Question 1: Naive Approach
Time Series data
weeks <- 1:6
values <- c(17, 13, 15, 11, 17, 14)
Part A. Most recent value as forecast
forecasts_a <- values[-length(values)]
actual_a <- values[-1]
mse_a <- mean((actual_a - forecasts_a)^2)
mse_a #result: 16.2
## [1] 16.2
forecast_week_7_a <- tail(values, 1)
forecast_week_7_a #result: 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 14.6
forecasts_b <- cumulative_averages
actual_b <- values[-1]
mse_b <- mean((actual_b - forecasts_b)^2)
mse_b #result: 8.27
## [1] 8.272
forecasts_week_7_b <- mean(values)
forecasts_week_7_b #result: 14.5
## [1] 14.5
Part C. Calculate MAE and MAPE
mae <- mean(abs(actual_a - forecasts_a))
mae
## [1] 3.8
mape <- mean(abs((actual_a - forecasts_a) / actual_a)) * 100
mape
## [1] 27.43778
Question 2
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)
##
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric
df <- data.frame(month=c(1,2,3,4,5,6,7,8,9,10,11,12),
data=c(240, 352, 230, 260, 280, 322, 220, 310,
240, 310, 240, 230))
Part A. Time series plot
plot(df$month, df$data, type = "o", col = "blue", xlab = "Month", ylab = "Contracts (in $ millions)",
main = "Monthly Percentage of Alabama building contracts")
#interpretation: The time series plot exhibits a horizontal pattern
Part B. Three-month moving average
df$avg_contract <- c(NA, NA, NA,
(df$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)
Calculate the squared errors (only for the months where moving
average is available)
df <- df %>%
mutate(
squared_error = ifelse(is.na(avg_contract), NA, (data - avg_contract)^2))
mse <- mean(df$squared_error, na.rm = TRUE)
mse #2040.44
## [1] 2040.444
Exponential Smoothing
alpha = 0.2
exp_smooth <- rep(NA, length(df$data))
exp_smooth[1] <- df$data[1]
for(i in 2: length(df$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 #2593.76
## [1] 2593.762
Determine the better method
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
) #Result: Better method is three-month moving average
## $MSE_moving_average
## [1] 2040.444
##
## $MSE_exponential_smoothing
## [1] 2593.762
##
## $Better_method
## [1] "Three-month moving average"
Question 3
#load the data
library(readxl)
library(ggplot2)
df1 <- read_excel("/Users/allysonguan/Downloads/Mortgage.xlsx")
Part A. Time Series Plot
ggplot(df1, aes(x = Year, y = Interest_Rate)) +
geom_line()+
geom_point() +
xlab("Year") +
ylab("Interest Rate(%)") +
ggtitle("Time Series Plot of Mortgage")
Part B. Develop a linear trend equation
model <- lm(Interest_Rate ~ Period, data = df1)
summary(model)
##
## Call:
## lm(formula = Interest_Rate ~ Period, data = df1)
##
## 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: Y = 6.70 - 0.13 * Period
Part C. Forecast the average interest rate for period 25
forecast_period_25 <- predict(model, newdata = data.frame(Period = 25))
forecast_period_25 #Result: 3.47%
## 1
## 3.472942