Add LFM kernel implementation backlog items

- Add code review task for existing LFM implementations
- Add design task for modern LFM kernel architecture
- Add implementation task for core LFM kernel functionality
- Establish clear task dependencies and acceptance criteria
- Link to CIP-0001 and relevant papers

backlog/features/2025-08-15_design-modern-lfm-kernel.md

@@ -0,0 +1,58 @@
+---
+id: "design-modern-lfm-kernel"
+title: "Design modern LFM kernel architecture"
+status: "Proposed"
+priority: "High"
+created: "2025-08-15"
+last_updated: "2025-08-15"
+owner: "Neil Lawrence"
+dependencies: ["lfm-kernel-code-review"]
+tags:
+- lfm
+- kernel
+- design
+- architecture
+---
+
+# Design modern LFM kernel architecture
+
+## Description
+Design a modern LFM kernel implementation that follows GPy's current architectural patterns and uses the multioutput kernel approach with output index as input.
+
+## Background
+- Current GPy LFM implementations don't use the modern multioutput kernel approach
+- Need to design a unified LFM kernel that integrates well with GPy's current framework
+- Should maintain backward compatibility while providing improved functionality
+
+## Design Requirements
+- [ ] Use GPy's multioutput kernel approach with output index as input
+- [ ] Follow consistent API design with other GPy kernels
+- [ ] Implement proper parameter handling and constraints
+- [ ] Support different base kernels for latent functions
+- [ ] Enable efficient gradient computation
+- [ ] Maintain backward compatibility with existing implementations
+
+## Design Tasks
+- [ ] Define kernel class structure and inheritance hierarchy
+- [ ] Design parameter handling for mass, damper, spring, sensitivity, delay
+- [ ] Plan integration with GPy's multioutput framework
+- [ ] Design cross-kernel computation methods
+- [ ] Plan parameter tying and constraint handling
+- [ ] Design efficient computation methods for large datasets
+
+## Acceptance Criteria
+- [ ] Complete design specification document
+- [ ] API design that follows GPy patterns
+- [ ] Integration plan with existing GPy infrastructure
+- [ ] Performance considerations documented
+- [ ] Backward compatibility strategy defined
+
+## Implementation Notes
+- Study how other multioutput kernels in GPy handle output indices
+- Consider parameter tying approaches from MATLAB implementation
+- Design for extensibility to different differential equation types
+- Plan for efficient computation of cross-kernel terms
+
+## Related
+- CIP: 0001 (LFM kernel implementation)
+- Backlog: lfm-kernel-code-review

backlog/features/2025-08-15_implement-lfm-kernel-core.md

@@ -0,0 +1,62 @@
+---
+id: "implement-lfm-kernel-core"
+title: "Implement core LFM kernel functionality"
+status: "Proposed"
+priority: "High"
+created: "2025-08-15"
+last_updated: "2025-08-15"
+owner: "Neil Lawrence"
+dependencies: ["design-modern-lfm-kernel"]
+tags:
+- lfm
+- kernel
+- implementation
+- core
+---
+
+# Implement core LFM kernel functionality
+
+## Description
+Implement the core LFM kernel class with basic functionality including kernel computation, parameter handling, and gradient computation.
+
+## Background
+- Design phase completed with modern LFM kernel architecture
+- Need to implement the core kernel computation methods
+- Should follow the mathematical foundations from the papers and MATLAB implementation
+
+## Implementation Tasks
+- [ ] Create `GPy.kern.LFM` class inheriting from appropriate base class
+- [ ] Implement parameter handling for mass, damper, spring, sensitivity, delay
+- [ ] Implement `K()` method for kernel matrix computation
+- [ ] Implement `Kdiag()` method for diagonal computation
+- [ ] Add parameter constraints and transformations
+- [ ] Implement basic gradient computation
+- [ ] Add support for different base kernels for latent functions
+
+## Core Methods to Implement
+- [ ] `__init__()` - Parameter initialization and validation
+- [ ] `K(X, X2=None)` - Kernel matrix computation
+- [ ] `Kdiag(X)` - Diagonal computation
+- [ ] `update_gradients_full()` - Gradient computation
+- [ ] `update_gradients_diag()` - Diagonal gradient computation
+- [ ] `parameters_changed()` - Parameter update handling
+
+## Acceptance Criteria
+- [ ] Core LFM kernel class implemented and functional
+- [ ] Basic kernel computation working correctly
+- [ ] Parameter handling and constraints implemented
+- [ ] Gradient computation implemented
+- [ ] Unit tests passing for core functionality
+- [ ] Integration with GPy's parameterization system
+
+## Implementation Notes
+- Follow the mathematical structure from the MATLAB implementation
+- Use GPy's parameterization system for constraints
+- Implement efficient computation methods
+- Ensure proper handling of edge cases and numerical stability
+- Add comprehensive docstrings and documentation
+
+## Related
+- CIP: 0001 (LFM kernel implementation)
+- Backlog: design-modern-lfm-kernel
+- Papers: Álvarez et al. (2009, 2012)

backlog/features/2025-08-15_lfm-kernel-code-review.md

@@ -0,0 +1,51 @@
+---
+id: "lfm-kernel-code-review"
+title: "Review existing LFM kernel implementations"
+status: "Ready"
+priority: "High"
+created: "2025-08-15"
+last_updated: "2025-08-15"
+owner: "Neil Lawrence"
+dependencies: []
+tags:
+- lfm
+- kernel
+- code-review
+- documentation
+---
+
+# Review existing LFM kernel implementations
+
+## Description
+Conduct a comprehensive review of existing LFM (Latent Force Model) kernel implementations in both GPy and MATLAB to understand the current state, design decisions, and limitations.
+
+## Background
+- GPy has existing ODE-based kernels (`EQ_ODE1`, `EQ_ODE2`) that implement LFM concepts
+- MATLAB implementation in GPmat provides a more complete LFM framework
+- Need to understand differences and identify modernization opportunities
+
+## Tasks
+- [ ] Review `GPy/kern/src/eq_ode1.py` and `eq_ode2.py` implementations
+- [ ] Analyze MATLAB LFM implementation structure and patterns
+- [ ] Document current limitations and inconsistencies
+- [ ] Identify reusable components and design patterns
+- [ ] Compare parameter handling approaches
+- [ ] Review cross-kernel computation methods
+- [ ] Document mathematical foundations and implementation details
+
+## Acceptance Criteria
+- [ ] Complete documentation of existing implementations
+- [ ] Clear understanding of design differences between GPy and MATLAB versions
+- [ ] Identified list of modernization opportunities
+- [ ] Documentation of mathematical foundations
+- [ ] Assessment of current limitations and bugs
+
+## Implementation Notes
+- Focus on understanding the mathematical foundations from the papers
+- Pay attention to parameter tying and multi-output handling
+- Document the differential equation structure and kernel computation
+- Identify opportunities for using GPy's modern multioutput kernel approach
+
+## Related
+- CIP: 0001 (LFM kernel implementation)
+- Papers: Álvarez et al. (2009, 2012), Lawrence et al. (2006)