#Class Excercise 16: Chapter 17
:Project Objective
Use the naive method to answer question 1
Use the smoothing approach to answer question 2
Use the linear trend approach to answer question3 :Question 1
:Install and load required libraries
library(Metrics)
#Time Series Data
week <- 1:6 #This is the independent variable- time
values <- c(17,13,15,11,17,14) #dependent variable :Part A. Most Recent Vaule as Forecast
forecast_a <- values [-length(values)] # Excludes the last value
actual_a <- values[-1] #Exclude the first sale
mse_a <- mean((actual_a- forecast_a)^2)
mse_a #Mean square error is 16.2## [1] 16.2mae <- mae(forecast_a ,actual_a ) # The mean absolute error is
mae## [1] 3.8mean(abs((actual_a - forecast_a) / actual_a) * 100)## [1] 27.43778:Forecast the sales for week 11
forecast_week7_a <- tail(values, 1)
forecast_week7_a #Interpretation: Forecast for week 7 is 14## [1] 14:Part B. Average all of the Data as Forecast
cumulative_averages <- cumsum(values[-length(values)]) / (1:(length(values) -1))
cumulative_averages## [1] 17.0 15.0 15.0 14.0 14.6forecast_b <- cumulative_averages
actual_b <- values[-1] #Exclude the first value
mse_b <- mean((actual_b - forecast_b)^2)
mse_b # Mean square error is 8.272## [1] 8.272#Forecast the sales for week 7
forecast_week7_b <- mean(values) #Average of all weeks as forecast for week 7
forecast_week7_b## [1] 14.5#Interpretation for week 7 is 14.5:Part C. Comparison
better_method <- ifelse(mse_a < mse_b, "Most Recent Value", "Average of all Data")
#Results
list(
MSE_Most_Recent_Value = mse_a,
Forecast_Week7_Most_Recent = forecast_week7_a,
MSE_Average = mse_b,
Forecast_Week7_Average = forecast_week7_b,
Better_Method= better_method
)## $MSE_Most_Recent_Value
## [1] 16.2
##
## $Forecast_Week7_Most_Recent
## [1] 14
##
## $MSE_Average
## [1] 8.272
##
## $Forecast_Week7_Average
## [1] 14.5
##
## $Better_Method
## [1] "Average of all Data"#** Question 2 : Moving Average and Exponential Smoothing Aproach**
:Part A. Moving Average
:Load the libraries
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(zoo)##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric:Import the Data
df <- data.frame(month=c(1,2,3,4,5,6,7,8,9,10,11,12),
contracts=c(240,352,230,260,280,322,220,310,240,310,240,230)):Descriptive statistics
summary(df)## month contracts
## Min. : 1.00 Min. :220.0
## 1st Qu.: 3.75 1st Qu.:237.5
## Median : 6.50 Median :250.0
## Mean : 6.50 Mean :269.5
## 3rd Qu.: 9.25 3rd Qu.:310.0
## Max. :12.00 Max. :352.0#Interpretation : The average value of Alabama building contracts:Time series plot
plot(df$month, df$contracts, type="o", col="red", xlab="month", ylab="contracts",
main= "Alabama Buiding Contracts Plot")
# Interpretation: The timer series plot exhibits a horizontal pattern:Manually calculate the three-week moving average
df$avg_contracts3 <- c(NA, NA, NA,
(df$contracts[1] + df$contracts[2] + df$contracts[3]) / 3,
(df$contracts[2] + df$contracts[3] + df$contracts[4])/ 3,
(df$contracts[3] + df$contracts[4] + df$contracts[5])/ 3,
(df$contracts[4] + df$contracts[5] + df$contracts[6])/ 3,
(df$contracts[5] + df$contracts[6] + df$contracts[7])/ 3,
(df$contracts[7] + df$contracts[8] + df$contracts[9])/ 3,
(df$contracts[8] + df$contracts[9] + df$contracts[10])/ 3,
(df$contracts[9] + df$contracts[10] + df$contracts[11])/ 3,
(df$contracts[10] + df$contracts[11] + df$contracts[12])/ 3
)###:Calculate the squared errors (only for months where moving average is available)
df <- df %>%
mutate(
squared_error= ifelse(is.na(avg_contracts3), NA, (contracts- avg_contracts3)^2)
):Compute MSE (excluding the initial weeks with NA)
mse <- mean(df$squared_error, na.rm = TRUE)
mse # Output the MSE - 1395.704## [1] 1395.704:#Part B. Exponential Smoothing
alpha <- 0.2
exp_smooth <- rep(NA, length(df$contracts))
exp_smooth[1] <- df$contracts[1] #starting point
for(i in 2: length(df$contracts)) {
exp_smooth[i] <- alpha * df$contracts[i-1] + (1-alpha) * exp_smooth[i-1]
}
mse_exp_smooth <- mean((df$contracts[2:12] - exp_smooth[2:12]) ^2)
mse_exp_smooth # OUTPUT THE MSE - 2593.763## [1] 2593.762:Comparison
better_method <- ifelse(mse < mse_exp_smooth, "Three-Month Moving Average", "exponential
smoothing"):Results
list(
MSE_Moving_Average = mse,
MSE_Exponential_Smoothing = mse_exp_smooth,
Better_Method = better_method
)## $MSE_Moving_Average
## [1] 1395.704
##
## $MSE_Exponential_Smoothing
## [1] 2593.762
##
## $Better_Method
## [1] "Three-Month Moving Average"Part 3 Method 3: Linear Trend Regression Aproach
###:Load the libraries
library(readxl) #use this library to import excel files
library(ggplot2):Load the data and clean
mortgage_df <- read_excel("mort.xlsx")
mort_df <- subset(mortgage_df, select= -(Year)) :Descriptive Statistics
summary(mort_df)## Period Interest_Rate
## Min. : 1.00 Min. :2.958
## 1st Qu.: 6.75 1st Qu.:3.966
## Median :12.50 Median :4.863
## Mean :12.50 Mean :5.084
## 3rd Qu.:18.25 3rd Qu.:6.105
## Max. :24.00 Max. :8.053# On average the interest rate (%) for a 30-year fixed-rate mortgage
#over a 20-year period is ~ 5.08:Construct a time series plot
ggplot(mort_df, aes(x= Period, y= Interest_Rate))+
geom_line() +
geom_point() +
xlab("Period") +
ylab("Interest_Rate") +
ggtitle(" Time Series Plot of Interest Rates")
#Interpretation : I observe a decreasing pattern or trend in the time series plot:Develop a linear trend equation
mortgage_model <- lm(Interest_Rate ~ Period, data= mortgage_df)
summary(mortgage_model)##
## Call:
## lm(formula = Interest_Rate ~ Period, data = mortgage_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.3622 -0.7212 -0.2823 0.5015 3.1847
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.69541 0.43776 15.295 3.32e-13 ***
## Period -0.12890 0.03064 -4.207 0.000364 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.039 on 22 degrees of freedom
## Multiple R-squared: 0.4459, Adjusted R-squared: 0.4207
## F-statistic: 17.7 on 1 and 22 DF, p-value: 0.0003637# Result- linear trend equation: interest rates = 6.70 + -0.13*Period or
# T_hat = 6.70 + -0.13*t
# The R-square is 0.45
#The overall model is significant as the p-value < 0.05:To find the MSE and values
# Calculate the fitted values from the model
mort_df$predicted_mortgage <- predict(mortgage_model):Calculate the residuals
mort_df$residuals <- mort_df$Interest_Rate - mort_df$predicted_mortgage:Calculate the mwan squared error (MSE)
mse_mortgage <- mean(mort_df$residuals^2)
cat("Mean Squared Error(MSE):", mse, "")## Mean Squared Error(MSE): 1395.704forecast_interest_rate_period_25 <- predict(mortgage_model, newdata= data.frame(Period=25))
forecast_interest_rate_period_25## 1
## 3.472942