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

## Sample size.
n = 100

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

## Design matrix.
X = np.ones((n,4))
X[:,1] = XV
X[:,2] = XV**2
X[:,3] = XV**3

## True coefficients.
beta = np.array([0, 0.1, -0.25, -0.25], 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 Scheffe interval.
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 confidence band.
D = s*np.sqrt(X.shape[1]*QF*V)
LB,UB = Yhat-D,Yhat+D

## Make the plot.
plt.clf()
plt.plot(XV, Y, 'o', ms=3, color='grey')
plt.plot(XV, EY, '-', color='blue', label = "Truth")
plt.plot(XV, Yhat, '-', color='green', label = "Estimate")
plt.plot(XV, LB, '-', color='red', label = "CB")
plt.plot(XV, UB, '-', color='red')
plt.legend(frameon=False)
plt.ylim([-25,20])
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("polyreg_scheffe.pdf")
plt.savefig("polyreg_scheffe.svg")