** Using the naive method (most recent value) as the forecast for the next week, compute the following measures of forecast accuracy.

##project Objective

to know the MSE of the values

##question 1 finding MSE

week <- 1:6
values <- c(17,13,15,11,17,14)
forcasts_a <- values[-length(values)]
actual_a <- values[-1]
mse_a <- mean((actual_a - forcasts_a)^2)
mse_a #result 16.2

## [1] 16.2

forecast_week_7_a <- tail(values,1)
forecast_week_7_a

## [1] 14

###part B average all of the data as forecast

cumulative_averages <- cumsum(values[-length(values)]) / (1:(length(values)-1))
forecasts_b <- cumulative_averages
actual_b <- values[-1]
mse_b <- mean((actual_b - forecasts_b)^2)
mse_b

## [1] 8.272

forecast_month_7_b <- mean(values)
forecast_month_7_b

## [1] 14.5

###part c best option

better_method <- ifelse(mse_a < mse_b, "most recent value" , "average of all data")
list(
  MSE_most_recent_value = mse_a,
  forecast_month_7_most_recent = forecast_week_7_a,
  MSE_average_of_all_data = mse_b,
  forecast_month_7_average = forecast_month_7_b,
  Better_Method = better_method

)

## $MSE_most_recent_value
## [1] 16.2
## 
## $forecast_month_7_most_recent
## [1] 14
## 
## $MSE_average_of_all_data
## [1] 8.272
## 
## $forecast_month_7_average
## [1] 14.5
## 
## $Better_Method
## [1] "average of all data"

##question 2 ### library and time series plot

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))
plot(df$month,df$data, type = "o", col = "blue", xlab = "Month" , ylab= "building contracts",
  main = "Monthly building contracts"   )

#the time-series plot exhibits a horizontal pattern

three month 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)

df <- df %>%
  mutate(
    squared_error = ifelse(is.na(avg_contract), NA, (data - avg_contract)^2)
  )
#compute MSE
mse <- mean((df$squared_error), na.rm = TRUE)
mse #2040.44

## [1] 2040.444

###exponetial soothing forecast

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

## [1] 2593.762

##example 3 #load the data

library(readxl)
library(ggplot2)
df1 <- read_excel("/Users/allenchen/Desktop/ttt/Mortgage.xlsx")

##time series plot

ggplot(df1, aes(x = Year, y = Interest_Rate )) +
  geom_line()+
  geom_point()+
  xlab("year")

  ylab("interest rate")

## $y
## [1] "interest rate"
## 
## attr(,"class")
## [1] "labels"

  ggtitle("Times Series Plot for Skechers Revenue")

## $title
## [1] "Times Series Plot for Skechers Revenue"
## 
## attr(,"class")
## [1] "labels"

##linear model

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

##forecast for period 25

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

##        1 
## 3.472942