Class Exercise 16

Load the required packages

#install.packages(zoo)
library(readxl)
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

library(ggplot2)

Question 1:

Input time series data

week <- 1:6 #Independent variable
valuesq1 <- c(17,13,15,11,17,14) #Dependent variables

# Most recent value
forecast_a <- valuesq1[-length(valuesq1)] #Exclude last value
actual <- valuesq1[-1] #Exclude first value
mse <- mean((actual - forecast_a)^2)
mse #Mean square error = 16.2

## [1] 16.2

mae <- mean(abs(actual - forecast_a))
mae #Mean absolute error

## [1] 3.8

mape <- mean(abs((actual - forecast_a)/actual)) * 100
mape #Mean absolute percentage error

## [1] 27.43778

forecast_week7 <- tail(valuesq1, 1)
forecast_week7 #Projected values

## [1] 14

Question 2:

## Moving Avg and Time Series Plot
df <- data.frame(month=c(1,2,3,4,5,6,7,8,9,10,11,12),
                 values=c(240, 352, 230, 260, 280, 322, 220, 310, 240, 310, 240, 230))
summary(df)# Average value is 269.5 over a 12 month period

##      month           values     
##  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$month, df$values, type = "o", col = "blue", xlab = "Month", ylab = "Values", 
     main = "Alabama Building Contracts Plot")

Manually calculate three-month moving average

df$avg_values <- c(NA, NA, NA,
                   (df$values[1] + df$values[2] + df$values[3]) / 3,
                   (df$values[2] + df$values[3] + df$values[4]) / 3,
                   (df$values[3] + df$values[4] + df$values[5]) / 3,
                   (df$values[4] + df$values[5] + df$values[6]) / 3,
                   (df$values[5] + df$values[6] + df$values[7]) / 3,
                   (df$values[6] + df$values[7] + df$values[8]) / 3,
                   (df$values[7] + df$values[8] + df$values[9]) / 3,
                   (df$values[8] + df$values[9] + df$values[10]) / 3,
                   (df$values[9] + df$values[10] + df$values[11]) / 3
)

Interpretation: The time series plot has a horizontal pattern.

Calculate the squared errors

df <- df %>%
  mutate(
    squared_error = ifelse(is.na(avg_values), NA, (values-avg_values)^2)
  )

Compute MSE not including intion weeks with NA

mse1 <- mean(df$squared_error, na.rm = TRUE)
mse1 #Output the MSE = 2040.44

## [1] 2040.444

Exponential smoothing

alpha <- 0.2
exp_smooth <- rep(NA, length(df$values))
exp_smooth[1] <- df$values[1] #starting point
for(i in 2:length(df$values)) {
  exp_smooth[i] <- alpha * df$values[i-1] + (1 - alpha) * exp_smooth[i-1]
}

mse_exp_smooth <- mean((df$values[2:12] - exp_smooth[2:12])^2) # Output the MSE= 2593.76
mse_exp_smooth

## [1] 2593.762

Comparison

better_method <- ifelse(mse < mse_exp_smooth, "Three-Month Moving Average", "Exponential Smoothing")

Result

list(
  MSE_Moving_Average = mse,
  MSE_Exponential_Smoothing = mse_exp_smooth,
  Better_Method = better_method
)

## $MSE_Moving_Average
## [1] 16.2
## 
## $MSE_Exponential_Smoothing
## [1] 2593.762
## 
## $Better_Method
## [1] "Three-Month Moving Average"

##Question 3:

## Question 3: Time Series Plot
data <- read_excel("/Applications/R Markdowns/Mortgage.xlsx")


df <- data.frame(period=data$Period,
                  rate=data$Interest_Rate)

summary(df)

##      period           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

Interpretation: Average mortgage rate over 20 years is 5.08%

Time Series Plot

ggplot(df, aes(x=period, y =rate)) +
  geom_line() +
  geom_point() +
  xlab("Period") +
  ylab("Interest Rate") +
  ggtitle("Time Series Plot of 30 Year Mortgage Rates")

Interpretation: Decreasing trend in the time series plot, seasonal patters could be the explanation 

##Question 4:

Develop a linear trend equation: 

model <- lm(rate ~ period, data = df)
summary(model)

## 
## Call:
## lm(formula = 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

Estimated liner equation= 6.69 + 0.13(period)

##Question5:

forecast_period_25 <- predict(model, newdata = data.frame(period=25))
forecast_period_25

##        1 
## 3.472942

Average interest rate for period 25 is 3.47