{
  "method": "VMD",
  "slug": "vmd",
  "title": "VMD: update narrow-band modes and their center frequencies",
  "summary": "VMD alternates between frequency-domain mode updates and spectral-energy centers. DeTime's grouping into trend and season is a separate step after this optimization.",
  "steps": [
    {
      "title": "Prepare the spectrum",
      "text": "Mirror the signal, compute its FFT, and retain the positive-frequency representation."
    },
    {
      "title": "Update a mode",
      "text": "Fit the remaining spectrum with a bandwidth penalty around the current center frequency."
    },
    {
      "title": "Update its center",
      "text": "Use the mode's positive-frequency energy to update its center, then repeat across modes."
    },
    {
      "title": "Reconstruct and group",
      "text": "Complete the conjugate spectrum, invert the FFT, crop the mirrored margins, and group modes."
    }
  ],
  "parameters": [
    {
      "name": "K",
      "effect": "Number of modes; the demonstration uses 3."
    },
    {
      "name": "alpha",
      "effect": "Penalizes spectral spread around each center; the demonstration uses 1500."
    },
    {
      "name": "tau",
      "effect": "Dual-update step; zero permits noise slack and does not enforce an exact sum of modes."
    },
    {
      "name": "init / tol",
      "effect": "Initialization and convergence tolerance affect the computed solution."
    }
  ],
  "cautions": [
    "The animation captures the optional vmdpy path; the native path has its own implementation and iteration control.",
    "Center frequencies shown here are raw solver frequencies in cycles/sample, not DeTime's converted metadata field.",
    "VMD modes need a grouping rule before being named trend or season; frequency ordering is not a physical-source guarantee."
  ],
  "references": [
    {
      "title": "Original VMD paper",
      "url": "https://doi.org/10.1109/TSP.2013.2288675"
    },
    {
      "title": "vmdpy reference implementation",
      "url": "https://github.com/vrcarva/vmdpy"
    },
    {
      "title": "Maintained sktime continuation",
      "url": "https://www.sktime.net/docs/api-reference/sktimetransformationsvmdvmdtransformer/"
    }
  ],
  "trace_kind": "upstream execution trace",
  "trace_note": "Captured vmdpy 0.2 loop arrays without changing solver updates; intermediate time-domain modes use its conjugate-completion and mirror-cropping convention.",
  "input_note": "Deterministic synthetic teaching signal; not a benchmark.",
  "example_config": {
    "method": "VMD",
    "params": {
      "K": 3,
      "alpha": 1500.0,
      "tau": 0.0,
      "init": 1,
      "tol": 1e-07,
      "primary_period": 24
    },
    "backend": "python"
  },
  "checks": {
    "recorded_final_iterate": 24,
    "trace_matches_upstream_modes": 0.0,
    "mode_sum_rmse": 0.05055252233370291
  },
  "source_files": [
    "src/detime/methods/vmd.py",
    "native/src/vmd.cpp"
  ],
  "frames": 32,
  "duration_ms": 12050,
  "poster_frame": 15,
  "width": 1280,
  "height": 820,
  "artifacts": {
    "gif": {
      "file": "vmd.gif",
      "bytes": 399253,
      "sha256": "847bfe96e5ebeb3695fe6d45a3af338f02b549724f28ab777f291b43346fc8cb"
    },
    "poster": {
      "file": "vmd.png",
      "bytes": 179243,
      "sha256": "1d69db3c4af2c1486d1a940875ab9b45b1636250df729557cfc10bb80f6c9237"
    },
    "data": {
      "file": "vmd.npz",
      "bytes": 117559,
      "sha256": "915fb7429247169e4055010b741c8ff459bc31b1388010075d8430ed0a899d0e"
    }
  },
  "source_arrays": {
    "time": [
      192
    ],
    "input": [
      192
    ],
    "mirrored_input": [
      384
    ],
    "frequency_grid": [
      384
    ],
    "iteration_ids": [
      8
    ],
    "mode_spectra": [
      8,
      384,
      3
    ],
    "center_frequency_trace": [
      25,
      3
    ],
    "mode_trace": [
      8,
      3,
      192
    ],
    "returned_modes": [
      3,
      192
    ]
  },
  "render_provenance": {
    "python_version": "3.11.3",
    "dependencies": {
      "numpy": "2.4.6",
      "scipy": "1.17.1",
      "statsmodels": "0.15.0",
      "matplotlib": "3.11.2",
      "Pillow": "13.0.0",
      "PyWavelets": "1.9.0",
      "EMD-signal": "1.10.0",
      "vmdpy": "0.2",
      "PySDKit": "0.5.0"
    },
    "source_hashes": {
      "native/src/vmd.cpp": "27b730744b4c17b2b0cfa4d2c4c0d29798d335294a71fb6a7a3e077e3012041b",
      "scripts/generate_method_animations.py": "f62989af55a7e48495c24afcfee31751d3785e5f78bb4b900bef23a429deaa84",
      "scripts/method_animations/adaptive.py": "7d728d88b4ba1a8dd82890c7a6b017d542728b50dc477c281e9bbfcbae164a1d",
      "scripts/method_animations/common.py": "73ad7e8b4e5b67f87d13e59adab2c93e983fe1aaf06ee627d22cb7c7e696016a",
      "src/detime/backends.py": "7432dcf59e382580dfb32f1f57e61ff71a30567c049b35243e24e3644523eda6",
      "src/detime/core.py": "d95b348cd8b16b56d18840965579c947a11628d969dd6c32c4af7723f5a027f4",
      "src/detime/methods/vmd.py": "249bdc7e67595c7f556d32dd70c64e6f8ee7bd7a0e6c3b0277a8010fc1f0cf67",
      "src/detime/registry.py": "e934ffca17880cefe5add728aa497e63ea811d7ae43abec1ea498de718094066"
    }
  }
}