exponential decay with AR(1)

decay_ar.R

# exponential decay estimation with AR model
N <- 30
y <- numeric(N)
first_obs <- 2  # where do we start?
asymptote <- .5 # where do we go?
ar1_param <- .6 # how slow do we go there? (between -1 and 1)
sigma  <- 0

y[1] <- first_obs
for (n in 2:N) {
  y[n] <- asymptote + ar1_param*(y[n-1] - asymptote) + rnorm(1, sd = sigma)
}
plot(y, type = "l")

# now estimate an AR(1) model
first_dummy <- c(1, rep(0, N-1)) # dummy variable for first
y_lag <- c(0, y[-N]) # lagged y
res <- lm(y ~ 1 + first_dummy + y_lag) # estimate

# give the coefficients nice names
alpha <- unname(coef(res)[1])
phi   <- unname(coef(res)[3])
first <- unname(coef(res)[2])

# now get the original parameters back
asymptote_estimated <- alpha / (1 - phi)
first_obs_estimated <- alpha + first
ar1_param_estimated <- phi

# 4pl model
# https://www.myassays.com/four-parameter-logistic-regression.html

# 4pl model
upper_asymp <- 2
lower_asymp <- 0.5
inflect_pnt <- 50
slope_param <- 4
sigma <- 0
N <- 100
x <- 1:N
y <- upper_asymp + (lower_asymp - upper_asymp) / (1 + (x/inflect_pnt)^slope_param) + rnorm(N, sd = sigma)
plot(x, y)


infl <- 50
ylag <- c(y[2:infl], 0, y[infl:(N-1)])
infl_dummy <- c(rep(1, infl), rep(0, N-infl))
mid_dummy <- numeric(N); mid_dummy[infl] <- 1
fit <- lm(y ~ 0 + infl_dummy + I(1 - infl_dummy + mid_dummy) + mid_dummy + ylag)
lines(1:N, predict(fit, newdata = list(x = 1:N)))

# all that's needed now is to make sure there is no discontinuity in the gradient at the inflection point