#Class Exercise 16: Chapter 17 Forecasting

Question 1: Naive Approach

Objective:

Using the naive method (most recent value) as the forecast for the next week, compute measures of forecast accuracy.

Time Series Data

weeks <- 1:6
values <- c(17, 13, 15, 11, 17, 14)

Part A: Compute MSE using the most recent value as the forecast for the next period. Forecast the next week.

forecasts_a <- values[-length(values)] #exclude last value
actual_a <- values[-1] # exclude the first value
mse_a <- mean((actual_a - forecasts_a)^2)
mse_a

## [1] 16.2

MAE (Mean Absolute Error)

mae_a <- mean(abs(actual_a - forecasts_a))
mae_a

## [1] 3.8

MAPE (Mean Absolute Percentage Error)

mape_a <- mean(abs((actual_a - forecasts_a) / actual_a)) * 100
mape_a

## [1] 27.43778

###Forecast Week 7 & Output accuracy measures

forecast_week_7_a <- tail(values, 1)
forecast_week_7_a

## [1] 14

list(
  MSE_most_recent_value = mse_a,
  MAE_most_recent_value = mae_a,
  MAPE_most_recent_value = mape_a,
  forecast_week_7_most_recent = forecast_week_7_a)

## $MSE_most_recent_value
## [1] 16.2
## 
## $MAE_most_recent_value
## [1] 3.8
## 
## $MAPE_most_recent_value
## [1] 27.43778
## 
## $forecast_week_7_most_recent
## [1] 14

Question 2: Moving Avergae and Exponential Smoothing

The values of Alabama building contracts (in $ millions) for a 12-month period is as follows: 240, 352, 230, 260, 280, 322, 220, 310, 240, 310, 240, 230.

Part A: Construct a Time Series Plot. What type of pattern exists in the data?

# Intall packages
# install.packages("dplyr")
# install.packages("zoo")

library(dplyr) # helps work w moving averages / data manipulation

## 
## 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) # helps w time series

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

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))

# Time Series Plot
plot(df$month, df$data, type = "o", col = "pink", xlab = "Month", ylab = "Alabama Building Contracts (in $ millions)", 
     main = "Monthly Value of Alabama Building Contracts")

Interpretation: The time series plot exhibits a horizontal pattern as it is stead on the mean

Part B: Compare

Compare the three-month moving average approach with the exponential smoothing approach for a = 0.2. Which provides more accurate forecasts using MSE as the measure of forecast accuracy?

# Three Month Moving Average
df$avg_contract <- c(NA, NA, NA,
                     (df$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 months that moving average is available)
df <- df %>%
  mutate(
    squared_error = ifelse(is.na(avg_contract), NA, (data - avg_contract)^2)
  )

# Compute MSE (excluding the initial months with NA)
mse <- mean(df$squared_error, na.rm = TRUE)
mse

## [1] 2040.444

# mse = 2040.444

# Exponantial 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

## [1] 2593.762

# Compare
better_method <- ifelse(mse < mse_exp_smooth, "Three-month moving average", 
                        "Exponantial Smoothing")
list(
  MSE_moving_avg = mse,
  MSE_exp_smoothing = mse_exp_smooth,
  Better_Method = better_method
)

## $MSE_moving_avg
## [1] 2040.444
## 
## $MSE_exp_smoothing
## [1] 2593.762
## 
## $Better_Method
## [1] "Three-month moving average"

# Result: Better method - "Three-month moving average"

Question 3: Linear Trend

library(readxl)
library(ggplot2)

# Load the data
df <- read_excel("Mortgage.xlsx")

# Part A: Time Series Plot
plot(df$Year, df$Interest_Rate, type = "o", col = "purple", xlab = "Year", ylab = "Interest Rate (%)", 
     main = "Average interest rate (%) for a 30-year fixed-rate mortgage over a 20-year period")

ggplot(df, aes(x = Year, y = Interest_Rate)) +
  geom_line(color = "lightblue") +
  #geom_point(col = "purple") +
  labs(title = "Skechers' Revenue",
       x = "Year", y = "ARevenue (in millions $)") +
  theme_minimal()

# Interpretation: We observe an decreasing pattern/trend in the time series plot 
# until an abrupt and steep rise in interest rates after 2020  

# Part B. 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

# Part C. Revenue for Period 15 (2024)

forecast_period_25 <- predict(model, newdata = data.frame(Period = 25))
forecast_period_25

##        1 
## 3.472942