TrendSurfaceAnalysis

Enhanced Trend Surface Analysis Plugin Documentation

Overview

The Enhanced Trend Surface Analysis Plugin is a powerful QGIS tool for spatial trend analysis using polynomial regression. It allows users to model spatial patterns in point data and generate trend surfaces with comprehensive statistical diagnostics.

Table of Contents

1. Features
2. Installation
3. Quick Start
4. User Interface
5. Parameters Explained
6. Outputs
7. Use Cases
8. Troubleshooting
9. Mathematical Background
10. FAQ

Features

Core Analysis

• Polynomial Trend Surfaces: Degrees 1-8 for flexible modeling
• Least Squares Regression: Robust mathematical foundation
• Multiple Output Formats: Raster surfaces and point residuals
• Automatic CRS Handling: Maintains coordinate reference systems

Advanced Capabilities

• Robust Regression: Outlier-resistant analysis
• Cross-Validation: Model performance validation
• Confidence Intervals: Uncertainty quantification
• Comprehensive Statistics: R², RMSE, AIC, BIC, and more

User Experience

• Intuitive Interface: Easy-to-use dialog-based workflow
• Automatic Field Detection: Smart field name recognition
• Real-time Feedback: Progress monitoring and logging
• Automatic Styling: Pre-configured layer symbology

Installation

System Requirements

• QGIS 3.16 or later
• Python 3.7+
• NumPy, SciPy libraries (included with QGIS)

Installation Steps

1. Download the Plugin

   # Clone from repository or download ZIP
   git clone https://github.com/your-repo/enhanced-trend-surface.git
   

2. Install in QGIS
   • Open QGIS
   • Go to Plugins → Manage and Install Plugins
   • Click Install from ZIP
   • Select the plugin directory or ZIP file
   • Click Install Plugin
3. Verify Installation
   • Check for “Enhanced Trend Surface” in the Plugins menu
   • Look for the plugin icon in the toolbar

Manual Installation

# Copy to QGIS plugins directory
cp -r EnhancedTrendSurface ~/.local/share/QGIS/QGIS3/profiles/default/python/plugins/

Quick Start

Basic Workflow

1. Load your point data with elevation/Z values
2. Click the plugin icon or go to Plugins → Enhanced Trend Surface → Enhanced Trend Surface Analysis
3. Select input layer and Z value field
4. Choose analysis parameters (polynomial degree, cell size)
5. Set output locations for results
6. Click “Run Analysis” and view results

Example: Topographic Analysis

1. Load a point layer with elevation data
2. Set polynomial degree to 2 (quadratic surface)
3. Use 100m cell size for output raster
4. Run analysis to visualize regional topography

User Interface

Main Dialog Components

Input Parameters Section

• Input Point Layer: Dropdown of all point layers in project
• Z Value Field: Numeric field containing values to analyze
• Weight Field (optional): Field for weighted regression
• Polynomial Degree: Complexity of trend surface (1-8)
• Cell Size: Resolution of output raster in map units
• Confidence Level: Statistical confidence for intervals (80-99.9%)

Analysis Options

• Cross-Validation: Enable k-fold validation for model assessment
• Robust Regression: Use outlier-resistant fitting method

Output Settings

• Trend Surface: Main output raster (required)
• Residuals Surface: Difference between observed and predicted (optional)
• Confidence Intervals: Uncertainty surface (optional)
• Statistics Folder: Location for detailed reports (optional)

Parameters Explained

Polynomial Degree

Degree	Equation Terms	Use Case
1	3 (planar)	Simple linear trends
2	6 (quadratic)	Gentle curvature, basic topography
3	10 (cubic)	Complex surfaces, geological features
4+	15+	Highly complex patterns (use with caution)

Recommendation: Start with degree 2-3 for most applications.

Cell Size

• Small values (1-10m): High detail, larger files
• Medium values (10-100m): Balanced detail and performance
• Large values (100-1000m): Regional trends, faster processing

Confidence Level

• 90%: Standard scientific use
• 95%: Common statistical threshold
• 99%: High certainty requirements

Outputs

Generated Layers

1. Trend Surface Raster

• Description: Continuous surface showing spatial trend
• Symbology: Blue-to-red color ramp (low to high values)
• Use: Primary visualization of spatial pattern

2. Residual Points Layer

• Description: Original points with residual values
• Fields:
  • Original attributes
  • residual: Observed - Predicted
  • predicted: Model-predicted value
• Use: Identify local anomalies and model fit

3. Residuals Surface Raster (Optional)

• Description: Interpolated residual values
• Symbology: Diverging blue-white-red (negative-zero-positive)
• Use: Spatial pattern of model errors

4. Confidence Intervals Raster (Optional)

• Description: Standard error of predictions
• Symbology: Grayscale (light to dark = low to high uncertainty)
• Use: Assess prediction reliability

Statistical Reports

Comprehensive Statistics

ENHANCED POLYNOMIAL TREND SURFACE ANALYSIS REPORT
==================================================

Analysis Date: 2024-01-15 14:30:25
Polynomial Degree: 2
Number of Observations: 250
Number of Parameters: 6

GOODNESS OF FIT STATISTICS:
R²: 0.8743
Adjusted R²: 0.8712
RMSE: 12.45
MAE: 8.67

MODEL SELECTION CRITERIA:
AIC: 1456.23
BIC: 1478.45

RESIDUAL ANALYSIS:
Residual Mean: 0.0234
Residual Std: 12.38
Normality Test p-value: 0.2345

CROSS-VALIDATION RESULTS:
CV R²: 0.8621
CV MSE: 156.34
Folds Completed: 5

Use Cases

1. Topographic Analysis

• Data: Elevation points, LiDAR data
• Degree: 2-3
• Output: Regional slope and aspect patterns
• Application: Watershed delineation, terrain analysis

2. Environmental Monitoring

• Data: Pollution measurements, temperature readings
• Degree: 1-2
• Output: Pollution gradients, thermal patterns
• Application: Environmental impact assessment

3. Geological Applications

• Data: Mineral concentrations, rock properties
• Degree: 2-4
• Output: Geological trend surfaces
• Application: Resource exploration, geological mapping

4. Climate Studies

• Data: Precipitation, temperature stations
• Degree: 1-3
• Output: Climate surfaces
• Application: Climate modeling, agricultural planning

Troubleshooting

Common Issues

“Insufficient points” Error

• Cause: Too few points for selected polynomial degree
• Solution:
  • Collect more data points
  • Reduce polynomial degree
  • Minimum points = (degree + 1) × (degree + 2) ÷ 2

“Matrix solving failed” Error

• Cause: Numerically unstable design matrix
• Solution:
  • Reduce polynomial degree
  • Check for duplicate points
  • Ensure adequate spatial distribution

Poor Model Fit (Low R²)

• Causes:
  • Insufficient polynomial complexity
  • High local variability
  • Outliers influencing model
• Solutions:
  • Increase polynomial degree
  • Enable robust regression
  • Remove spatial outliers

Performance Tips

For Large Datasets

• Use larger cell sizes initially
• Start with lower polynomial degrees
• Enable robust regression for noisy data

For Complex Patterns

• Gradually increase polynomial degree
• Use cross-validation to prevent overfitting
• Compare multiple degree results

Mathematical Background

Polynomial Trend Surface

A polynomial trend surface of degree d is defined as:

z(x,y) = β₀ + β₁x + β₂y + β₃x² + β₄xy + β₅y² + ... + βₙyᵈ

Where:

• z(x,y) is the predicted value at coordinates (x,y)
• βᵢ are the coefficients determined by regression
• The number of terms = (d + 1) × (d + 2) ÷ 2

Least Squares Regression

The coefficients are determined by minimizing the sum of squared residuals:

min Σ(z_observed - z_predicted)²

Solved using the normal equations:

β = (AᵀA)⁻¹Aᵀz

Where A is the design matrix of polynomial terms.

Robust Regression

Uses iteratively reweighted least squares with Huber weighting:

w_i = min(1, k / |r_i|)

Where r_i are residuals and k is a tuning constant.

FAQ

Q: What’s the difference between degree 1, 2, and 3?

• Degree 1: Planar surface (flat plane)
• Degree 2: Curved surface with one inflection
• Degree 3: Complex surface with multiple inflection points

Q: When should I use robust regression?

Use robust regression when your data contains outliers or when you want to reduce their influence on the trend surface.

Q: How many points do I need?

Minimum points = number of polynomial terms + 10. For degree 2 (6 terms), aim for at least 16 points.

Q: Can I use this for categorical data?

No, the plugin is designed for continuous numeric data. For categorical data, consider classification methods.

Q: Why are my residuals clustered spatially?

Spatial clustering of residuals suggests local effects not captured by the global trend. Consider local interpolation methods for these areas.

Q: How do I interpret R² values?

• 0.0-0.3: Weak fit
• 0.3-0.7: Moderate fit
• 0.7-0.9: Strong fit
• 0.9-1.0: Excellent fit

Advanced Usage

Batch Processing

For multiple datasets, consider using the Python API:

from EnhancedTrendSurface.core_analysis import TrendSurfaceAnalyzer

analyzer = TrendSurfaceAnalyzer()
results = analyzer.analyze(layer, 'elevation', degree=3, cell_size=50)

Custom Styling

Modify the automatic styling by editing the style functions in the plugin code.

Support and Development

Getting Help

• Documentation: This guide
• Issues: GitHub issue tracker
• Community: QGIS user forums

Contributing

We welcome contributions! Areas for improvement:

• Additional regression methods
• 3D visualization support
• Time-series trend analysis
• Enhanced export options

Citation

If you use this plugin in research, please cite:

S. S. Goveia. Enhanced Trend Surface Analysis Plugin for QGIS. Version 1.0. 
Spatial Analysis Tools. 2025.

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Note: This documentation is for version 1.0 of the Enhanced Trend Surface Analysis Plugin. Check for updates and new features in subsequent releases.

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