# loess example
span = 0.8
model = loess(dist ~ speed, cars, span = span, degree = 1, control = loess.control(surface = "direct"))
new.input.value = 21
predict(model, new.input.value)

# simple loess implementation
X = cbind(rep(1, length(cars$speed)), cars$speed)
y = cars$dist
neighbor.count = as.integer(span * nrow(X))
distance.threshold = sort(abs(X[,2] - new.input.value))[neighbor.count]
weights = rep(0, nrow(X))
selected = abs(X[,2] - new.input.value) <= distance.threshold
# weights derived using the tricube weight function
weights[selected] = (1 - (abs(X[selected,2] - new.input.value)/distance.threshold)^3)^3
W = diag(weights)
coefficients = solve(t(X) %*% W %*% X, diag(ncol(X))) %*% t(X) %*% W %*% y
coefficients[1] + coefficients[2] * new.input.value