Class Exercise 16: Chapter 17
library(readxl)
library(dplyr)
##
## 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)
##
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric
library(lubridate)
##
## Attaching package: 'lubridate'
## The following objects are masked from 'package:base':
##
## date, intersect, setdiff, union
library(forecast)
## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo
library(ggplot2)
Question 1: Naive Forecasting Time Series Data
# Time Series Data
week <- 1:6
values <- c(17, 13, 15, 11, 17, 14)
# Naive Forecasting Approach
forecast_a <- values[-length(values)]
actual_a <- values[-1]
# Calculate MSE
mse_a <- mean((actual_a - forecast_a)^2)
mse_a # MSE is 16.2
## [1] 16.2
# Calculate MAE
mae_a <- mean(abs(actual_a - forecast_a))
mae_a # MAE is 3.8
## [1] 3.8
# Calculate MAPE
mape_a <- mean(abs((actual_a - forecast_a) / actual_a)) * 100
mape_a # MAPE is 27.44%
## [1] 27.43778
# Forecast the value for week 7
forecast_week7_a <- tail(values,1)
forecast_week7_a
## [1] 14
# Interpretation: The value will be 14 in week 7.
Question 2: Moving Average vs Exponential Smoothing Forecasting
# Time Series Data
df_b <- 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))
# Descriptive Statistics
summary(df_b)
## month values
## Min. : 1.00 Min. :220.0
## 1st Qu.: 3.75 1st Qu.:237.5
## Median : 6.50 Median :250.0
## Mean : 6.50 Mean :269.5
## 3rd Qu.: 9.25 3rd Qu.:310.0
## Max. :12.00 Max. :352.0
# Time series plot
plot(df_b$month, df_b$values, type = "o", col = "blue", xlab = "Month", ylab = "Values (in $ millions)",
main = "The Values of Alabama Building Contracts for a 12-Month Period")
# Interpretation: Time series plot data is horizontal.
# Manually calculate the 3-Month Moving Average
df_b$avg_values3 <- c(NA, NA, NA,
(df_b$values[1] + df_b$values[2] + df_b$values[3])/3,
(df_b$values[2] + df_b$values[3] + df_b$values[4])/3,
(df_b$values[3] + df_b$values[4] + df_b$values[5])/3,
(df_b$values[4] + df_b$values[5] + df_b$values[6])/3,
(df_b$values[5] + df_b$values[6] + df_b$values[7])/3,
(df_b$values[6] + df_b$values[7] + df_b$values[8])/3,
(df_b$values[7] + df_b$values[8] + df_b$values[9])/3,
(df_b$values[8] + df_b$values[9] + df_b$values[10])/3,
(df_b$values[9] + df_b$values[10] + df_b$values[11])/3
)
df_b <- df_b %>%
mutate(
squared_error_b_ma = ifelse(is.na(avg_values3), NA, (df_b$values - avg_values3)^2)
)
# Compute MSE for 3 Month Moving Average
mse_b_ma <- mean(df_b$squared_error_b_ma, na.rm = TRUE)
mse_b_ma # MSE for Moving Average is 2040.44
## [1] 2040.444
# Compute Exponential Smoothing
alpha <- 0.2
exp_smooth <- rep(NA, length(df_b$values))
exp_smooth[1] <- df_b$values[1]
for(i in 2: length(df_b$values)) {
exp_smooth[i] <- alpha * df_b$values[i-1] + (1 - alpha) * exp_smooth[i-1]
}
mse_b_exp_smooth <- mean((df_b$values[2:length(df_b$values)] - exp_smooth[2:length(exp_smooth)])^2, na.rm = TRUE)
mse_b_exp_smooth # MSE for EXP Smoothing is 2593.76
## [1] 2593.762
# Comparison
better_method <- (ifelse(mse_b_ma < mse_b_exp_smooth, "Three-Month Moving Average", "Exponential Smoothing"))
# Results
list(
MSE_B_Moving_Average = mse_b_ma,
MSE_B_Exponential_Smoothing = mse_b_exp_smooth,
Better_Method = better_method
)
## $MSE_B_Moving_Average
## [1] 2040.444
##
## $MSE_B_Exponential_Smoothing
## [1] 2593.762
##
## $Better_Method
## [1] "Three-Month Moving Average"
# Interpretation: The Three-Month Moving Average method is more accurate because it's mean square error is lower at 2040, when compared to the mean
# square error of the Exponential Smoothing method whose mean square error is 2593.76.
Question 3: Mortgage Data
# Time Series Data
mortgage_df <- read_excel("Mortgage.xlsx")
# Descriptive Statistics
summary(mortgage_df)
## Year Period Interest_Rate
## Min. :2000-01-01 00:00:00 Min. : 1.00 Min. :2.958
## 1st Qu.:2005-10-01 18:00:00 1st Qu.: 6.75 1st Qu.:3.966
## Median :2011-07-02 12:00:00 Median :12.50 Median :4.863
## Mean :2011-07-02 18:00:00 Mean :12.50 Mean :5.084
## 3rd Qu.:2017-04-02 06:00:00 3rd Qu.:18.25 3rd Qu.:6.105
## Max. :2023-01-01 00:00:00 Max. :24.00 Max. :8.053
# On average, the interest rate is 5.08% over a 23-year period (24 periods). The median interest rate is 4.86%.
# Time series plot
ggplot(mortgage_df, aes(x = Period, y = Interest_Rate,)) +
geom_line() +
geom_point() +
xlab("Period") +
ylab("Interest Rate") +
ggtitle("Mortgage Interest Rates Time Series Plot")
# Interpretation: This time series data plot exhibits a decreasing pattern or trend.
# Develop a 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 - estimated linear trend equation: Interest Rate = 6.70 - 0.13 * Period OR
# T_hat = 6.70 - 0.13 * t
# The R-squared is 0.45, which shows it moderately fits the data
# The overall model is significant because the p-value is less than alpha at 0.5
# Calculate MSE and MAPE values and begin prediction
mortgage_df$predicted_interest_rate <- predict(model)
# Calculate the residuals
mortgage_df$residuals <- mortgage_df$Interest_Rate - mortgage_df$predicted_interest_rate
# Calculate the MSE
mse_mortgage <- mean(mortgage_df$residuals^2)
cat("MSE:", mse_mortgage, "\n") # MSE for this model is 0.99
## MSE: 0.989475
# Calculate the MAPE
mortgage_df$percentage_error <- abs(mortgage_df$residuals / mortgage_df$Interest_Rate) * 100
mape_mortgage <- mean(mortgage_df$percentage_error)
cat("MAPE:", mape_mortgage, "%\n") # MAPE is 15.79%
## MAPE: 15.79088 %
# Forecast the interest rate for Period 25 (2024)
forecast_period_25 <- predict(model, newdata = data.frame(Period = 25))
forecast_period_25
## 1
## 3.472942
# Interpretation: The interest rate for 30-year mortgages will be ~3.47% in 2024 (Period 25).