## Diagram: Partition of Input Space ### Overview This image displays a two-dimensional Cartesian plot titled "Partition of input space". It shows a rectangular region divided into multiple smaller, non-uniform rectangular cells by a series of blue horizontal and vertical lines. The axes are labeled with mathematical expressions defining the range of two variables, `x₁` and `x₂`. ### Components/Axes **Title:** * "Partition of input space" (located at the top-center of the plot area). **X-axis (Horizontal Axis):** * **Title:** `π/3 < x₁ < 2π/3` (located below the x-axis, centered). * Numerically, `π/3` is approximately 1.047, and `2π/3` is approximately 2.094. * **Range:** The visible plot area for `x₁` spans from approximately 1.05 to 2.05. * **Tick Markers:** 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0. **Y-axis (Vertical Axis):** * **Title:** `π/3 < x₂ < 2π/3` (located to the left of the y-axis, rotated vertically). * Numerically, `π/3` is approximately 1.047, and `2π/3` is approximately 2.094. * **Range:** The visible plot area for `x₂` spans from approximately 1.05 to 2.05. * **Tick Markers:** 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0. **Legend:** * No legend is present in the image. **Plot Area:** * The background of the plot area is white. * The partitioning lines are solid blue. ### Detailed Analysis The plot area represents a 2D input space defined by `x₁` and `x₂`, bounded by the approximate ranges [1.047, 2.094] for both variables. This space is divided into a grid of rectangular cells by a series of blue lines. The partitioning is not uniform. **Horizontal Partitioning Lines (y-coordinates):** The space is primarily divided horizontally by lines at the following y-coordinates, which span the entire width of the plot (from x ≈ 1.047 to x ≈ 2.094): * y = 1.30 * y ≈ 1.57 (between 1.5 and 1.6) * y ≈ 1.83 (between 1.8 and 1.9) **Vertical Partitioning Lines (x-coordinates):** The space is divided vertically by lines at the following x-coordinates: * x ≈ 1.18 (between 1.1 and 1.2): This line only extends from y ≈ 1.047 (bottom boundary) to y = 1.30. * x = 1.30: This line extends across the full height of the plot (from y ≈ 1.047 to y ≈ 2.094). * x ≈ 1.57 (between 1.5 and 1.6): This line extends across the full height of the plot. * x ≈ 1.83 (between 1.8 and 1.9): This line extends across the full height of the plot. * x ≈ 1.95 (between 1.9 and 2.0): This line only extends from y ≈ 1.047 (bottom boundary) to y = 1.30. **Resulting Rectangular Cells:** The combination of these lines creates 18 distinct rectangular regions (cells) within the input space: * **Bottom Row (y from ~1.047 to 1.30):** This row is divided into 6 cells by vertical lines at x ≈ 1.18, x = 1.30, x ≈ 1.57, x ≈ 1.83, and x ≈ 1.95. * Cell 1: x ∈ [~1.047, 1.18], y ∈ [~1.047, 1.30] * Cell 2: x ∈ [1.18, 1.30], y ∈ [~1.047, 1.30] * Cell 3: x ∈ [1.30, 1.57], y ∈ [~1.047, 1.30] * Cell 4: x ∈ [1.57, 1.83], y ∈ [~1.047, 1.30] * Cell 5: x ∈ [1.83, 1.95], y ∈ [~1.047, 1.30] * Cell 6: x ∈ [1.95, ~2.094], y ∈ [~1.047, 1.30] * **Middle Two Rows (y from 1.30 to 1.57, and y from 1.57 to 1.83):** Each of these rows is divided into 4 cells by vertical lines at x = 1.30, x ≈ 1.57, and x ≈ 1.83. * For y ∈ [1.30, 1.57]: * Cell 7: x ∈ [~1.047, 1.30], y ∈ [1.30, 1.57] * Cell 8: x ∈ [1.30, 1.57], y ∈ [1.30, 1.57] * Cell 9: x ∈ [1.57, 1.83], y ∈ [1.30, 1.57] * Cell 10: x ∈ [1.83, ~2.094], y ∈ [1.30, 1.57] * For y ∈ [1.57, 1.83]: * Cell 11: x ∈ [~1.047, 1.30], y ∈ [1.57, 1.83] * Cell 12: x ∈ [1.30, 1.57], y ∈ [1.57, 1.83] * Cell 13: x ∈ [1.57, 1.83], y ∈ [1.57, 1.83] * Cell 14: x ∈ [1.83, ~2.094], y ∈ [1.57, 1.83] * **Top Row (y from 1.83 to ~2.094):** This row is also divided into 4 cells by vertical lines at x = 1.30, x ≈ 1.57, and x ≈ 1.83. * Cell 15: x ∈ [~1.047, 1.30], y ∈ [1.83, ~2.094] * Cell 16: x ∈ [1.30, 1.57], y ∈ [1.83, ~2.094] * Cell 17: x ∈ [1.57, 1.83], y ∈ [1.83, ~2.094] * Cell 18: x ∈ [1.83, ~2.094], y ∈ [1.83, ~2.094] ### Key Observations * The input space is a square region defined by `x₁` and `x₂` both ranging from approximately 1.047 to 2.094. * The partitioning is non-uniform, meaning the resulting rectangular cells have varying widths and heights. * Some vertical partition lines (at x ≈ 1.18 and x ≈ 1.95) do not span the entire height of the plot, indicating a hierarchical or localized refinement of the partition, specifically in the bottom-most region (y from ~1.047 to 1.30). * The major horizontal and vertical partition lines (at x=1.30, x≈1.57, x≈1.83 and y=1.30, y≈1.57, y≈1.83) divide the space into a coarser 4x4 grid, with the bottom-left and bottom-right cells of this coarser grid further subdivided. * The axis labels `π/3` and `2π/3` suggest that the input space might represent angular or phase information, or parameters derived from such quantities. ### Interpretation This diagram illustrates a "partition of input space," which is a common technique in various fields such as numerical analysis, machine learning, control systems, and optimization. The non-uniformity of the cells suggests an adaptive partitioning strategy. Instead of dividing the space into equally sized bins, certain regions (specifically the bottom strip of the `x₂` range, from approximately 1.047 to 1.30) have been further subdivided along the `x₁` axis. This could imply: 1. **Higher Resolution/Sensitivity:** The system or model being analyzed requires finer granularity or more detailed exploration in the region where `x₂` is between ~1.047 and 1.30. This might be an area where the system's behavior is more complex, sensitive to changes in `x₁`, or where critical operating points are located. 2. **Adaptive Sampling/Meshing:** The partition might be the result of an adaptive algorithm that refines the grid based on some criteria, such as error estimates, density of data points, or gradients of a function being optimized. 3. **Feature Importance:** The variables `x₁` and `x₂` are likely input features or parameters. The partitioning indicates that their combined influence on an output or system state is not uniform across the entire domain. The use of `π/3` and `2π/3` in the axis labels strongly hints at applications involving periodic functions, angles, or phase spaces, where these specific values often represent significant points (e.g., 60 degrees and 120 degrees). This type of partitioning could be used for tasks like function approximation, state-space discretization for reinforcement learning, or defining regions for different control strategies. The diagram effectively communicates a structured, yet adaptively refined, decomposition of a 2D parameter domain.