Class Exercise 16

Question 1 Naive method A. Mean Absolute Error B. MSE C. MAPE D. Week 7 Forcast

###Step 1 Time Series Data

weeks <- 1:6
values <- c(17, 13, 15, 11, 17, 14)

Step 2 Most Recent Value as forecast

forecasts_a <- values[-length(values)] #exclude the last value
actual_a <- values[-1] #exclude the first value
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

Step 3: Average all of the data as a 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] #exclue the first value
mse_b <- mean((actual_b - forecasts_b)^2)
mse_b #result: 8.272

## [1] 8.272

forecast_week_7_b <- mean(values) #average of all weeks forcast for week 7
forecast_week_7_b # result: 14.5

## [1] 14.5

Step 4: find the best 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
)# results said average of all data is better

## $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] 14.5
## 
## $Better_method
## [1] "Average of All Data"

Question 2: Alabama Building Contracts

Step 1: Install Packages

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

###Step 2: Import Data

dff <- 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))

###Step 3: Make a time series plot

plot(dff$month,dff$data, type =  "o", col ="blue", xlab = "Month", 
     ylab = "Contract Revenue", main = "Alabama building contracts in millions")

# Interpretation: The time-series plot exhibits a horizontal pattern. Because of up and down flucuations

###Step 4: Three Month Moving Average

dff$avg_contract <- c(NA, NA, NA,
                     (dff$data[1] + dff$data[2] + dff$data[3]) / 3,
                     (dff$data[2] + dff$data[3] + dff$data[4]) / 3,
                     (dff$data[3] + dff$data[4] + dff$data[5]) / 3,
                     (dff$data[4] + dff$data[5] + dff$data[6]) / 3,
                     (dff$data[5] + dff$data[6] + dff$data[7]) / 3,
                     (dff$data[6] + dff$data[7] + dff$data[8]) / 3,
                     (dff$data[7] + dff$data[8] + dff$data[9]) / 3,
                     (dff$data[8] + dff$data[9] + dff$data[10]) / 3,
                     (dff$data[9] + dff$data[10] + dff$data[11]) / 3)

###Step 5: Calculate the square errors

dff <- dff %>%
  mutate(
    squared_error = ifelse(is.na(avg_contract), NA, (data - avg_contract)^2)
  )

###Step 6: Compute the MSE

mse <- mean(dff$squared_error ,na.rm = TRUE)
mse #results:mse = 2040.444

## [1] 2040.444

###Step 7: Exponetial Smoothing

alpha = 0.2
exp_smooth <- rep(NA, length(dff$data))
exp_smooth[1] <- dff$data[1] #starting point
for(i in 2:length(dff$data)) {
  exp_smooth[i] <- alpha * dff$data[i-1] + (1 - alpha) * exp_smooth[i-1]
}
mse_exp_smooth <- mean((dff$data[2:12] - exp_smooth[2:12])^2)
mse_exp_smooth #results 2593.762

## [1] 2593.762

###Step 8: Find out which method is better

better_method <- ifelse(mse < mse_exp_smooth, "Three month moving average", "Exponetial smoothing")
list(
  MSE_moving_average = mse,
  MSE_exponetial_smoothing = mse_exp_smooth,
  Better_method = better_method
)# results say that "Three month moving average"

## $MSE_moving_average
## [1] 2040.444
## 
## $MSE_exponetial_smoothing
## [1] 2593.762
## 
## $Better_method
## [1] "Three month moving average"

##Question 3: Interest Rate Data

###Step 1 Load Data

library(readxl)
library(ggplot2)

df1 <- read_excel("/Users/tyhershberger/Downloads/R Markdown/Mortgage.xlsx")

###Step 2 Make a time series plot

ggplot(df1, aes(x= Year, y= Interest_Rate)) +
  geom_line() +
  geom_point() +
  xlab("Year") +
  ylab("Intredst Rate") + 
  ggtitle("Average Intrest rate")

#overall decreasing and going down until about 2021 then the graph shoots back up.

###Step 3: 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 - linear trend equation: 6.70 - 0.13

###Step 4: Forecast for period 25

forecast_period_25 <- predict(model, newdata = data.frame(Period = 25))
forecast_period_25 # 3.47 interest rate. 

##        1 
## 3.472942