# How to fit observed data by assuming an equation beforehand?

Consider the following equation:

where,
OPDV = speed difference between a vehicle and its lead vehicle in the same lane
(delta x - L) = H = distance between the front center of vehicle and rear end of its lead vehicle
CC6 and delta.CC5 are parameters.

# Sample data

Now, consider the following data frame, that contains all parameters of the equation:

df <- data.frame(w = rep(1.83, 17),
vrate = 0.0006,
CC4 = -0.35,
dCC5 = 0.35,
H = c(300, 200, 100, 70, 60, 50, 40, 30, 20,
10, 5, 2, 1, 0.5, 0.005, 0.0005, 0))
library(dplyr)

df <- df %>%
mutate(CC6 = 1*vrate/w) %>%
mutate(
OPDV = -CC6*H^2-dCC5,
DV2 = -CC6*H^2)


Note that I'm considering CC6/17000 as one parameter CC6.

This will result in following plot:

library(ggplot2)
ggplot(data = df)+
geom_line(aes(x = (OPDV), y = H, color = "OPDV"))


Here, I assumed the values of CC6 and CC5. But in observed data I don't know the values of these parameters

# Observed Data

I have observed values of OPDV and H. When I plot them, I want to use a smoothing method that is similar to this equation, i.e. a quadratic part and a constant subtracted from it. Following is what I get if I use quadratic equation only:

ggplot(data = ud8 %>% filter(svelkm.level == "(15,20]",
abs(slo_v)>0.1,
abs(slo_p)>0.1))+
geom_point(aes(x = OPDV, y = frspacing_OPDV)) +
geom_smooth(aes(x = OPDV, y = frspacing_OPDV),
method="lm", formula = y ~ x + I(x^2))


# Question

Which smoothing method is best to use that can replicate the equation? I use R.

Your equation appears to posit a straight line relationship between $y$ (i.e. $\text{OPDV}$) and $x$ (=$H^2$).
It's not clear what you intend to do with smoothing beyond just fitting the equation), unless you're considering whether your model might be incorrect, in which case some locally linear model in $H^2$ might be a good thing to lok at. It's also not clear why you're progressing the way you are -- it suggests there may be missing context.