lm r (correlation coefficient), r-squared(coefficient of determination), SSR, SST, SSE

# lm  -- -- -- -- -- -- -- -- -- --  604 Linear Regression
# from Statistics 101: Linear Regression, Residual Analysis
# youtube.com/watch?v=gLENW2AdJWg

bill <- c( 34,	 108, 64	, 88	, 99	, 51	 )
mean(bill)
tip <-c ( 5, 17, 11, 8, 14, 5 )
mean(tip)

#correlation coefficient = covariance / co-standard-deviation
sample_bill_to_tip <- length(bill)
covariance_bill_to_tip <- sum((bill - mean(bill)) * (tip - mean(tip))) / (sample_bill_to_tip-1) # 123
co_standard_deviation <- sd(bill) * sd(tip) # 142.0873
r_correlation_coefficient <- covariance_bill_to_tip/co_standard_deviation # 0.865665
# prove the above
cor(bill,tip) == r_correlation_coefficient
r_sqrd_bill_to_tip <- r_correlation_coefficient^2
# or r^2 is SSR/SST, requires a linear model from #604 - - - - - - - - - - 
meal_to_tip <- data.frame(bill,tip)
meal_model <- lm(tip~bill, data = meal_to_tip)
#SSR predicted y - mean of observed y
sr_meal_model <- predict(meal_model) - mean(tip)
SSR_meal_model <- sum(sr_meal_model^2)

round(SST_meal_model, digits = 0) == round(SSR_meal_model + SSE_meal_model, digits = 0)

meal_to_tip <- data.frame(bill,tip)
meal_model <- lm(tip ~ bill, data = meal_to_tip)
summary(meal_model)
plot(meal_model)
meal_to_tip %>% ggplot(aes(x=bill, y=tip)) +
# observed in diamonds from 597(R-Graphics Cookbook)::Sect5-6
  geom_point(col="orange", shape=18, size=2.5) +
# predicted linear model
  geom_smooth(method = "lm", se=F, color="lightgrey") +
# view predicteed
# predict(meal_model)  
  geom_point(aes(y=predict(meal_model)), col="purple", size=2.5) +
# y_bar is the overall observed mean of y
  geom_line(aes(y=mean(tip)), linetype="dashed")


#SSE observed y - predicted y squared
se_meal_model <- tip - predict(meal_model)
SSE_meal_model = sum((tip - predict(meal_model))^2)
# above are the same as below
rm(SST)
(se_meal_model^2 == sse_meal_model) # TRUE
#SST observed y - mean of observed y squared
st_meal_model <- tip - mean(tip)
SST_meal_model <- sum(st_meal_model^2)
#SSR predicted y - mean of observed y
sr_meal_model <- predict(meal_model) - mean(tip)
SSR_meal_model <- sum(sr_meal_model^2)