Question 1: Naive approach
Time series data
week <- 1:6
value <- c(17, 13, 15, 11, 17, 14)
Most recent value as forecast
forecast_1 <- value[-length(value)] # exclude the last value
actual_1 <- value[-1] # exclude the first value
mae_1 <- mean(abs(actual_1 - forecast_1))
mae_1 # Result: 3.8
## [1] 3.8
mse_1 <- mean((actual_1 - forecast_1)^2)
mse_1 # Result: 16.2
## [1] 16.2
mape_1 <- mean(abs((mae_1/actual_1)*100))
mape_1 # Result: 27.72
## [1] 27.72107
forecast_week7_1 <- tail(value, 1)
forecast_week7_1 # Result: 14
## [1] 14
Interpretation:
The mean absolute error is 3.8.
The mean squared error is 16.2.
The mean absolute percentage error is 27.72%.
Using the naive method (most recent value), the forecast for week 7 is 14
Question 2: Average approach
Install and load packages
#install.packages("dplyr")
#install.packages("zoo")
library(dplyr)
## Warning: package 'dplyr' was built under R version 4.3.3
##
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
##
## filter, lag
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union
library(zoo)
## Warning: package 'zoo' was built under R version 4.3.3
##
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric
Time series data
df <- data.frame(month=c(1:12), values=c(240, 352, 230, 260, 280, 322, 220,
310, 240, 310, 240, 230))
Time series plot
plot(df$month, df$values, type = "o", col = "blue",
xlab = "Month", ylab = "Building Contracts (in $ millions)",
main = "Alabama Building Contracts")
mean(df$values)
## [1] 269.5
Interpretation: The time series plot exhibits a horizontal pattern, fluctuating around the mean of 269.5.
Three-month moving average approach
df$avg_contracts <- c(NA, NA, NA,
(df$values[1] + df$values[2] + df$values[3]) / 3,
(df$values[2] + df$values[3] + df$values[4]) / 3,
(df$values[3] + df$values[4] + df$values[5]) / 3,
(df$values[4] + df$values[5] + df$values[6]) / 3,
(df$values[5] + df$values[6] + df$values[7]) / 3,
(df$values[6] + df$values[7] + df$values[8]) / 3,
(df$values[7] + df$values[8] + df$values[9]) / 3,
(df$values[8] + df$values[9] + df$values[10]) / 3,
(df$values[9] + df$values[10] + df$values[11]) / 3)
df <- df %>%
mutate(
squared_error = ifelse(is.na(avg_contracts), NA, (values - avg_contracts)^2)
) # calculate squared errors
mse_moving <- mean(df$squared_error, na.rm = TRUE)
mse_moving # Result: 2040.44
## [1] 2040.444
Interpretation: Using the three-month moving average approach, the mean squared error is 2040.44.
Exponential smoothing forecast
alpha = 0.02
exp_smooth <- rep(NA, length(df$values))
exp_smooth[1] <- df$values[1] # Starting point
for(i in 2:length(df$values)) {
exp_smooth[i] <- alpha * df$values[i-1] + (1 - alpha) * exp_smooth[i-1]
}
mse_exp_smooth <- mean((df$values[2:12] - exp_smooth[2:12])^2)
mse_exp_smooth # Result: 2738.34
## [1] 2738.335
Interpretation: Using the exponential smoothing forecast, the mean squared error is 2738.34.
Which method is better
better_method <- ifelse(mse_moving < mse_exp_smooth,
"Three-month moving average", "Exponential smoothing")
list(
MSE_moving_average = mse_moving,
MSE_exponential_smoothing = mse_exp_smooth,
Better_Method = better_method
) # Result: Better method = three-month moving average
## $MSE_moving_average
## [1] 2040.444
##
## $MSE_exponential_smoothing
## [1] 2738.335
##
## $Better_Method
## [1] "Three-month moving average"
Based on the MSE, the three-month moving average provides more accurate forecasts because its MSE of 2040.44 is lower than the exponential smoothing's MSE of 2738.34. The three-month moving average has a smaller error, making it a better method.
Questions 3-5
Install and load packages
#install.packages("readxl")
#install.packages("ggplot2")
library(readxl)
library(ggplot2)
## Warning: package 'ggplot2' was built under R version 4.3.3
Load the data
mortgage_df <- read_excel("Mortgage.xlsx")
Time series plot
ggplot(mortgage_df, aes(x = Year, y = Interest_Rate)) +
geom_line() +
geom_point() +
xlab("Year") +
ylab("Interest Rate") +
ggtitle("Mortgage Average Interest Rate")
Interpretation: The time series plot exhibits a cyclical pattern.
Linear trend equation
model <- lm(Interest_Rate ~ Period, data = mortgage_df)
summary(model)
##
## Call:
## lm(formula = Interest_Rate ~ Period, data = mortgage_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.3622 -0.7212 -0.2823 0.5015 3.1847
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.69541 0.43776 15.295 3.32e-13 ***
## Period -0.12890 0.03064 -4.207 0.000364 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.039 on 22 degrees of freedom
## Multiple R-squared: 0.4459, Adjusted R-squared: 0.4207
## F-statistic: 17.7 on 1 and 22 DF, p-value: 0.0003637
Result - linear trend equation: 6.70 - 0.13*Period OR y = 6.70 - 0.13t
Forecast for period 25 (i.e., 2024)
forecast_period_25 <- predict(model, newdata = data.frame(Period = 25))
forecast_period_25 # Result: 3.47
## 1
## 3.472942
Interpretation: Using the linear trend equation, the average interest rate for period 25, or 2024, is 3.47