Advanced Curve Fitting App

1. App Overview

The Advanced Curve Fitting App is a self-contained, browser-based tool that lets you enter or import X–Y data, automatically fit one or more mathematical curves to that data, visualize all curves on an interactive chart, and export polished statistical reports.

Key capabilities at a glance:

• Fit up to four curve families simultaneously: Polynomial, Exponential, Logarithmic, and Power
• Polynomial degree 1–5 (linear through quintic)
• Instant R² goodness-of-fit scoring with quality labels
• Predict Y at any arbitrary X value across all fitted curves
• Import/export data in CSV or JSON; copy to clipboard
• Two print reports: a Basic report and an Advanced Statistical report with descriptive stats, AIC/BIC model comparison, confidence intervals, and Cook's distance

2. Interface Layout

The app uses a two-column layout. The left column contains all controls; the right column shows the chart and results.

3. Entering Data

3.1 Manual entry

Type or paste your data directly into the large text area labeled Enter X, Y data. Each row represents one data point. The required format is:

x_value, y_value

Example — six data points:

1, 2.1
2, 4.8
3, 11.0
4, 20.9
5, 35.5
6, 55.2

3.2 Load built-in example

Click 📊 Example Data to instantly load an 11-point exponential-growth dataset (x = 0 to 10). This is a good starting point to explore all features before working with your own data.

3.3 Import from file or clipboard

Click Import / Export to open the Import/Export modal, then use one of these methods:

• Upload a file — drag-and-drop or browse for a .csv or .json file
• Paste text — paste CSV or JSON into the "Copy/Paste Data" textarea and click Import from Text

Accepted CSV format (first line with an x header is optional and ignored):

# Title: My Experiment
x,y
1, 3.2
2, 7.8
3, 18.1

Accepted JSON format:

[{"x": 1, "y": 3.2}, {"x": 2, "y": 7.8}, {"x": 3, "y": 18.1}]

Or the titled JSON format (preserves the dataset name):

{"title": "My Experiment", "data": [{"x": 1, "y": 3.2}, ...]}

3.4 Data title

Enter a descriptive name in the Data Title field above the data textarea. This title appears in all exported files and printed reports.

4. Curve Types Explained

Check one or more of the four curve families before clicking Fit Curves. You can run all four simultaneously and compare R² values to find the best model.

Polynomial degree selector

When Polynomial is checked, choose the degree from the drop-down:

• 1 — Linear: y = a₁x + a₀ (straight line)
• 2 — Quadratic: y = a₂x² + a₁x + a₀ (parabola)
• 3 — Cubic: one inflection point
• 4 — Quartic: up to three turning points
• 5 — Quintic: up to four turning points (needs ≥ 6 data points)

5. Fitting Curves & Reading Results

Understanding the result card

Each card looks like this:

R² — goodness of fit

R² (coefficient of determination) measures how well the curve explains the variance in your data. It ranges from 0 to 1 (or can be negative for very poor fits).

6. Chart Interpretation

The canvas chart auto-scales to fit your data range with 10% padding on each axis. Grid lines, axis tick labels, and all fitted curve overlays are drawn automatically.

• Black dots — your actual data points
• Blue line — Polynomial fit
• Green line — Exponential fit
• Orange line — Logarithmic fit
• Purple line — Power fit

Each curve is plotted with 300 sample points across the data range, so even highly curved models render smoothly. Logarithmic and Power curves are only plotted for x > 0.

7. Evaluate at a Custom Point

The Evaluate at Custom Point panel (bottom of the left column) lets you predict the Y value for any X across all currently fitted curves — useful for interpolation and extrapolation.

How to use it

1. Fit your curves first (click Fit Curves).
2. In the X value box, enter the X coordinate you want to predict at.
3. Optionally enter an Expected Y value if you have a measured result you want to compare against.
4. Click Calculate.

Output

The results box shows the predicted Y from every fitted curve. If you supplied an Expected Y, it also shows the absolute difference and percentage error for each curve — handy for ranking which model is most accurate at that specific point.

8. Import & Export Data

Open the Import/Export modal by clicking Import / Export in the control panel.

Exporting

• Export as CSV — copies a CSV string to your clipboard, including the dataset title as a comment on the first line. Paste into a text file and save as .csv.
• Export as JSON — copies a JSON object with title and data keys to your clipboard.
• Copy to Clipboard — copies raw CSV (x,y per line) without a title comment.

Importing

• Upload a file — click the dashed file area, choose a .csv or .json file from your computer. The data is parsed, loaded into the textarea, and curves are automatically fitted.
• Paste text — paste CSV or JSON into the textarea inside the modal and click Import from Text. The app auto-detects which format you pasted.

9. Print Reports

You must fit at least one curve before printing. Two report formats are available:

Basic Report

A two-page print layout containing:

• Page 1 — chart image, regression equations, R² values, and fit quality ratings for all fitted curves
• Page 2 — data table showing actual Y values alongside predicted Y and residual error from each fitted curve; followed by SSE, MAE, and RMSE summary statistics

Advanced Statistical Report

A comprehensive multi-page report that adds:

• Descriptive statistics for both X and Y: mean, median, standard deviation, variance, min, max, Q1, Q3, skewness, kurtosis, Pearson correlation r and r²
• Model comparison table: R², adjusted R², AIC, BIC, parameter count, and RMSE for every fitted curve — use this to objectively select the best model
• Residual analysis: residuals, squared residuals, and Cook's distance per data point
• Confidence intervals: 95% prediction interval for each data point under the polynomial model

10. Example Data Sets

The following ready-to-use data sets demonstrate each curve type. Copy any block below and paste it into the data textarea, then click Fit Curves.

Example A — Exponential Growth (built-in example)

Simulates a bacterial population doubling roughly every 2 time steps. Exponential fit should score ≥ 0.99.

0, 1.2
1, 2.5
2, 5.1
3, 10.2
4, 17.8
5, 26.5
6, 37.1
7, 51.2
8, 66.8
9, 85.1
10, 105.5

Example B — Quadratic (Projectile Height)

Represents the height of a ball thrown upward over time. Quadratic polynomial should dominate.

0, 0
0.5, 12.3
1.0, 19.8
1.5, 22.6
2.0, 20.1
2.5, 12.9
3.0, 1.2

Example C — Logarithmic (Learning Curve)

Diminishing performance gains with practice — a classic logarithmic shape. All X values must be > 0.

1, 15.0
2, 21.0
4, 27.5
8, 34.0
16, 40.5
32, 47.0
64, 53.5
128, 60.0

Example D — Power Law (Tool Wear)

Tool wear rate modelled by a power law. Both X and Y must be positive.

1, 0.50
2, 0.71
3, 0.87
5, 1.12
8, 1.41
12, 1.73
20, 2.24
50, 3.54

Example E — Noisy Real-World Data (Multi-model comparison)

A realistic dataset with measurement noise. Use this to practice comparing all four models and using AIC/BIC for selection.

1, 3.8
2, 8.1
3, 14.7
4, 18.3
5, 29.2
6, 39.5
7, 52.1
8, 68.4
9, 87.3
10, 112.0
11, 131.5
12, 165.2

Try evaluating at x = 13 with expected Y = 200 using the Evaluate panel to see which model predicts best beyond the data range.

11. Tips & Best Practices

Choosing the right model

Avoiding overfitting with polynomial

A degree-5 polynomial through 6 data points will achieve R² = 1 by passing through every point exactly — but it will oscillate wildly between them and make terrible predictions. As a rule:

• Keep degree ≤ 3 for most practical use cases
• Check the Advanced Report's adjusted R² — it penalises extra parameters
• Compare AIC values: if degree 3 gives lower AIC than degree 4, stop at 3

When a curve can't be fitted

• Exponential / Power — if any Y ≤ 0 (exponential) or any X ≤ 0 or Y ≤ 0 (power), the log transform needed for fitting fails. Remove those points or shift your data.
• Logarithmic — requires all X > 0. An X value of 0 is undefined for ln(x).
• Insufficient points — polynomial of degree D needs at least D + 1 points. With only 3 data points you can fit at most a degree-2 curve.

Saving your work

The app has no internal save function. To preserve work between sessions:

1. Click Import / Export → Export as JSON. The JSON (including your title) is copied to the clipboard.
2. Paste it into a text file and save it with a .json extension.
3. Next session, use Import → Upload File or paste the JSON text to restore everything.

12. FAQ

Why is my R² negative?

A negative R² means the fitted curve is a worse predictor than simply using the mean of Y. This usually happens when the selected curve type is completely wrong for the data shape — for example, fitting a logarithm to data that clearly grows exponentially.

Why does the Power curve disappear from the chart?

Power and Logarithmic curves are undefined for x ≤ 0. If your data range includes zero or negative X values, those portions of the curve are simply not drawn. The regression coefficients are still calculated from the valid (positive X) portion of your data.

Can I fit more than four curves?

The app supports exactly four model families. To explore additional model families (e.g., Gompertz, logistic, Gaussian), the data would need to be analysed in a dedicated statistics package.

The Print button opens a print dialog but the chart is missing.

Some browser pop-up blockers interfere with the print-preview window. Ensure pop-ups are allowed for the file URL. Alternatively, use Print Advanced Report, which embeds the chart as a canvas copy.

I pasted CSV but got "No valid data points found".

Check that: (1) commas separate X and Y (not semicolons or tabs); (2) there are no empty lines at the start; (3) decimal numbers use a period, not a comma (i.e., 3.14 not 3,14).

What algorithm is used for polynomial regression?

The app builds the normal equations (Xᵀ·X)·β = Xᵀ·Y and solves them using Gaussian elimination with partial pivoting. Exponential, Logarithmic, and Power regressions are linearised via logarithmic transformation and then solved as degree-1 polynomial problems.