File:Regression confidence band.svg

import numpy as np
import matplotlib.pyplot as plt
import scipy.special as sp

## Sample size.
n = 50

## Predictor values.
XV = np.random.uniform(low=-4, high=4, size=n)
XV.sort()

## Design matrix.
X = np.ones((n,2))
X[:,1] = XV

## True coefficients.
beta = np.array([0, 1.], dtype=np.float64)

## True response values.
EY = np.dot(X, beta)

## Observed response values.
Y = EY + np.random.normal(size=n)*np.sqrt(20)

## Get the coefficient estimates.
u,s,vt = np.linalg.svd(X,0)
v = np.transpose(vt)
bhat = np.dot(v, np.dot(np.transpose(u), Y)/s)

## The fitted values.
Yhat = np.dot(X, bhat)

## The MSE and RMSE.
MSE = ((Y-EY)**2).sum()/(n-X.shape[1])
s = np.sqrt(MSE)

## These multipliers are used in constructing the intervals.
XtX = np.dot(np.transpose(X), X)
V = [np.dot(X[i,:], np.linalg.solve(XtX, X[i,:])) for i in range(n)]
V = np.array(V)

## The F quantile used in constructing the Scheffe interval.
QF = sp.fdtri(X.shape[1], n-X.shape[1], 0.95)

## The lower and upper bounds of the Scheffe band.
D = s*np.sqrt(X.shape[1]*QF*V)
LB,UB = Yhat-D,Yhat+D

## The lower and upper bounds of the pointwise band.
D = s*np.sqrt(2*V)
LBP,UBP = Yhat-D,Yhat+D

## Make the plot.
plt.clf()
plt.plot(XV, Y, 'o', ms=3, color='grey')
plt.hold(True)
a = plt.plot(XV, EY, '-', color='black')
b = plt.plot(XV, LB, '-', color='red')
plt.plot(XV, UB, '-', color='red')
c = plt.plot(XV, LBP, '-', color='blue')
plt.plot(XV, UBP, '-', color='blue')
d = plt.plot(XV, Yhat, '-', color='green')
B = plt.legend( (a,d,b,c), ("Truth", "Estimate", "95% simultaneous CB",\
                            "95% pointwise CB"), 'lower left')
B.draw_frame(False)
plt.ylim([-20,15])
plt.gca().set_yticks([-20,-10,0,10,20])
plt.xlim([-4,4])
plt.gca().set_xticks([-4,-2,0,2,4])
plt.xlabel("X")
plt.ylabel("Y")
plt.savefig("regression_confidence_band.png")
plt.savefig("regression_confidence_band.svg")