Use bayesian inference to find relationship between variables

Question

I have a lot of data, like the image below, which have a relationship with each other. I want to make an equation to describe this relationship, something like Power = a * WindSpeed ^ b . How can I use Bayesian Inference to find a and b? I want to use R for this.

Answer 1

Welcome to SO and good luck. Do not forget to pay attention to the comment (they are really relevant & you can increase probability to get an answer). Please see below example of using Bayesian univariate regression using Bolstadt package.

library(Bolstad)

# Simulation of Power vs Wind
# Power = 5 * Windspeed ^ 2
set.seed(123)
n <- 100

# y = Power
# x = WindSpeed
# e = error term

x <- (1:(25 * n))/ n
e <- rnorm(length(x)) / 10

# y = a * x ^ b
# log(y) = log(a) + b * log(x) + e  
# or
# in exponential form
y <- exp(log(5) + e) * x ^ 2

# bayes univariate linear regression model
z <- bayes.lin.reg(log(y), log(x))
# Standard deviation of residuals:  0.0943 
# Posterior Mean Posterior Std. Deviation
# -------------- ------------------------
#   Intercept:  6.076          0.0059657               
#   Slope:      1.996          0.0062209    

# ------------------------------------------------
# pay attention the result of bayession regression
# are shifted for intercept by the mean
# is is accouted as below
intercept_shifted <- z$intercept$mean - z$slope$mean * mean(log(x))
intercept_shifted
# [1] 1.617218

# validate by standar linear model:
lm(log(y) ~ log(x))
# Coefficients:
#  (Intercept)       log(x)  
#  1.617        1.996  

a = exp(intercept_shifted) 
a
# [1] 5.039051
b = z$slope$mean
b
# [1] 1.996134