Quartic Regression Calculator
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Results
Enter data and click calculate.
Understanding Quartic Regression
Quartic regression is a form of polynomial regression where the relationship between a dependent variable (Y) and an independent variable (X) is modeled using a fourth-degree polynomial equation. This means the curve that best fits the data points has a shape influenced by terms up to X4. The general form of a quartic regression equation is:
Y = β₀ + β₁X + β₂X² + β₃X³ + β₄X⁴ + ε
Here:
- Y is the dependent variable.
- X is the independent variable.
- β₀, β₁, β₂, β₃, β₄ are the regression coefficients, representing the parameters of the quartic curve. These are estimated from the data.
- ε represents the error term (the difference between the observed and predicted Y values).
The goal of quartic regression is to find the coefficients (β₀ to β₄) that minimize the sum of the squared differences between the observed Y values and the Y values predicted by the polynomial equation. This is typically achieved using methods like the method of least squares.
When to Use Quartic Regression
Quartic regression is used when the relationship between variables is complex and appears to have a "W" or "M" shape, or multiple turning points. While higher-order polynomials can fit data very closely, they can also lead to overfitting, where the model fits the training data extremely well but performs poorly on new, unseen data. Therefore, quartic regression is usually considered only when simpler models (linear, quadratic, cubic) do not adequately capture the underlying trend.
Common applications include:
- Physics and Engineering: Modeling complex physical phenomena where the forces or behaviors are not linear, such as certain material deformation studies or fluid dynamics.
- Economics: Analyzing economic trends that exhibit multiple peaks and troughs over time, though often simpler models are preferred for interpretability.
- Biology: Describing biological growth patterns or responses that show non-monotonic behavior.
- Environmental Science: Modeling pollutant concentration or ecological population dynamics that can fluctuate significantly.
It's crucial to visually inspect the data and the regression curve to ensure that the quartic model is appropriate and not simply overfitting noise in the data.
How the Calculator Works
This calculator takes pairs of X and Y data points. It uses a numerical method (specifically, solving a system of linear equations derived from the normal equations of least squares) to estimate the coefficients β₀, β₁, β₂, β₃, β₄ for the quartic equation:
Y ≈ b₀ + b₁X + b₂X² + b₃X³ + b₄X⁴
Once the coefficients are determined, the calculator can predict the Y value for a new, specified X value using the derived equation. It also calculates the equation itself.
function calculateQuarticRegression() {
var xDataStr = document.getElementById("xData").value.trim();
var yDataStr = document.getElementById("yData").value.trim();
var predictX = parseFloat(document.getElementById("predictX").value);
if (xDataStr === "" || yDataStr === "") {
document.getElementById("result").innerHTML = "Please enter both X and Y data.";
return;
}
var xValues = xDataStr.split(',').map(function(x) { return parseFloat(x.trim()); });
var yValues = yDataStr.split(',').map(function(y) { return parseFloat(y.trim()); });
if (xValues.length !== yValues.length) {
document.getElementById("result").innerHTML = "Error: X and Y data must have the same number of points.";
return;
}
if (xValues.length < 5) {
document.getElementById("result").innerHTML = "Error: At least 5 data points are required for quartic regression.";
return;
}
// Check for valid numbers
for (var i = 0; i < xValues.length; i++) {
if (isNaN(xValues[i]) || isNaN(yValues[i])) {
document.getElementById("result").innerHTML = "Error: Invalid number found in data.";
return;
}
}
if (isNaN(predictX)) {
predictX = null; // Allow calculation without prediction value
}
// Prepare data for matrix calculation
var n = xValues.length;
var X = []; // Matrix for powers of X
var Y = []; // Vector for Y values
for (var i = 0; i < n; i++) {
var x = xValues[i];
X.push([1, x, x*x, x*x*x, x*x*x*x]); // 1, X, X^2, X^3, X^4
Y.push(yValues[i]);
}
// Calculate sums for the normal equations
// We need to solve the system of linear equations:
// S(y) = b0*S(1) + b1*S(x) + b2*S(x^2) + b3*S(x^3) + b4*S(x^4)
// S(xy) = b0*S(x) + b1*S(x^2) + b2*S(x^3) + b3*S(x^4) + b4*S(x^5)
// S(x^2y) = b0*S(x^2) + b1*S(x^3) + b2*S(x^4) + b3*S(x^5) + b4*S(x^6)
// S(x^3y) = b0*S(x^3) + b1*S(x^4) + b2*S(x^5) + b3*S(x^6) + b4*S(x^7)
// S(x^4y) = b0*S(x^4) + b1*S(x^5) + b2*S(x^6) + b3*S(x^7) + b4*S(x^8)
var sums = {
'1': n,
'x': 0, 'x2': 0, 'x3': 0, 'x4': 0, 'x5': 0, 'x6': 0, 'x7': 0, 'x8': 0,
'y': 0, 'xy': 0, 'x2y': 0, 'x3y': 0, 'x4y': 0
};
for (var i = 0; i < n; i++) {
var xi = xValues[i];
var yi = yValues[i];
var xi2 = xi * xi;
var xi3 = xi2 * xi;
var xi4 = xi3 * xi;
var xi5 = xi4 * xi;
var xi6 = xi5 * xi;
var xi7 = xi6 * xi;
var xi8 = xi7 * xi;
sums['x'] += xi;
sums['x2'] += xi2;
sums['x3'] += xi3;
sums['x4'] += xi4;
sums['x5'] += xi5;
sums['x6'] += xi6;
sums['x7'] += xi7;
sums['x8'] += xi8;
sums['y'] += yi;
sums['xy'] += xi * yi;
sums['x2y'] += xi2 * yi;
sums['x3y'] += xi3 * yi;
sums['x4y'] += xi4 * yi;
}
// Construct the matrix A and vector B for the system Ax = B
// A is a 5×5 matrix, B is a 5×1 vector
var A = [
[sums['1'], sums['x'], sums['x2'], sums['x3'], sums['x4']],
[sums['x'], sums['x2'], sums['x3'], sums['x4'], sums['x5']],
[sums['x2'], sums['x3'], sums['x4'], sums['x5'], sums['x6']],
[sums['x3'], sums['x4'], sums['x5'], sums['x6'], sums['x7']],
[sums['x4'], sums['x5'], sums['x6'], sums['x7'], sums['x8']]
];
var B = [sums['y'], sums['xy'], sums['x2y'], sums['x3y'], sums['x4y']];
// Solve the system Ax = B using Gaussian elimination
var coeffs = gaussianElimination(A, B);
if (!coeffs) {
document.getElementById("result").innerHTML = "Error: Could not solve the system of equations. Check your data.";
return;
}
var b0 = coeffs[0];
var b1 = coeffs[1];
var b2 = coeffs[2];
var b3 = coeffs[3];
var b4 = coeffs[4];
var equation = "Y = " + b0.toFixed(6) + " + " + b1.toFixed(6) + "X + " + b2.toFixed(6) + "X² + " + b3.toFixed(6) + "X³ + " + b4.toFixed(6) + "X⁴";
var resultText = "
Quartic Regression Equation:" + equation + "";
if (predictX !== null) {
var predictedY = b0 + b1 * predictX + b2 * Math.pow(predictX, 2) + b3 * Math.pow(predictX, 3) + b4 * Math.pow(predictX, 4);
resultText += "
Predicted Y for X = " + predictX + ": " + predictedY.toFixed(6);
} else {
resultText += "
(Enter an 'X' value in 'Predict Y for X' to see a specific prediction.)";
}
document.getElementById("result").innerHTML = resultText;
}
// Helper function for Gaussian elimination to solve Ax = B
// Modifies matrix A and vector B in place.
function gaussianElimination(A, B) {
var n = A.length;
// Forward elimination
for (var i = 0; i < n; i++) {
// Find pivot row
var maxRow = i;
for (var k = i + 1; k Math.abs(A[maxRow][i])) {
maxRow = k;
}
}
// Swap rows
var temp = A[i];
A[i] = A[maxRow];
A[maxRow] = temp;
temp = B[i];
B[i] = B[maxRow];
B[maxRow] = temp;
// Check for singularity
if (A[i][i] === 0) {
return null; // Matrix is singular
}
// Eliminate below
for (var k = i + 1; k < n; k++) {
var factor = A[k][i] / A[i][i];
B[k] -= factor * B[i];
for (var j = i; j = 0; i–) {
var sum = 0;
for (var j = i + 1; j < n; j++) {
sum += A[i][j] * x[j];
}
x[i] = (B[i] – sum) / A[i][i];
}
return x;
}