Ex16.knit

Question 1: Using Naive Method(Most recent value)

Step 1: Enter Data

# Time Series Data
week <- 1:6 # Independent variable - time
value <- c(17, 13, 15, 11, 17, 14) # Dependent Variable

Step 2: Set up actual and forecast data

forecast <- value[-length(value)] # Excludes the last value
actual <- value[-1] # Exclude the first value

Step 3: Mean absolute error

error <- actual - forecast 
mae <- mean(abs(error))
mae #Mean absolute error is 3.8

## [1] 3.8

Step 4: Mean squared error

mse <- mean((actual - forecast)^2)
mse #Mean squared error is 16.2

## [1] 16.2

Step 5: Mean absolute percentage error

mape <- mean(abs((actual - forecast)/ actual)) * 100
mape #Mean absolute percentage error is 27.44%

## [1] 27.43778

Step 6: Forecast for week 7

forecast_week11_a <- tail(value, 1)
forecast_week11_a

## [1] 14

# Interpretation: The projected number for week 7 is 14

Question 2: Smoothing Approach

Step 1: Install and load packages

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

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)

## 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: Enter Data

df <- data.frame(week=c(1,2,3,4,5,6,7,8,9,10,11,12),
contract=c(240,352,230,260,280,322,220,310,240,310,240,230))

Step 3: Time Series Plot

plot(df$week, df$contract, type = "o", col = "blue", xlab = "Week", ylab = "Contracts in millions", main = "Alabama Building Contract Plot")

# Interpretation: The plot exhibits a fluctuating pattern as it jumps above and below the mean.

Step 4: Three Month Average

df$avg <- c(NA, NA, NA,
            (df$contract[1] + df$contract[2] + df$contract[3])/ 3,
            (df$contract[2] + df$contract[3] + df$contract[4])/ 3,
            (df$contract[3] + df$contract[4] + df$contract[5])/ 3,
            (df$contract[4] + df$contract[5] + df$contract[6])/ 3,
            (df$contract[5] + df$contract[6] + df$contract[7])/ 3,
            (df$contract[6] + df$contract[7] + df$contract[8])/ 3,
            (df$contract[7] + df$contract[8] + df$contract[9])/ 3,
            (df$contract[8] + df$contract[9] + df$contract[10])/ 3,
            (df$contract[9] + df$contract[10] + df$contract[11])/ 3
)

Step 5: Calculate the squared errors (only for weeks were moving average is available)

df <- df %>%
  mutate(squared_error = ifelse(is.na(avg), NA, (contract - avg)^2))

Step 6: Compute MSE (excluding the initial weeks with NA)

mse <- mean(df$squared_error, na.rm = TRUE)
mse #Output the MSE - 2040.44

## [1] 2040.444

Step 7: Exponential Smoothing using alpha = 0.2

# Exponential Smoothing 
alpha <- 0.2
exp_smooth <- rep(NA, length(df$contract))
exp_smooth[1] <- df$contract[1] # Starting point 
for(i in 2: length(df$contract)) {
  exp_smooth[i] <- alpha * df$contract[i-1] + (1-alpha) * exp_smooth[i-1]}

Step 8: Compute MSE (excluding the initial weeks with NA)

mse_exp_smooth <- mean((df$contract[2:12] - exp_smooth[2:12])^2)
mse_exp_smooth #Output the MSE - 2593.76

## [1] 2593.762

Step 9: Compare

# MSE for three month moving average is 2040.44
# MSE for exponential smoothing using alpha 0.2 = 2593.76
# Exponential smoothing provides more accurate forecasts based on MSE

Question 3: Linear Trend Approach

Step 1: Install and load packages

# install.packages("ggplot2")
# install.packages("readxl")
# install.packages("dplyr")
library(ggplot2)
library(readxl)

## Warning: package 'readxl' was built under R version 4.4.2

library(dplyr)

Step 2: Import and Load data

mortgage_data <- read_excel(file.choose())

Step 3: Construct Time Series Plot

data <- subset(mortgage_data, select = -c(Year)) # eliminates Year column

ggplot(data, aes(x = Period, y = Interest_Rate)) + 
  geom_line() + 
  geom_point() + 
  xlab("Period") +
  ylab("Interest Rate") + 
  ggtitle("Time Series Plot of Fixed-Rate Mortgage")

# Interpretation: Decreasing pattern or trend then started increasing when it entered period 23

Step 4: Develop Linear trend equation

model <- lm(Interest_Rate ~ Period, data = data)
summary(model)

## 
## Call:
## lm(formula = Interest_Rate ~ Period, data = data)
## 
## 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 = Interest Rate = 6.70 - 0.13*Period or 
# T_hat = 6.70 - 0.13*t
# R-squared is 0.45 (Moderate fit)
# Overall model is significant as p-value < 0.05

Step 5: Forecast period 25

forecast_period25 <- predict(model, newdata = data.frame(Period = 25))
forecast_period25

##        1 
## 3.472942

# Interpretation: The forecasted number for interest rate in 2024 is ~3.47%