I have two variables (x and y) that have a somewhat sigmoidal relationship with each other, and I need to find some sort of prediction equation that will enable me to predict the value of y, given any value of x. My prediction equation needs to show the somewhat sigmoidal relationship between the two variables. Therefore, I cannot settle for a linear regression equation that produces a line. I need to see the gradual, curvilinear change in slope that occurs at both the right and left of the graph of the two variables.
I started using numpy.polyfit after googling curvilinear regression and python, but that gave me the awful results you can see if you run the code below. Can anyone show me how to re-write the code below to get the type of sigmoidal regression equation that I want?
If you run the code below, you can see that it gives a downward facing parabola, which is not what the relationship between my variables should look like. Instead, there should be more of a sigmoidal relationship between my two variables, but with a tight fit with the data that I am using in the code below. The data in the code below are means from a large-sample research study, so they pack more statistical power than their five data points might suggest. I do not have the actual data from the large-sample research study, but I do have the means below and their standard deviations(which I am not showing). I would prefer to just plot a simple function with the mean data listed below, but the code could get more complex if complexity would offer substantial improvements.
How can I change my code to show a best fit of a sigmoidal function, preferably using scipy, numpy, and python? Here is the current version of my code, which needs to be fixed:
import numpy as np
import matplotlib.pyplot as plt
# Create numpy data arrays
x = np.array([821,576,473,377,326])
y = np.array([255,235,208,166,157