Tuesday, 23 February 2021

Calculating R^2 value for the data that fall within 95% prediction interval

I have data as follows.

import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats

x = np.array([50,52,53,54,58,60,62,64,66,67,68,70,72,74,76,55,50,45,65])
y = np.array([25,50,55,75,80,85,50,65,85,55,45,45,50,75,95,65,50,40,45])

I can calculate overall R^2 as follows.

slope, intercept = np.polyfit(x, y, 1)  # linear model adjustment

y_model = np.polyval([slope, intercept], x)   # modeling...

x_mean = np.mean(x)
y_mean = np.mean(y)
n = x.size                        # number of samples
m = 2                             # number of parameters
dof = n - m                       # degrees of freedom
t = stats.t.ppf(0.975, dof)       # Students statistic of interval confidence

residual = y - y_model

std_error = (np.sum(residual**2) / dof)**.5   # Standard deviation of the error


numerator = np.sum((x - x_mean)*(y - y_mean))
denominator = ( np.sum((x - x_mean)**2) * np.sum((y - y_mean)**2) )**.5
correlation_coef = numerator / denominator
r2 = correlation_coef**2

# mean squared error
MSE = 1/n * np.sum( (y - y_model)**2 )

# to plot the adjusted model
x_line = np.linspace(np.min(x), np.max(x), 100)
y_line = np.polyval([slope, intercept], x_line)

# confidence interval
ci = t * std_error * (1/n + (x_line - x_mean)**2 / np.sum((x - x_mean)**2))**.5
# predicting interval
pi = t * std_error * (1 + 1/n + (x_line - x_mean)**2 / np.sum((x - x_mean)**2))**.5  

############### Ploting
plt.rcParams.update({'font.size': 14})
fig = plt.figure()
ax = fig.add_axes([.1, .1, .8, .8])

ax.plot(x, y, 'o', color = 'royalblue')
ax.plot(x_line, y_line, color = 'royalblue')
ax.fill_between(x_line, y_line + pi, y_line - pi, color = 'lightcyan', label = '95% prediction interval')
ax.fill_between(x_line, y_line + ci, y_line - ci, color = 'skyblue', label = '95% confidence interval')

ax.set_xlabel('x')
ax.set_ylabel('y')

# rounding and position must be changed for each case and preference
a = str(np.round(intercept))
b = str(np.round(slope,2))
r2s = str(np.round(r2,2))
MSEs = str(np.round(MSE))

ax.text(45, 110, 'y = ' + a + ' + ' + b + ' x')
ax.text(45, 100, '$r^2$ = ' + r2s + '     MSE = ' + MSEs)

plt.legend(bbox_to_anchor=(1, .25), fontsize=12)

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I want to calculate R^2 value for the data that fall within 95% prediction interval. How can I do it ?

Credit: code adapted from, Show confidence limits and prediction limits in scatter plot



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