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Summary

Description
English: Plot of a cubic polynomial fit using multiple least squares to a simulated data set. Along with the point estimate, a 95% simultaneous confidence band (CB) constructed using Scheffe's method is shown.
Date
Source Own work
Author Skbkekas
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Source code
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Python code

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")

Licensing

I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution
This file is licensed under the Creative Commons Attribution 3.0 Unported license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.

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9 April 2009

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Date/TimeThumbnailDimensionsUserComment
current15:27, 8 March 2021Thumbnail for version as of 15:27, 8 March 2021540 × 360 (40 KB)Olexa RiznykFixed Confidence band ("CB") label
21:00, 9 April 2009Thumbnail for version as of 21:00, 9 April 2009341 × 288 (51 KB)SkbkekasAdded axis labels.
20:54, 9 April 2009Thumbnail for version as of 20:54, 9 April 2009341 × 288 (51 KB)Skbkekas{{Information |Description={{en|1=Plot of a cubic polynomial fit using multiple least squares to a simulated data set.}} |Source=Own work by uploader |Author=Skbkekas |Date=April 9, 2009 |Permission= |other_versions= }} <!--{{ImageUploa

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