hmnoidk
0
I'm trying to get the A and B terms for an exponential model written as:
mod <- lm(log(y) ~ x)
When I call summary(mod), I understand I should take exp() of x to get B. What do I do with the intercept to get A so that I can write it in the form:
Y = A*B^x
In order to determine the coefficients of the linearized form of Y=AB^x, you need to know a little about log rules. First, we take the log of both sides, which gives log(Y)=log(AB^x). Multiplication with in a log is the same as addition, so we split A and B^x, log(Y)=log(A)+log(B^x). Lastly, exponentials in a log are the same as multiplication, so log(Y)=log(A)+xlog(B). This gives the general linear equation y=mx+b, where m = log(B), and b = log(A). When you run a linear regression, you need to calculate the A as exp(intercept) and B as exp(slope). Here is an example:
library(tidyverse)
example_data <- tibble(x = seq(1, 5, by = 0.1),
Y = 10*(4^{x}) +runif(length(x),min = -1000, max = 1000))
example_data |>
ggplot(aes(x, Y))+
geom_point()
model <- lm(log(Y) ~ x, data = example_data)
summary(model)
#>
#> Call:
#> lm(formula = log(Y) ~ x, data = example_data)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -1.9210 -0.3911 0.1394 0.3597 1.9107
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 3.6696 0.4061 9.036 2.55e-10 ***
#> x 1.0368 0.1175 8.825 4.40e-10 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.7424 on 32 degrees of freedom
#> (7 observations deleted due to missingness)
#> Multiple R-squared: 0.7088, Adjusted R-squared: 0.6997
#> F-statistic: 77.88 on 1 and 32 DF, p-value: 4.398e-10
A <- exp(summary(model)$coefficients[1,1]) #intercept
B <- exp(summary(model)$coefficients[2,1]) #slope
example_data |>
ggplot(aes(x, Y))+
geom_point()+
geom_line(data = tibble(x = seq(1,5, by = 0.1),
Y = A*B^x), color = "blue") # plot model as check
