE MPE MAPE MASE
Training set -0.02408487 1.692347 1.361851 -0.03265661 1.146471 0.5447402
ACF1
Training set 0.0400953
Forecasts:
Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
2023 129.6481 127.0414 132.2547 125.6616 133.6345
2024 132.1166 127.7101 136.5230 125.3775 138.8557
2025 134.5851 128.2731 140.8971 124.9317 144.2384
2026 137.0536 128.6908 145.4164 124.2638 149.8433
2027 139.5221 128.9594 150.0847 123.3679 155.6763
2028 141.9906 129.0828 154.8983 122.2499 161.7313
2029 144.4591 129.0667 159.8515 120.9185 167.9997
2030 146.9276 128.9169 164.9383 119.3826 174.4726
2031 149.3961 128.6390 170.1532 117.6509 181.1414
2032 151.8646 128.2382 175.4911 115.7311 187.9982autoplot(cpits) + ylab("CPI (2010 = 100)") + xlab("Year") +
autolayer(forecast(holt(cpits, alpha = 0.9922, beta = 0.3708), h = 5),
series = "Holt Forecast", PI = TRUE) +
labs(title = "Malaysia CPI: Holt's Method Forecast") +
theme_ts()
Excel Tutorial: Holt’s Method in Excel (α = 0.6, β = 0.2)
In SES, \(\alpha\) is constant. But structural changes in the data may make a fixed \(\alpha\) unrealistic. ARRES adapts \(\alpha\) at each period based on the ratio of smoothed error to smoothed absolute error.
\[F_{t+1} = \alpha_t y_t + (1-\alpha_t)F_t\]
\[\alpha_t = \left|\frac{E_t}{AE_t}\right|, \quad e_t = y_t - F_t\]
\[E_t = \beta e_t + (1-\beta)E_{t-1} \qquad \text{(smoothed average error)}\]
\[AE_t = \beta|e_t| + (1-\beta)AE_{t-1} \qquad \text{(smoothed absolute error)}\]
where \(0 < \beta < 1\).
beta_a <- 0.2
F_ar <- numeric(n_p); e_ar <- numeric(n_p)
E_ar <- numeric(n_p); AE_ar <- numeric(n_p)
alp_ar <- numeric(n_p)
alp_ar[1] <- 0.6; F_ar[1] <- price[1]
e_ar[1] <- 0; E_ar[1] <- 0; AE_ar[1] <- 0
for (i in 2:n_p) {
F_ar[i] <- alp_ar[i-1] * price[i-1] + (1 - alp_ar[i-1]) * F_ar[i-1]
e_ar[i] <- price[i] - F_ar[i]
E_ar[i] <- beta_a * e_ar[i] + (1 - beta_a) * E_ar[i-1]
AE_ar[i] <- beta_a * abs(e_ar[i]) + (1 - beta_a) * AE_ar[i-1]
alp_ar[i] <- ifelse(AE_ar[i] == 0, 0, abs(E_ar[i] / AE_ar[i]))
}
e2_ar <- e_ar^2
mse_ar <- mean(e2_ar[-1])
knitr::kable(
data.frame(Year = year, Price = price,
F_t = round(F_ar, 4), e = round(e_ar, 4),
E = round(E_ar, 4), AE = round(AE_ar, 4),
alpha = round(alp_ar, 4), e2 = round(e2_ar, 4)),
col.names = c("Year", "Price (RM)", "$F_t$", "$e_t$",
"$E_t$", "$AE_t$", "$\\alpha_t$", "$e_t^2$"),
align = "c"
)| Year | Price (RM) | \(F_t\) | \(e_t\) | \(E_t\) | \(AE_t\) | \(\alpha_t\) | \(e_t^2\) |
|---|---|---|---|---|---|---|---|
| 2018 | 6 | 6 | 0 | 0.0000 | 0.0000 | 0.6 | 0 |
| 2019 | 8 | 6 | 2 | 0.4000 | 0.4000 | 1.0 | 4 |
| 2020 | 9 | 8 | 1 | 0.5200 | 0.5200 | 1.0 | 1 |
| 2021 | 12 | 9 | 3 | 1.0160 | 1.0160 | 1.0 | 9 |
| 2022 | 15 | 12 | 3 | 1.4128 | 1.4128 | 1.0 | 9 |
| 2023 | 16 | 15 | 1 | 1.3302 | 1.3302 | 1.0 | 1 |
cat("MSE (ARRES, β = 0.2):", round(mse_ar, 4), "\n")MSE (ARRES, <U+03B2> = 0.2): 4.8 Excel Tutorial: ARRES in Excel (β = 0.2)
When a seasonal component exists, Holt-Winters is the most suitable method. Two model variants:
\[L_t = \alpha\frac{y_t}{S_{t-s}} + (1-\alpha)(L_{t-1}+T_{t-1})\] \[T_t = \beta(L_t-L_{t-1}) + (1-\beta)T_{t-1}\] \[S_t = \gamma\frac{y_t}{L_t} + (1-\gamma)S_{t-s}\] \[F_{t+m} = (L_t + T_t \times m)\cdot S_{t-s+m}\]
\[L_t = \alpha(y_t - S_{t-s}) + (1-\alpha)(L_{t-1}+T_{t-1})\] \[T_t = \beta(L_t-L_{t-1}) + (1-\beta)T_{t-1}\] \[S_t = \gamma(y_t - L_t) + (1-\gamma)S_{t-s}\] \[F_{t+m} = L_t + T_t \times m + S_{t-s+m}\]
where \(0\leq\alpha,\beta,\gamma\leq 1\); \(s\) = season length (\(s=4\) quarterly, \(s=12\) monthly).
hw_data <- data.frame(Year = c(rep(2022,4), rep(2023,4)),
Quarter = rep(1:4, 2),
Price = c(6, 8, 3, 1, 8, 10, 5, 3))
knitr::kable(hw_data, align = "c")| Year | Quarter | Price |
|---|---|---|
| 2022 | 1 | 6 |
| 2022 | 2 | 8 |
| 2022 | 3 | 3 |
| 2022 | 4 | 1 |
| 2023 | 1 | 8 |
| 2023 | 2 | 10 |
| 2023 | 3 | 5 |
| 2023 | 4 | 3 |
hw_ts <- ts(hw_data$Price, start = c(2022,1), frequency = 4)
hw_fit <- hw(hw_ts, seasonal = "additive",
alpha = 0.6, beta = 0.2, gamma = 0.4)
summary(hw_fit)
Forecast method: Holt-Winters' additive method
Model Information:
Holt-Winters' additive method
Call:
hw(y = hw_ts, seasonal = "additive", alpha = 0.6, beta = 0.2,
gamma = 0.4)
Smoothing parameters:
alpha = 0.6
beta = 0.2
gamma = 0.4
Initial states:
l = 4.5
b = 0.5
s = -3.5 -1.5 3.5 1.5
sigma: 0.7929
Error measures:
ME RMSE MAE MPE MAPE MASE
Training set -0.1462946 0.7929204 0.6676439 -11.58299 17.89702 0.333822
ACF1
Training set 0.1531588
Forecasts:
Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
2024 Q1 9.321220 8.305051 10.337388 7.7671242 10.875315
2024 Q2 11.050246 9.748915 12.351576 9.0600325 13.040459
2024 Q3 6.147005 4.495927 7.798083 3.6218986 8.672111
2024 Q4 4.438229 2.385669 6.490788 1.2991102 7.577348
2025 Q1 10.759448 8.010154 13.508743 6.5547660 14.964131
2025 Q2 12.488474 9.294407 15.682542 7.6035694 17.373380
2025 Q3 7.585234 3.904515 11.265952 1.9560608 13.214406
2025 Q4 5.876458 1.671927 10.080989 -0.5538171 12.306733HW_est <- hw(est_elec)
summary(HW_est)
Forecast method: Holt-Winters' additive method
Model Information:
Holt-Winters' additive method
Call:
hw(y = est_elec)
Smoothing parameters:
alpha = 0.9253
beta = 4e-04
gamma = 1e-04
Initial states:
l = 102.4914
b = 0.1712
s = -0.233 -2.2881 3.3367 -0.561 3.7136 1.0978
-3.7209 3.5383 -0.1766 4.6854 -8.8695 -0.5228
sigma: 3.8523
AIC AICc BIC
631.3403 640.3403 673.0642
Error measures:
ME RMSE MAE MPE MAPE MASE
Training set 0.01529911 3.475488 2.522562 -0.03638605 2.255034 0.4535252
ACF1
Training set 0.06272114
Forecasts:
Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
Mar 2022 123.3920 118.45517 128.3289 115.84175 130.9423
Apr 2022 118.7020 111.97466 125.4294 108.41339 128.9907
May 2022 122.5889 114.45510 130.7227 110.14932 135.0285
Jun 2022 115.5018 106.17037 124.8333 101.23060 129.7730
Jul 2022 120.4914 110.09866 130.8841 104.59708 136.3857
Aug 2022 123.2787 111.92278 134.6347 105.91130 140.6462
Sep 2022 119.1760 106.93167 131.4203 100.44993 137.9020
Oct 2022 123.2463 110.17328 136.3193 103.25285 143.2397
Nov 2022 117.7928 103.94001 131.6456 96.60677 138.9789
Dec 2022 120.0196 105.42804 134.6111 97.70374 142.3354
Jan 2023 119.9012 104.60604 135.1964 96.50927 143.2932
Feb 2023 111.7258 95.75746 127.6941 87.30434 136.1472
Mar 2023 125.4525 108.83772 142.0674 100.04236 150.8627
Apr 2023 120.7626 103.52511 138.0000 94.40016 147.1249
May 2023 124.6494 106.81062 142.4882 97.36734 151.9315
Jun 2023 117.5623 99.14137 135.9833 89.38991 145.7347
Jul 2023 122.5519 103.56620 141.5376 93.51579 151.5880
Aug 2023 125.3392 105.80473 144.8738 95.46378 155.2147
Sep 2023 121.2365 101.16772 141.3052 90.54397 151.9290
Oct 2023 125.3068 104.71729 145.8963 93.81787 156.7957
Nov 2023 119.8533 98.75553 140.9511 87.58704 152.1196
Dec 2023 122.0801 100.48560 143.6746 89.05417 155.1060
Jan 2024 121.9617 99.88133 144.0421 88.19268 155.7307
Feb 2024 113.7863 91.23011 136.3424 79.28959 148.2830HW_eva <- hw(eva_elec, alpha = 0.9522, beta = 0.0002, gamma = 0.0008)
summary(HW_eva)
Forecast method: Holt-Winters' additive method
Model Information:
Holt-Winters' additive method
Call:
hw(y = eva_elec, alpha = 0.9522, beta = 2e-04, gamma = 8e-04)
Smoothing parameters:
alpha = 0.9522
beta = 2e-04
gamma = 8e-04
Initial states:
l = 123.1084
b = 0.238
s = -11.6018 -3.1742 -1.905 -2.8557 2.5453 -1.8175
3.9752 4.0803 1.2517 6.6506 -0.5222 3.3732
sigma: 2.9547
AIC AICc BIC
115.0885 175.0885 130.3631
Error measures:
ME RMSE MAE MPE MAPE MASE
Training set 0.006215554 1.543055 0.9930067 -0.004220897 0.7778301 0.2679319
ACF1
Training set -0.3706544
Forecasts:
Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
Jan 2024 125.5331 121.7465 129.3198 119.74199 131.3243
Feb 2024 117.3436 112.1144 122.5729 109.34626 125.3410
Mar 2024 132.5567 126.2040 138.9093 122.84116 142.2722
Apr 2024 128.8994 121.5937 136.2051 117.72633 140.0725
May 2024 136.3103 128.1619 144.4586 123.84847 148.7721
Jun 2024 131.1494 122.2374 140.0614 117.51971 144.7791
Jul 2024 134.2160 124.6006 143.8315 119.51047 148.9216
Aug 2024 134.3490 124.0779 144.6201 118.64071 150.0573
Sep 2024 128.7944 117.9068 139.6820 112.14325 145.4456
Oct 2024 133.3954 121.9241 144.8666 115.85160 150.9391
Nov 2024 128.2324 116.2056 140.2592 109.83893 146.6259
Dec 2024 129.4212 116.8631 141.9793 110.21526 148.6271
Jan 2025 128.3900 115.3212 141.4588 108.40298 148.3770
Feb 2025 120.2005 106.6408 133.7602 99.46272 140.9383
Mar 2025 135.4135 121.3799 149.4472 113.95097 156.8761
Apr 2025 131.7563 117.2640 146.2485 109.59229 153.9202
May 2025 139.1671 124.2302 154.1041 116.32300 162.0113
Jun 2025 134.0063 118.6372 149.3753 110.50136 157.5111
Jul 2025 137.0729 121.2835 152.8623 112.92504 161.2207
Aug 2025 137.2059 121.0068 153.4049 112.43149 161.9802
Sep 2025 131.6513 115.0524 148.2501 106.26556 157.0369
Oct 2025 136.2522 119.2629 153.2415 110.26934 162.2351
Nov 2025 131.0893 113.7181 148.4604 104.52236 157.6561
Dec 2025 132.2781 114.5331 150.0230 105.13949 159.4166autoplot(electricts) + ylab("Output Index") + xlab("Year") +
autolayer(forecast(hw(electricts, alpha = 0.9522,
beta = 0.0002, gamma = 0.0008), h = 12),
series = "HW Forecast", PI = TRUE) +
labs(title = "Electricity Output Index: Holt-Winters Forecast") +
theme_ts()
Excel Tutorial: Holt-Winters in Excel (α = 0.6, β = 0.2, γ = 0.1)
Assumes past patterns repeat in the future. Suitable for data with both trend and seasonal components. Cyclical and irregular components are excluded.
| Relationship | Forecast |
|---|---|
| Multiplicative | \(F_t = \hat{T}_t \times S_t\) |
| Additive | \(F_t = \hat{T}_t + S_t\) |
| Step | Action |
|---|---|
| 1 | Compute trend (\(\hat{T}_t\)) using linear regression or moving average |
| 2 | Compute seasonal indices (\(S_t\)) |
| 3 | Combine trend and seasonal index to obtain forecasts |
Seasonal indices for motorcycle production (’000), 2021–2023:
| Quarter | Q1 | Q2 | Q3 | Q4 |
|---|---|---|---|---|
| Seasonal Index | 75 | 96 | 91 | 138 |
Trend: \(\hat{T}_t = 3.87 + 1.14t\), where \(t = 1\) for Q1 2021.
Forecast Q1 and Q4 of 2024:
SI <- c(75, 96, 91, 138)
t_vals <- c(13, 16) # Q1 2024 = t13, Q4 2024 = t16
T_hat <- 3.87 + 1.14 * t_vals
F_vals <- T_hat * SI[c(1,4)] / 100
knitr::kable(
data.frame(Period = c("Q1 2024","Q4 2024"),
t = t_vals, T_hat = round(T_hat, 4),
SI = SI[c(1,4)], Forecast = round(F_vals, 4)),
col.names = c("Period","$t$","$\\hat{T}_t$","SI","Forecast ('000)"),
align = "c"
)| Period | \(t\) | \(\hat{T}_t\) | SI | Forecast (’000) |
|---|---|---|---|---|
| Q1 2024 | 13 | 18.69 | 75 | 14.0175 |
| Q4 2024 | 16 | 22.11 | 138 | 30.5118 |
decomp_fit <- decompose(electricts, type = "multiplicative")
t_vec <- seq_along(electricts)
lm_elec <- lm(electricts ~ t_vec)
t_future <- (length(electricts) + 1):(length(electricts) + 12)
T_future <- predict(lm_elec, newdata = data.frame(t_vec = t_future))
SI_elec <- decomp_fit$seasonal
SI_future <- as.numeric(SI_elec[((t_future - 1) %% 12) + 1])
F_future <- ts(T_future * SI_future, start = c(2024,1), frequency = 12)
autoplot(electricts) +
autolayer(F_future, series = "Decomposition Forecast") +
labs(title = "Electricity Output: Decomposition-Based Forecast (2024)",
y = "Output Index", x = "Year") +
theme_ts()
# Compute HW MSE from the quarterly example (hw_ts / hw_fit already in environment)
e2_hw <- (hw_ts - fitted(hw_fit))^2
mse_hw <- mean(e2_hw) # all 8 quarterly observations
# n = number of errors contributing to each MSE
# (rows with NA forecast are excluded; SES/ARRES exclude init row t=1)
n_errors <- c(
sum(!is.na(F_naive)), # 5
sum(!is.na(F_trend)), # 4
sum(!is.na(F_ac)), # 3
sum(!is.na(F_apc)), # 3
length(F_ses) - 1, # 5 (exclude init)
sum(!is.na(F_des)), # 5
sum(!is.na(F_holt)), # 5
length(F_ar) - 1, # 5 (exclude init)
length(hw_ts) # 8 (quarterly data)
)
data.frame(
Method = c("Naive","Naive with Trend","Avg Change",
"Avg % Change","SES","DES","Holt's","ARRES",
"Holt-Winters *"),
Trend = c("No","Yes","No","No","No","Yes","Yes","No","Yes"),
Seasonal = c("No","No","No","No","No","No","No","No","Yes"),
MultiStep = c("Yes","Yes","Yes","Yes","No","Yes","Yes","No","Yes"),
n = n_errors,
MSE = round(c(mse_naive, mse_trend, mse_ac, mse_apc,
mse_ses, mse_des, mse_h, mse_ar,
mse_hw), 4)
) |>
knitr::kable(col.names = c("Method","Handles Trend","Handles Seasonal",
"Multi-step","n","MSE"),
align = "c")| Method | Handles Trend | Handles Seasonal | Multi-step | n | MSE |
|---|---|---|---|---|---|
| Naive | No | No | Yes | 5 | 4.8000 |
| Naive with Trend | Yes | No | Yes | 4 | 3.7140 |
| Avg Change | No | No | Yes | 3 | 2.4167 |
| Avg % Change | No | No | Yes | 3 | 4.1107 |
| SES | No | No | No | 5 | 6.6164 |
| DES | Yes | No | Yes | 5 | 2.2267 |
| Holt’s | Yes | No | Yes | 5 | 0.7241 |
| ARRES | No | No | No | 5 | 4.8000 |
| Holt-Winters * | Yes | Yes | Yes | 8 | 0.6287 |
Note on Holt-Winters MSE: Holt-Winters requires seasonal data and therefore uses a different dataset (quarterly, \(n = 8\)) compared to all other methods (annual price data, \(n = 6\)). The MSE value of Holt-Winters marked with * should not be directly compared with other methods — different dataset means the MSE not directly comparable with the others.
Data: https://drive.google.com/drive/folders/1SlWHeWAAVXre62j9VC-nKDdKiVd1KaM3?usp=sharing