LSA Engine — Least Squares Adjustment

A native desktop GUI for geodetic and general least squares adjustment — four estimators, residual plot, and variance-covariance heatmap. Built in Rust with egui.

Screenshots

Input Tab	Results Tab

Residual Plot	VCV Heatmap

Estimators

Method	Formula	Use Case
Unweighted LS	x̂ = (AᵀA)⁻¹Aᵀb	Equal-precision observations
Weighted LS	x̂ = (AᵀWA)⁻¹AᵀWb	Different observation precisions
Constrained LS	min ‖v‖² s.t. Cx=d	Fixed constraints on unknowns
Robust IRLS	Iterative Huber reweighting	Data with outliers

Problem Type

General Ax = b — you supply the design matrix A, observation vector b, and optionally the weight matrix W and constraint matrix C. Works for any linear adjustment problem: levelling networks, traverse, polynomial fitting, coordinate transformation, sensor fusion, etc.

Outputs

• Estimated unknowns x̂ with standard deviations σ(x̂)
• Reference variance σ₀² and σ₀
• Residuals v = Ax̂ − b with standardised values v/σ₀
• Colour-coded outlier detection: green |v/σ₀|<2, yellow 2–3, red >3
• Residual bar chart with hover tooltip
• VCV heatmap — visual variance-covariance matrix

Math

Unweighted LS:   x̂ = (AᵀA)⁻¹Aᵀb
Weighted LS:     x̂ = (AᵀWA)⁻¹AᵀWb
Constrained LS:  augmented system via Lagrange multipliers
σ₀²:             vᵀWv / r       r = n − u  (degrees of freedom)
Σ_x̂:             σ₀² · (AᵀWA)⁻¹

Robust IRLS — Huber ψ-function:

ψ(t) = t              |t| ≤ k   (Gaussian)
ψ(t) = k · sign(t)    |t| > k   (outlier)
wᵢ   = k·σ̂ / |vᵢ|    for downweighted obs
σ̂    = median(|vᵢ|) / 0.6745   (MAD robust estimate)

Default k = 1.345 → 95% efficiency under normality.

Built-in Examples

Example	n	u	r	Notes
Levelling Network	4	3	1	Height differences
Polynomial Fit	5	3	2	Quadratic y = a + bx + cx²
2D Position	5	2	3	Weighted observations
Outlier Demo	7	2	5	One large outlier — use Robust IRLS

Build & Run

# Prerequisites: Rust ≥ 1.75
cargo build --release
cargo run --release
# or double-click run.bat on Windows

Tests:

cargo test
# 9 passed; 0 failed

References

• Mikhail, E.M. & Ackermann, F. (1976). Observations and Least Squares. IEP.
• Koch, K.-R. (1999). Parameter Estimation and Hypothesis Testing in Linear Models. Springer.
• Huber, P.J. (1964). Robust estimation of a location parameter. Ann. Math. Stat. 35(1).
• Leick, A. (2004). GPS Satellite Surveying, 3rd ed. Wiley.

Author

Dr. Mosab Hawarey

PhD, Geodetic & Photogrammetric Engineering (ITU) | MSc, Geomatics (Purdue) | MBA (Wales) | BSc, MSc (METU)

• GitHub: https://github.com/mhawarey
• Personal: https://hawarey.org/mosab
• ORCID: https://orcid.org/0000-0001-7846-951X

License

MIT License