Class Exercise 16
#Time Series Analysis
#Question 1 #Time series data
week <- 1:6 #This is the independent value variable
value <- c(17,13,15,11,17,14) #dependent variable#Forecast values
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## [1] 3.8#Mean absolute percent error
mape <- mean(abs(actual_a - forecast_a)/ actual_a) * 100
mape## [1] 27.43778#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## [1] 16.2#forecast sales for week 7
forecast_week7 <- tail(value, 1)
forecast_week7## [1] 14#Interpretation: the value for week 7 is 14#Question2: moving average
#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, 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),
data = c(240, 352, 230, 260, 280, 322, 220, 310, 240, 310, 240, 230))#Descriptive statistics
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#time series plot
plot(df$month, df$data, type = "o", col = "blue",
xlab = "Month", ylab = "Data",
main = "Alabama buiding contracts value plot")
#Interpretation: the time series plot exhbits a horizontal pattern
#Manually calculate the three month moving average
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
)#calc the squared errors ( only for months were moving average is available)
df <- df %>%
mutate(
squared_error = ifelse(is.na(avg_data3), NA, (data - avg_data3)^2)
)#computing MSE
mse <- mean(df$squared_error, na.rm = TRUE)
mse #output the MSE = 2040.44## [1] 2040.444#Exponential smoothing
alpha <- 0.2
exp_smooth <- rep(NA, length(df$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 #output the MSE = 2593.76## [1] 2593.762#comparison
better_method <- ifelse(mse < mse_exp_smooth, "Three-Month moving average", "Exponential smoothing")#final 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:
#Load the libraries
library(readxl)
library(ggplot2)#load the data
df <- read_excel("~/Downloads/RMarkdown/Mortgage.xlsx")
df## # A tibble: 24 × 3
## Year Period Interest_Rate
## <dttm> <dbl> <dbl>
## 1 2000-01-01 00:00:00 1 8.05
## 2 2001-01-01 00:00:00 2 6.97
## 3 2002-01-01 00:00:00 3 6.54
## 4 2003-01-01 00:00:00 4 5.83
## 5 2004-01-01 00:00:00 5 5.84
## 6 2005-01-01 00:00:00 6 5.87
## 7 2006-01-01 00:00:00 7 6.41
## 8 2007-01-01 00:00:00 8 6.34
## 9 2008-01-01 00:00:00 9 6.03
## 10 2009-01-01 00:00:00 10 5.04
## # ℹ 14 more rows#Descriptive statistics
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. :8.053#construct a 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 fixed-rate mortgage")
#interpretation: we observe a downward trend pattern #develop a linear trend 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: 0.0003637#result- estimated linear trend equation: interest rate = 6.70 - 0.13*period#Forecast the average interest rate for period 25
forecast_period_25 <- predict(model, newdata = data.frame(Period = 25))
forecast_period_25## 1
## 3.472942