Time Series Data
### Question 1
months <- 1:6
values <- c(17,13,15,11,17,14)
A(Most Recent)
# A
forecasts_a <- values[-length(values)]
actual_a <- values[-1]
mse_a <- mean((actual_a -forecasts_a)^2)
mse_a # 16.2
## [1] 16.2
forecast_week_7_a <- tail(values,1)
forecast_week_7_a # 14
## [1] 14
cumulative_averages <- cumsum(values[-length(values)])/ (1:(length(values)-1))
cumulative_averages # 17.0, 15.0, 15.0, 14.0, 14.6
## [1] 17.0 15.0 15.0 14.0 14.6
B(All)
# B
forecasts_b <- cumulative_averages
actual_b <- values[-1]
mse_b <- mean((actual_b- forecasts_b)^2)
mse_b
## [1] 8.272
forecast_week_7_b <- tail(values)
forecast_week_7_b
## [1] 17 13 15 11 17 14
Better Method
#Better Method
better_method <- ifelse(mse_a < mse_b, "Most Recent Value", "Average of all Data")
list(
MSE_most_recent_value = mse_a,
forecast_week_7_most_recent = forecast_week_7_a,
MSE_average_of_all_data = mse_b,
forecast_week_7_average = forecast_week_7_b,
Better_Method = better_method
)
## $MSE_most_recent_value
## [1] 16.2
##
## $forecast_week_7_most_recent
## [1] 14
##
## $MSE_average_of_all_data
## [1] 8.272
##
## $forecast_week_7_average
## [1] 17 13 15 11 17 14
##
## $Better_Method
## [1] "Average of all Data"
Question 2
library(dplyr)
## Warning: package 'dplyr' was built under R version 4.3.3
##
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
##
## filter, lag
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union
library(zoo)
## Warning: package 'zoo' was built under R version 4.3.3
##
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric
df <- data.frame(month=c(1,2,3,4,5,6,7,8,9,10,11,12),
data=c(240, 352, 230, 260, 280, 322, 220, 310, 240, 310, 240, 230))
Time Series Plot
plot(df$month, df$data, type = "o", col= "blue", xlab = "Month", ylab = "Contracts",
main= "Monthly Contracts")
Horizontal pattern observed in the Graph
# 3 Months
df$avg_shipment <- 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$data[11])/3)
Square Errors
df<- df %>%
mutate(
squared_error= ifelse(is.na(avg_shipment), NA, (data- avg_shipment)^2)
)
Compute MSE
mse <- mean(df$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] # Starting p
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# 2593.762
## [1] 2593.762
Comparison
better_method <- ifelse(mse < mse_exp_smooth, "Three Month average", "Exponential smoothing")
list(
MSE_moving_average= mse,
MSE_exponential_smoothing= mse_exp_smooth,
Bette_method= better_method
)
## $MSE_moving_average
## [1] 2040.444
##
## $MSE_exponential_smoothing
## [1] 2593.762
##
## $Bette_method
## [1] "Three Month average"
# The three month average approach is better because it has a lower mean
Questions 3 to 5
library(readxl)
library(ggplot2)
## Warning: package 'ggplot2' was built under R version 4.3.3
mort <- read_excel(file.choose())
# Plot it
ggplot(mort, aes(x= Year, y= Interest_Rate))+
geom_line()+
geom_point()+
xlab("Year")+
ylab("Interest Rate")+
ggtitle("Mortgage intrest rate")
# Graph Showing a Decreasing Trend that Increases After 2020
Develop an Equation
model <- lm(Interest_Rate ~ Period, data = mort )
summary(model)
##
## Call:
## lm(formula = Interest_Rate ~ Period, data = mort)
##
## 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
# y = 6.70 + (-0.13t) or 6.70 - 0.13*Period
# Forecasting the Average Interest Rate for Period 25 (i.e. 2024)
forecast_period_25 <- predict(model, newdata = data.frame(Period= 25))
forecast_period_25 # 3.47
## 1
## 3.472942