{
  "method": "STDR",
  "slug": "stdr",
  "title": "Average normalized cycle shapes",
  "summary": "STDR uses a phase-wise average of normalized cycle shapes, then restores each cycle's dispersion and keeps a remainder.",
  "steps": [
    {
      "title": "Partition and center cycles",
      "text": "Split at the configured period, compute each block's arithmetic mean, and subtract it. DeTime assigns that mean to every sample of the block, producing a piecewise-constant trend. A final partial block uses the samples that exist."
    },
    {
      "title": "Separate dispersion and shape",
      "text": "Compute the Euclidean norm of each centered block as its dispersion. Divide by that norm to obtain a unit-length seasonal shape. A block whose norm is below eps receives zero dispersion and a zero shape."
    },
    {
      "title": "Average shapes across cycles",
      "text": "Average normalized shape entries at the same phase across cycles. For a partial cycle, increment counts only for available phases. Multiply this shared average by each cycle's own dispersion; the average shape is not normalized again."
    },
    {
      "title": "Compute the common result",
      "text": "Return trend, the scaled seasonal term, and residual = observed - trend - season, plus dispersion and seasonal-shape arrays. Deviations from the common cycle shape remain in the residual. In the cited formulation, R denotes a remainder; the code does not perform robust reweighting."
    }
  ],
  "parameters": [
    {
      "name": "period",
      "effect": "Block length and seasonal phase convention; a wrong period changes levels, scales, and shape alignment."
    },
    {
      "name": "eps",
      "effect": "Threshold below which a centered block is treated as having zero dispersion."
    },
    {
      "name": "max_period_search",
      "effect": "Upper bound on the FFT-based period heuristic used only when period is omitted; this animation supplies the period explicitly."
    }
  ],
  "cautions": [
    "DeTime's STD/STDR use block means and L2 norms. Dispersion is not the sample standard deviation.",
    "STDR uses ordinary phase-wise means of normalized shapes. No robust loss, trimming, or outlier weights are used in this implementation.",
    "A partial cycle has its own mean and norm and contributes only available phases; its scale is not directly comparable to a full cycle without accounting for length.",
    "STD retains within-cycle irregular variation in its shape. STDR assigns deviations from a shared shape to the remainder.",
    "The supplied period and the starting phase define block boundaries; shifting the start changes the decomposition."
  ],
  "references": [
    {
      "title": "Dudek (2022), STD: A Seasonal-Trend-Dispersion Decomposition of Time Series",
      "url": "https://arxiv.org/abs/2204.10398"
    }
  ],
  "trace_kind": "computed operator trace",
  "trace_note": "Every stage follows compute_std_components and the corresponding native kernel: block means, Euclidean norms, phase counts, and scaling. Both direct-source and public Python outputs are compared. The final short block is intentional.",
  "input_note": "Deterministic synthetic teaching signal; not a benchmark.",
  "example_config": {
    "method": "STDR",
    "backend": "python",
    "params": {
      "period": 24
    }
  },
  "checks": {
    "traced_block_levels_max_abs_error": 0.0,
    "traced_dispersion_max_abs_error": 0.0,
    "traced_shape_max_abs_error": 0.0,
    "traced_season_max_abs_error": 0.0,
    "source_trend_max_abs_error": 0.0,
    "source_season_max_abs_error": 0.0,
    "source_residual_max_abs_error": 0.0,
    "additive_reconstruction_max_abs_error": 4.440892098500626e-16,
    "phase_average_trace_max_abs_error": 0.0
  },
  "source_files": [
    "src/detime/methods/std.py",
    "native/src/std.cpp"
  ],
  "frames": 32,
  "duration_ms": 12050,
  "poster_frame": 15,
  "width": 1280,
  "height": 820,
  "artifacts": {
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      "file": "stdr.gif",
      "bytes": 590036,
      "sha256": "7e4fc040e546acf0c56a10d193de27d60f51cb298d201e03c7d07c79bcbdfc59"
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    "poster": {
      "file": "stdr.png",
      "bytes": 104021,
      "sha256": "9b00955566a630496fcf5bd390d754b1fb0e44ac15285ac877b33e40c0487772"
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    "data": {
      "file": "stdr.npz",
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  },
  "source_arrays": {
    "t": [
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    ],
    "observed": [
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    ],
    "block_means": [
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    ],
    "block_dispersion": [
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    ],
    "normalized_shapes_zero_padded": [
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    ],
    "phase_available_mask": [
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    ],
    "running_phase_counts": [
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    ],
    "running_average_shapes": [
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    ],
    "sample_dispersion": [
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    ],
    "returned_seasonal_shape": [
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    ],
    "trend": [
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    "season": [
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    ],
    "residual": [
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    ]
  },
  "render_provenance": {
    "python_version": "3.12.14",
    "dependencies": {
      "numpy": "2.3.5",
      "scipy": "1.17.0",
      "statsmodels": "0.15.0",
      "matplotlib": "3.10.8",
      "Pillow": "12.3.0",
      "PyWavelets": "1.10.0",
      "EMD-signal": "1.10.0",
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