Least Squares Sine Fit Calculator Online

Calculator

Data points

Use one row per point. Accepted formats: x, y or x, y, weight.

Frequency mode

Search best angular frequency

Use fixed angular frequency

Fixed angular frequency ω

Minimum ω for search

Maximum ω for search

Search steps

Phase display

Optional prediction x

Weighted fitting

Use third column as point weight

Submit

Formula Used

The fitted model is:

y = A sin(ωx + φ) + D

For a fixed angular frequency, the model is rewritten as:

y = a sin(ωx) + b cos(ωx) + D

The weighted least squares system is:

β = (XᵀWX)⁻¹XᵀWy

Here, β = [a, b, D]. Then:

A = √(a² + b²)

φ = atan2(b, a)

frequency = ω / 2π

period = 2π / ω

SSE = Σ wᵢ(yᵢ - ŷᵢ)²

How to Use This Calculator

Enter x and y data pairs in the data box.
Add a third weight column when weighted fitting is needed.
Select fixed frequency if angular frequency is already known.
Select search mode when the best frequency must be estimated.
Enter a realistic minimum and maximum angular frequency.
Press calculate to view the result above the form.
Check residuals, RMSE, and R squared before using the fit.
Use CSV or PDF export for records and reports.

Example Data Table

x	y	weight
0	2.18	1
0.5	3.02	1
1.0	3.52	1
1.5	3.60	1
2.0	3.22	1
2.5	2.42	1
3.0	1.48	1
3.5	0.55	1

Least Squares Sine Fit Guide

A sine pattern appears in many measured systems. It can describe sound, tides, vibration, seasons, electrical waves, and rotating motion. Real data is rarely perfect. Sensors add noise. Sampling may be uneven. A least squares sine fit finds the smooth curve that stays closest to all points.

What the Fit Finds

The calculator models data with amplitude, angular frequency, phase, and vertical offset. Amplitude shows half the wave height. Angular frequency controls how fast the cycle repeats. Phase shifts the curve left or right. Offset moves the center line up or down.

Why Least Squares Works

Least squares minimizes squared residuals. A residual is the measured value minus the fitted value. Squaring makes positive and negative errors add together. Large errors also receive more attention. This gives a stable fit for noisy mathematical data.

Frequency Handling

If the angular frequency is known, enter it directly. The page then solves a linear regression for sine, cosine, and offset terms. If the frequency is unknown, use the search option. The calculator tests many angular frequencies and selects the one with the smallest error. A refinement pass improves the final value.

Useful Output

The result includes amplitude, phase, offset, frequency, period, and fitted equation. It also reports error measures, such as SSE, RMSE, MAE, and R squared. These values help judge whether the wave explains the data well. A high R squared usually means a strong repeating pattern.

Practical Tips

Use several points across more than one cycle. Avoid fitting only one small section of a wave. Choose a realistic frequency search range. Too wide a range can find a false local answer. Weights can raise the influence of trusted points. Keep weak or uncertain points with lower weights.

When data contains gaps, keep the x values unchanged. Uneven spacing is allowed. The method uses each recorded coordinate directly. That makes it useful for lab records and daily field logs.

Interpreting the Model

The fitted equation is best viewed as an approximation. It should not replace subject knowledge. Check residuals before using the curve for decisions. Random residuals are a good sign. Clear residual patterns may mean the data needs another term, a trend, or a different model.

FAQs

What is a least squares sine fit?

It is a method that finds the sine curve with the smallest squared error from measured data. The calculator estimates amplitude, frequency, phase, and offset.

What model does this calculator use?

It uses y = A sin(ωx + φ) + D. For solving, it converts the equation into sine, cosine, and offset coefficients.

Can it estimate unknown frequency?

Yes. Select search mode, enter a minimum and maximum angular frequency, and set search steps. The calculator chooses the frequency with the lowest error.

When should I use fixed frequency?

Use fixed frequency when you already know the angular frequency from theory, equipment settings, or a previous experiment. This gives a more controlled fit.

What does phase mean?

Phase shows the horizontal shift of the fitted sine wave. It can be displayed in radians or degrees, depending on your selected output option.

What is RMSE?

RMSE means root mean square error. It measures typical fitting error in the same unit as y. Lower RMSE usually means a better fit.

Why use weights?

Weights let trusted points influence the fit more strongly. Use larger weights for reliable observations and smaller weights for uncertain measurements.

Can I export the results?

Yes. Use the CSV button for spreadsheet work. Use the PDF button for a compact report containing the equation, statistics, and residual table.