Project Objective: To forecast for the next week, using Naive
method
# Install necessary package
library(ggplot2)
# Time Series Data
week <- 1:6
values <- c(17, 13, 15, 11, 17, 14)
# Exclude the Last and First Value
forecast_a <- values[-length(values)]
actual_a <- values[-1]
# Find the MSE (Mean Squared Error)
mse_a <- mean((actual_a - forecast_a)^2)
print(mse_a)
## [1] 16.2
# Forecast the sales for the week 7
forecast_week7_a <- tail(values, 1)
print(forecast_week7_a)
## [1] 14
# Find the MAE (Mean Absolute Error)
mae_a <- mean(abs(actual_a - forecast_a))
print(mae_a)
## [1] 3.8
# Find the Mean Absolute Percentage Error (MAPE)
mape_a <- mean(abs((actual_a - forecast_a) / actual_a)) * 100
print(mape_a)
## [1] 27.43778
# Question 2. Moving Average and Smoothing Exponential
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
# Import the data
df <- data.frame(month = 1:12,
values = c(240, 352, 230, 260, 280, 322, 220, 310, 240, 310, 240, 230))
# Descriptive Statistics
summary(df)
## 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$month, df$values, type = "o", col = "green", xlab = "Month", ylab = "$ Millions",
main = "Values of Alabama Building contracts for 12 months")
# Calculate the Three-Month Moving Average
df$avg_values3 <- zoo::rollmean(df$values, k = 3, fill = NA, align = "right")
# Calculate Squared Errors
df <- df %>%
mutate(squared_error = ifelse(is.na(avg_values3), NA, (values - avg_values3)^2))
# Compute Mean Squared Error (MSE)
mse <- mean(df$squared_error, na.rm = TRUE)
print(mse)
## [1] 996.8
# Exponential Smoothing
alpha <- 0.2
exp_smooth <- rep(NA, length(df$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[-1] - exp_smooth[-1])^2, na.rm = TRUE)
print(mse_exp_smooth)
## [1] 2593.762
# Comparison
better_method <- ifelse(mse < mse_exp_smooth, "Three-Month Moving Average", "Exponential Smoothing")
print(better_method)
## [1] "Three-Month Moving Average"
# Question 3. Time Series Plot and Pattern
library(ggplot2)
library(readxl)
# Load the data
df <- read_excel("Mortgage1.xlsx")
df
## # A tibble: 24 × 3
## Year Period Interest_Rate
## <dttm> <dbl> <dbl>
## 1 2000-01-01 00:00:00 1 8.05
## 2 2001-01-01 00:00:00 2 6.97
## 3 2002-01-01 00:00:00 3 6.54
## 4 2003-01-01 00:00:00 4 5.83
## 5 2004-01-01 00:00:00 5 5.84
## 6 2005-01-01 00:00:00 6 5.87
## 7 2006-01-01 00:00:00 7 6.41
## 8 2007-01-01 00:00:00 8 6.34
## 9 2008-01-01 00:00:00 9 6.03
## 10 2009-01-01 00:00:00 10 5.04
## # ℹ 14 more rows
# Descriptive Statistics
summary(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
# Construct a time series plot
ggplot(df, aes(x = Period, y = `Interest_Rate`)) +
geom_line() +
geom_point() +
xlab("Period") +
ylab("Interest Rate") +
ggtitle("Interest Rate of Mortgage")
# Question 4. 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
# Question 5. Forecast for Period 25
df$predicted_interest_rate <- predict(model)
# Calculate the residuals
df$residuals <- df$Interest_Rate - df$predicted_interest_rate
# Calculate MSE and MAPE
mse <- mean(df$residuals^2, na.rm = TRUE)
cat("Mean Squared Error (MSE):", mse, "\n")
## Mean Squared Error (MSE): 0.989475
mape <- mean(abs(df$residuals / df$Interest_Rate) * 100, na.rm = TRUE)
cat("Mean Absolute Percentage Error (MAPE):", mape, "%\n")
## Mean Absolute Percentage Error (MAPE): 15.79088 %
# Forecast for Period 25
forecast_period_25 <- predict(model, newdata = data.frame(Period = 25))
print(forecast_period_25)
## 1
## 3.472942