〰️ 二次回归计算器Quadratic Regression Calculator

使用说明与常见问题How-to & FAQ

「二次回归计算器」是什么？What is the Quadratic Regression Calculator tool?

输入若干对 (x, y) 数据，用最小二乘法拟合二次函数 y = ax² + bx + c，给出系数、决定系数 R²、残差平方和，并可代入 x 求预测值。Fit a quadratic y = ax² + bx + c to (x, y) data by least squares, with the coefficients, R², residual sum of squares and a prediction for any x.

如何使用「二次回归计算器」？How do I use the Quadratic Regression Calculator tool?

在「二次回归计算器」中，填写「预测 x（可选）」等参数，结果会随输入实时计算并显示在下方。In the Quadratic Regression Calculator: Fill in "Predict at x (optional)"; results are recalculated instantly as you type.

怎么拟合的？How is the fit computed?

最小二乘法：求 a、b、c 使 Σ(yᵢ − (axᵢ² + bxᵢ + c))² 最小。工具先把 x 标准化再解正规方程，减少大数 x 时的舍入误差。Least squares: choose a, b and c to minimise Σ(yᵢ − (axᵢ² + bxᵢ + c))². The tool standardises x before solving the normal equations to limit rounding error for large x.

R² 是什么？What is R²?

R² = 1 − SSE ÷ SST，表示模型解释了 y 的多少变化，越接近 1 拟合越好。但 R² 高不代表二次模型是正确的模型，最好再看看残差。R² = 1 − SSE ÷ SST, the share of the variation in y the model explains; closer to 1 is a tighter fit. A high R² does not prove the quadratic is the right model, so check the residuals too.

至少要几个点？How many points are needed?

至少 3 个不同的 x。只有 3 个点时曲线必然穿过它们。At least 3 different x values. With exactly 3, the curve always passes through them.

计算精度如何？How precise are the results?

「二次回归计算器」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Quadratic Regression Calculator uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.

「二次回归计算器」免费吗？需要注册吗？Is the Quadratic Regression Calculator free? Do I need an account?

「二次回归计算器」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Quadratic Regression Calculator is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.