Question 2
Step 1: Install and load the packages
install.packages("dplyr",repos = "http://cran.us.r-project.org")
## Installing package into 'C:/Users/User/AppData/Local/R/win-library/4.4'
## (as 'lib' is unspecified)
## package 'dplyr' successfully unpacked and MD5 sums checked
## Warning: cannot remove prior installation of package 'dplyr'
## Warning in file.copy(savedcopy, lib, recursive = TRUE): problem copying
## C:\Users\User\AppData\Local\R\win-library\4.4\00LOCK\dplyr\libs\x64\dplyr.dll
## to C:\Users\User\AppData\Local\R\win-library\4.4\dplyr\libs\x64\dplyr.dll:
## Permission denied
## Warning: restored 'dplyr'
##
## The downloaded binary packages are in
## C:\Users\User\AppData\Local\Temp\RtmpkVHbti\downloaded_packages
install.packages("zoo",repos = "http://cran.us.r-project.org")
## Installing package into 'C:/Users/User/AppData/Local/R/win-library/4.4'
## (as 'lib' is unspecified)
## package 'zoo' successfully unpacked and MD5 sums checked
##
## The downloaded binary packages are in
## C:\Users\User\AppData\Local\Temp\RtmpkVHbti\downloaded_packages
library(dplyr)
## Warning: package 'dplyr' was built under R version 4.4.2
##
## 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 4.4.2
##
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric
Step 2: Import the data
df <- data.frame(month=c(1,2,3,4,5,6,7,8,9,10,11,12),
values=c(240,352,230,260,280,322,220,310,240,310,240,230))
Part 2A: Time series plot
plot(df$month, df$values, type = "o", col = "blue", xlab = "Month", ylab = "Values",
main = "Alabama Building Contracts Values Plot")
Interpretation: The time series plot exhibits a a seasonal pattern as there are recurring patterns over the 12-month period.
Part 2B: Three-month moving average
Step 1: Manually calculate the three-month moving average
df$avg_values3 <- c(NA, NA, NA,
(df$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
)
Step 2: Calculate the squared errors
df <- df %>%
mutate(
squared_error = ifelse(is.na(avg_values3), NA, (values - avg_values3)^2)
)
Step 3: Compute the MSE
mse <- mean(df$squared_error, na.rm = TRUE)
mse
## [1] 2040.444
Step 4: Exponential smoothing
alpha <- 0.2
exp_smooth <- rep(NA, length(df$values))
exp_smooth[1] <- df$values[1]
for(i in 2: length(df$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
## [1] 2593.762
better_method <- ifelse(mse < mse_exp_smooth, "Three-Month Moving Average", "Exponential Smoothing")
Results
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"
Interpretation: The three-month moving average provides more accurate forecasts based on MSE than exponential smoothing because it has a smaller MSE and overall less error.
Question 3
Step 1: Install and load the packages
install.packages("ggplot2",repos = "http://cran.us.r-project.org")
## Installing package into 'C:/Users/User/AppData/Local/R/win-library/4.4'
## (as 'lib' is unspecified)
## package 'ggplot2' successfully unpacked and MD5 sums checked
##
## The downloaded binary packages are in
## C:\Users\User\AppData\Local\Temp\RtmpkVHbti\downloaded_packages
library(readxl)
## Warning: package 'readxl' was built under R version 4.4.2
library(ggplot2)
## Warning: package 'ggplot2' was built under R version 4.4.2
Step 2: Import the data
df <- read_excel(file.choose())
Part 3A: Time series plot
ggplot(df, aes(x = Period, y = Interest_Rate)) +
geom_line() +
geom_point() +
xlab("Period") +
ylab("Interest Rate") +
ggtitle("Time Series Plot of FreddieMan Mortgage Interest Rates")
Interpretation: We observe a decreasing pattern or trend in the time series plot, with a peak around the 23rd period.
Part 3B: Linear trend equation
model <- lm(Interest_Rate ~ Period, data = df)
summary(model)
##
## Call:
## lm(formula = Interest_Rate ~ Period, data = 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
Interpretation: The linear trend equation is Interest Rate = 6.70 - 0.13Period.
Part 3C: New data for prediction
new_data <- data.frame(Period = 25)
prediction <- predict(model, newdata = new_data)
prediction
## 1
## 3.472942