Question 1: Naive Method
Time Series Data
week <- 1:6 #This is the independent value variable
value <- c(17,13,15,11,17,14) #dependent variable
Part A: Most Recent Values as Forecast
forecast_a <- value[-length(value)] # Excludes the last value
actual_a <- value [-1] #Exclude the first sale
Mean Absolute Error
mae_a <- mean(abs(actual_a - forecast_a))
mae_a #3.8
## [1] 3.8
Mean Squared Error
forecast_a <- value[-length(value)] # Excludes the last value
actual_a <- value [-1] #Exclude the first sale
mse_a <- mean((actual_a - forecast_a)^2)
mse_a #16.2
## [1] 16.2
Mean Absolute Percentage Error
mape <- mean(abs(actual_a - forecast_a)/ actual_a) * 100
mape #27.44
## [1] 27.43778
Forecast Sales for Week 7
forecast_week7 <- tail(value, 1)
forecast_week7
## [1] 14
###Interpretation: the value forecast for week 7 is 14.
Question 2: Moving Average and Exponential Smoothing Approach
Part A: Moving Average
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
Time Series 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)) # in $ millions
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 is 269.5
Time Series Plot
plot(df$month,df$contracts,type = "o", col = "blue",xlab = "Month", ylab = "Values of Contracts (in $ million)", main = "Alabama Building Contracts")
Compare Three-Month Moving Average Approach
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$contracts[6] + df$contracts[7] + df$contracts[8])/ 3,
(df$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
)
Calculate the squared errors (only for weeks moving average is
available)
df <- df %>%
mutate(
squared_error = ifelse(is.na(avg_contracts3), NA, (contracts - avg_contracts3)^2)
)
Compute MSE (excluding the initla weeks with NA)
mse <- mean(df$squared_error,na.rm = TRUE)
mse # Output MSE - 2040.44
## [1] 2040.444
Exponential Smoothing
alpha <- 0.2
exp_smooth <- rep(NA, length(df$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 MSE - 2593.76
## [1] 2593.762
Comparison
better_method <- ifelse(mse < mse_exp_smooth, "Three-Month Moving Average", "Exponential Smoothing")
better_method # The "Three-Month Moving Average" is a better method
## [1] "Three-Month Moving Average"
Results
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"
Question 3: Linear Trend Approach
Load the Libraries
library(readxl)
library(ggplot2)
Load the data
Mortgage <- read_excel("~/Downloads/Mortgage.xlsx")
Descriptive Statistics
summary(Mortgage)
## 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
Construct a Time Series Plot
ggplot(Mortgage, aes(x = Year, y = Interest_Rate)) +
geom_line() +
geom_point() +
xlab("Year") +
ylab("Interest Rates") +
ggtitle("US Mortgage Interest Rate Average")
#Interpretation: We observe a decreasing pattern in 2000 and a small spike in 2005 and another decreasing pattern from 2005 until 2020 where there is a sharp increase.
Develop a Linear Trend Equation
model <- lm(Interest_Rate ~ Period, data = Mortgage)
summary(model)
##
## Call:
## lm(formula = Interest_Rate ~ Period, data = Mortgage)
##
## 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 - estimated liner trend equation: Average Interest = 6.70 + -.13 * Period
Forecast the average interest for period 25
Mortgage_prediction <- predict(model, newdata = data.frame(Period = 25))
Mortgage_prediction
## 1
## 3.472942
# Interpretation: The average interest for period 25 is 3.47.