Question 1
Using the naive method (most recent value) as the forecast for the next week, compute the following measures of forecast accuracy.
Load the Data
week <- 1:6
values <- c(17, 13, 15, 11, 17, 14)
Part A: MAE
forecasts_a <- values[-length(values)] #exclude the last value
actual_a <- values[-1] #exclude the first value
mae_a <- mean(abs(actual_a - forecasts_a))
round((mae_a), 2)
## [1] 3.8
Part B: MSE
mse_a <- mean((actual_a - forecasts_a)^2)
round((mse_a), 2)
## [1] 16.2
Part C: MAPE
Import libraries
library(MLmetrics)
## Warning: package 'MLmetrics' was built under R version 4.3.3
##
## Attaching package: 'MLmetrics'
## The following object is masked from 'package:base':
##
## Recall
MAPE
mape_a <- MAPE(forecasts_a, actual_a)
round((mape_a), 2)
## [1] 0.27
Part D: Forecast for Week 7
forecast_week_7_a <- tail(values, 1)
round((forecast_week_7_a), 2)
## [1] 14
Question 2
Part A: Construct a time series plot. What type of pattern exists in the data? Upload a screenshot of your time series plot.
Part B: Compare the three-month moving average approach with the exponential smoothing forecast using alpha = 0.2. Which provides more accurate forecasts based on MSE? Upload a screenshot of your code outputs with proper explanation of your answers.
Load the packages
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
Load the Data
df <- data.frame(month=c(1,2,3,4,5,6,7,8,9,10,11,12),
data=c(240, 352, 230, 260, 280, 322, 220, 310, 240, 310, 240, 230))
Part A: Time Series Plot
plot(df$month, df$data, type = "o", col = "blue", xlab = "Month", ylab = "Values of Building Contracts ($ millions)",
main = "Monthly Percentage of Values of Building Contracts")
Interpretation: The time-series plot exhibits a seasonal pattern
Part B: Three-month moving average
df$avg_value <- c(NA, NA, NA,
(df$data[1] + df$data[2] + df$data[3]) / 3,
(df$data[2] + df$data[3] + df$data[4]) / 3,
(df$data[3] + df$data[4] + df$data[5]) / 3,
(df$data[4] + df$data[5] + df$data[6]) / 3,
(df$data[5] + df$data[6] + df$data[7]) / 3,
(df$data[6] + df$data[7] + df$data[8]) / 3,
(df$data[7] + df$data[8] + df$data[9]) / 3,
(df$data[8] + df$data[9] + df$data[10]) / 3,
(df$data[9] + df$data[10] + df$data[11]) / 3)
Calculate the squared errors (only for the months where moving
average is avaiable)
df <- df %>%
mutate(
squared_error = ifelse(is.na(avg_value), NA, (data - avg_value)^2)
)
Compute MSE (excluding the initial months with NA)
mse <- mean(df$squared_error, na.rm = TRUE)
mse #2040.444
## [1] 2040.444
Exponential smoothing
alpha = 0.2
exp_smooth <- rep(NA, length(df$data))
exp_smooth[1] <- df$data[1] #starting point
for(i in 2:length(df$data)) {
exp_smooth[i] <- alpha * df$data[i-1] + (1 - alpha) * exp_smooth[i-1]
}
mse_exp_smooth <- mean((df$data[2:12] - exp_smooth[2:12])^2)
mse_exp_smooth #3.56
## [1] 2593.762
better_method <- ifelse(mse < mse_exp_smooth, "Three month moving average", "Exponential Smoothing")
list(
MSE_moving_average = mse,
MSE_exponential_smoothing = mse_exp_smooth,
Better_Method = better_method
)
## $MSE_moving_average
## [1] 2040.444
##
## $MSE_exponential_smoothing
## [1] 2593.762
##
## $Better_Method
## [1] "Three month moving average"
Result: Better method is three month moving average
Question 3
Construct a time series plot. What type of pattern exists in the data?
Load the packages
library(readxl)
library(ggplot2)
## Warning: package 'ggplot2' was built under R version 4.3.3
Load the data
df1 <- read_excel("Mortgage.xlsx")
Time series plot
ggplot(df1, aes(x = Year, y = Interest_Rate)) +
geom_line() +
geom_point() +
xlab("Year") +
ylab("Interest Rate (%)") +
ggtitle("Time Series Plot of Average Interest Rate for a 30-Year Fixed-Rate Mortgage")
Interpretation: The time-series plot exhibits a decreasing trend pattern.
Question 4
Develop the linear trend equation for this time series.
Develop the Model
model <- lm(Interest_Rate ~ Period, data = df1)
summary(model)
##
## Call:
## lm(formula = Interest_Rate ~ Period, data = df1)
##
## 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: Interest_Rate = 6.70 - 0.13*Period
Question 5
Using the linear trend equation from question 3B, forecast the average interest rate for period 25 (i.e., 2024).
forecast_period_25 <- predict(model, newdata = data.frame(Period = 25))
round((forecast_period_25), 2)
## 1
## 3.47