WEBVTT

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Choosing a section boundary. Six equal 100 m intervals: 2.2, 2.4, 2.3, 5.8, 6.2, 6.0 m/km.

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MATHEMATICS · define the relationship and the meaning of each symbol before calculating.

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yi: interval roughness in m/km; k: candidate split after an interval. All six intervals are 100 m. The objective measures within-section variation.

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INPUT · six interval measurements · read the axis units and the numerical values.

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Split at 100 m: within-section SSE = 16.072 (m/km)².

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Split at 200 m: within-section SSE = 10.368 (m/km)².

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Split at 300 m: within-section SSE = 0.100 (m/km)².

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Split at 400 m: within-section SSE = 9.227 (m/km)².

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Split at 500 m: within-section SSE = 16.528 (m/km)².

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Minimum at 300 m: means 2.30 and 6.00 m/km; SSE = 0.100 (m/km)².

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OUTPUT · error for each candidate boundary · read the axis units and the numerical values.

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The minimum is 0.100 at 300 m. Means are 2.30 and 6.00 m/km; engineering evidence must still justify the boundary.

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CALCULATION · substitute the defined inputs; keep units through each intermediate step.

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Left mean: (2.2 + 2.4 + 2.3) / 3 = 2.3 m/km

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Right mean: (5.8 + 6.2 + 6.0) / 3 = 6.0 m/km

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Left squared residuals: 0.01 + 0.01 + 0 = 0.02 (m/km)²

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Right squared residuals: 0.04 + 0.04 + 0 = 0.08 (m/km)²

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Total objective: 0.02 + 0.08 = 0.10 (m/km)²

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SENSITIVITY · change one declared input · read the axis units and the numerical values.

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Boundary 100 m: 16.072 (m/km)² SSE. Same six observations; change only the candidate split.

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Boundary 200 m: 10.3675 (m/km)² SSE. Same six observations; change only the candidate split.

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Boundary 300 m: 0.1 (m/km)² SSE. Same six observations; change only the candidate split.

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Boundary 400 m: 9.2275 (m/km)² SSE. Same six observations; change only the candidate split.

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Boundary 500 m: 16.528 (m/km)² SSE. Same six observations; change only the candidate split.

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Check: Recompute every candidate rather than selecting a visually attractive split. Retain the no-split error as a baseline and examine whether the chosen boundary coincides with located construction, drainage or distress evidence.

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ONE ROW · A neat vertical line on a chart is a hypothesis. Does the evidence actually prefer it?

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PARAMETER MAP · 6100 evaluated cells. The colour limits stay fixed across every row.

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Generate 101 equal intervals with rᵢ = 2.3 + Δ·1(i ≥ 50) + 0.2 sin(i), using zero-based i. At each candidate split, recompute both section means and sum squared residuals. As the

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prescribed step grows, a clearer trough develops near the actual transition. At small Δ the oscillation competes with the boundary signal.

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VALUE LANDSCAPE · Height and colour encode SSE ((m/km)²). This is a graph over the two input axes, not automatically a physical pavement shape.

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This optimises one split in one equally spaced synthetic signal. Minimum section lengths, multiple splits and corroborating material, drainage and construction evidence remain separate decisions.

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The lowest SSE is a statistical candidate, not a design boundary.