Time Series Analysis

Project Objective

Forecasting Trends and Statistics

Method 1: Naive Approach

#Step 1: Load Data

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

#Step 2: Find following measures to forecast accuracy

forecast_a <- values[-length(values)] #Excludes the last value
actual_a <- values[-1] #Excludes the first sale

error_a <- actual_a - forecast_a 
mae <- mean(abs(error_a))
mae #Mean Absolute Error is 3.8

## [1] 3.8

mse_a <- mean((actual_a - forecast_a)^2)
mse_a #Mean Square Error is 16.2

## [1] 16.2

mape <- mean(abs((actual_a - forecast_a)/ actual_a)) * 100
mape #Mean Absolute Percentage Error is 27.44%

## [1] 27.43778

#Step 3: Forecasting Week 7

forecast_week7_a <- tail(values, 1)
forecast_week7_a #Week 7 Forecast is 14

## [1] 14

Method 2: Smoothing Approach

#Step 1: Load Libraries

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

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: Load Data

df <- data.frame(month = c(1:12),
                 data = c(240,352,230,260,280,322,220,310,240,310,240,230))

#Step 3: Summarize Data

summary(df)

##      month            data      
##  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

#Interpretation: Average sales over the 12 week period is 269.5


#Step 4: Create Time Series Plot

``` r
plot(df$month, df$data, type = "o", col = "blue", xlab = "Month", ylab = "Contract",
     main = "Alabama building contracts")

#Interpretation: The time series plot exhibits fluctuations going above and below the mean.

#Step 5: Create new column in data averaging three months

df$avg_data3 <- c(NA, NA, NA,
                  (df$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)

#Step 6: Calculate the squared_error for weeks with three-month moving average

df <- df %>% 
  mutate(
    squared_error = ifelse(is.na(avg_data3), NA, (data - avg_data3)^2)
    )

#Step 7: Compute MSE (Excluding initial weeks with NA)

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

## [1] 2040.444

#Step 8: Use Exponential 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 #MSE = 2593.76

## [1] 2593.762

#Step 9: Compare Whether Three-Month Moving Average or Exponential Smoothing is better

better_method <- ifelse(mse < mse_exp_smooth, "Three-Month Moving Average", "Exponential Smoothing")

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"

##Method 3: Linear Trend Approach

#Step 1: Install Packages

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

library(ggplot2)
library(readxl)
library(dplyr)

#Step 2: Import Data and only use relevant columns

mort_data <- read_excel(file.choose())

data <- data.frame(
  period <- mort_data$Period,
  rate <- mort_data$Interest_Rate
)

#Step 3: Summarize Data

summary(mort_data)

##       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

#Interest rate on average is 5.08%

#Step 4: Create Time Series Plot

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

#Step 5: Develop the linear trend equation

model <- lm(rate ~ period, data = data)
summary(model)

## 
## Call:
## lm(formula = 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

#Results - Estimated Linear Trend Equation Rate = 6.70 - 0.13*Period

#Step 6: Forecast Period 25

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

##        1 
## 3.472942

** Time Series Analysis **

Project Objective

Method 1: Naive Approach

Method 2: Smoothing Approach

Time Series Analysis