Calculator inputs
Example data table
| Point | x | y | Comment |
|---|---|---|---|
| 1 | 0 | 1.0 | Starting observation |
| 2 | 1 | 2.2 | Small upward trend |
| 3 | 2 | 4.1 | Growth continues |
| 4 | 3 | 6.5 | Curvature appears |
| 5 | 4 | 9.2 | Higher response |
| 6 | 5 | 12.1 | Useful for fitting |
Formula used
1. Piecewise linear interpolation: for two surrounding points, the estimate is y = y₀ + (y₁ - y₀)(x - x₀)/(x₁ - x₀). It matches known points exactly and estimates values between them.
2. Polynomial least squares: the fitted model is y ≈ a₀ + a₁x + a₂x² + ... + aₙxⁿ. Coefficients are found by minimizing the sum of squared residuals across all points.
3. Exponential fit: the model is y ≈ ae^(bx). It is solved by converting to ln(y) = ln(a) + bx, then applying linear regression.
4. Logarithmic fit: the model is y ≈ a + b ln(x). This suits data that changes rapidly at first and then levels off.
5. Power fit: the model is y ≈ ax^b. It becomes linear after transformation: ln(y) = ln(a) + b ln(x).
Accuracy metrics: RMSE measures typical error size, MAE gives average absolute error, MAPE shows percentage error, and R² indicates how much variation the model explains.
How to use this calculator
- Enter sample points with one x,y pair on each line.
- Select the approximation method that best suits your data.
- For polynomial fitting, choose a degree from 1 to 5.
- Enter the target x value you want to estimate.
- Set graph limits if you want a custom plotting range.
- Click Approximate Function to generate the result.
- Review the predicted value, equation, fit metrics, and residual table.
- Use the CSV and PDF buttons to export the analysis.
FAQs
1. What does this calculator estimate?
It estimates a function value at a chosen x by fitting or interpolating from your sample points. It also shows the model equation and fit quality metrics.
2. When should I use interpolation?
Use interpolation when you trust the nearby measured points and want estimates between them. It is best for local estimates rather than broad trend modeling.
3. How do I choose the polynomial degree?
Start with degree 2 or 3. Higher degrees can fit data more closely, but they may overreact to noise and create unstable predictions outside the data range.
4. Why are some methods restricted to positive values?
Exponential, logarithmic, and power models use logarithms during fitting. Logarithms require positive inputs, so the calculator blocks invalid x or y values.
5. What does R² tell me?
R² shows how much variation in y is explained by the fitted model. Values closer to 1 usually indicate a stronger fit to the provided data.
6. What is the difference between RMSE and MAE?
MAE gives the average absolute error. RMSE penalizes larger errors more strongly, making it useful when large misses matter more in your analysis.
7. Can I use it for extrapolation?
Yes, but extrapolation is riskier than interpolation. Predictions beyond the data range depend heavily on the chosen model and may become unreliable quickly.
8. What should my input format look like?
Enter one point per line, such as 1, 2.5 or 1 2.5. The first number is x, and the second is y.