Class Exercise 16: Time Series
Project Objective
To Practice using Time Series Forecasting Methods and to Determine Types of Patterns from the Plots.Question 1: Use the Naive Method as the Forecast and Compute the Following Measures of Forecast Accuracy
Step 1: Input the Time Series Data
week <- 1:6
sales <- c(17,13,15,11,17,14)Step 2: Create Actual and Forecast Variables to Compute Error
forecast_a <- sales[-length(sales)]
actual_a <- sales[-1]Step 3: Compute MAE, MSE, Mape, and Forecasted Value of Week 7
mse <- mean((actual_a - forecast_a)^2)
mse## [1] 16.2mae <- mean(abs(actual_a - forecast_a))
mae## [1] 3.8mape <- mean(abs((actual_a - forecast_a) / actual_a) * 100)
mape## [1] 27.43778forecast_week7_a <- tail(sales, 1)
forecast_week7_a## [1] 14Interpretation: The Values are Computed for the Following Errors. MAE = 3.8 | MSE = 16.2 | MAPE = 27.44 | Week 7 Forecast = 14Question 2: Construct a Time Series Plot for the Values of Alabama Building Contracts Over a 12 Month Period and Utilize the Moving Average and Exponential Smoothening Forecasts Methods
Step 1: Install & Load Required Libraries
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, unionlibrary(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.numericStep 2: Input Alabama Building Contracts Data and Create the Plot
df <- data.frame(months = 1:12, building_contracts = c(240, 352, 230, 260, 280, 322, 220, 310, 240, 310, 240, 230))
summary(df)## months building_contracts
## 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. :352.0plot(df$months, df$building_contracts, type = "o", col = "blue", xlab = "Months", ylab = "Building Contractions (In Millions)", main ="Alabama Building Contracts" )
Interpretation: This is a horizontal pattern as the price of building contracts fluctuate around the mean of the 12 month period. There is no signs of a trend or seasonal pattern as it does not consistently move up or down and looks to react similarly across the year.Step 3: Set-up the Three-Month Moving Average and Compute the MSE
df$avg_sales3 <- c(NA, NA, NA,
(df$building_contracts[1] + df$building_contracts[2] + df$building_contracts[3]) / 3,
(df$building_contracts[2] + df$building_contracts[3] + df$building_contracts[4]) / 3,
(df$building_contracts[3] + df$building_contracts[4] + df$building_contracts[5]) / 3,
(df$building_contracts[4] + df$building_contracts[5] + df$building_contracts[6]) / 3,
(df$building_contracts[5] + df$building_contracts[6] + df$building_contracts[7]) / 3,
(df$building_contracts[6] + df$building_contracts[7] + df$building_contracts[8]) / 3,
(df$building_contracts[7] + df$building_contracts[8] + df$building_contracts[9]) / 3,
(df$building_contracts[8] + df$building_contracts[9] + df$building_contracts[10]) / 3,
(df$building_contracts[9] + df$building_contracts[10] + df$building_contracts[11]) / 3
)
df <- df %>%
mutate(
squared_error = ifelse(is.na(avg_sales3), NA, (building_contracts - avg_sales3)^2)
)
mse <- mean(df$squared_error, na.rm = TRUE)
mse## [1] 2040.444Interpretation: The Computed MSE Value Using This Method is 2040.44Step 4: Set-up Exponential Smoothing Function and Compute the MSE
alpha <- 0.2
exp_smooth <- rep(NA, length(df$building_contracts))
exp_smooth[1] <- df$building_contracts[1]
for(i in 2: length(df$building_contracts)) {
exp_smooth[i] <- alpha * df$building_contracts[i-1] + (1 - alpha) * exp_smooth[i-1]
}
mse_exp_smooth <- mean((df$building_contracts[2:12] - exp_smooth[2:12])^2)
mse_exp_smooth## [1] 2593.762Interpretation: The Computed MSE Value Using Exponential Smoothing is 2593.76Step 5: Set-up ifelse Function to Determine Method With the Lowest Error
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"Interpretation: The moving average method is more accurate because it has less error than the exponential smoothening method which is significantly higher.Question 3: Construct a Time Series Plot for the Average Interest Rate of a 30-year Fixed Mortgage Rate Over a 20 Year Period
Step 1: Install Packages & Load Data
library(readxl)
library(ggplot2)
df <- read_excel(file.choose())
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. :8.053Step 2: Construct Time Series Plot
ggplot(df, aes(x = Period, y = Interest_Rate)) +
geom_line() +
geom_point() +
xlab("Period") +
ylab("Interest Rate") +
ggtitle("Time Series Plot of Mortgage Interest Rate")
Interpretation: Over the 20 year period, there seems to be a consistent decline in interest rate from 2000 to 2020Step 3: Create Linear 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: 0.0003637Interpretation: The Equation is 6.70 - 0.13*PeriodStep 4: Forecast the Average Interest Rate For Period 25
period_25 <- predict(model, newdata = data.frame(Period = 25))
period_25## 1
## 3.472942Interpretation: The Forecasted Average Interest Rate of Period 25 is 3.47