Compute ceneter time of particle transit in laser beam (tao) using max likelihood

pysp2/util/__init__.py

@@ -18,4 +18,4 @@
 from .particle_properties import calc_diams_masses, process_psds
 from .deadtime import deadtime
 from .leo_fit import beam_shape,leo_fit
-from .normalized_derivative_method import central_difference
+from .normalized_derivative_method import central_difference, plot_normalized_derivative, mle_tau_moteki_kondo

pysp2/util/normalized_derivative_method.py

@@ -1,6 +1,12 @@
+from __future__ import annotations
+
 import numpy as np
 import xarray as xr
 
+from dataclasses import dataclass
+from typing import Optional, Union
+
+
 def central_difference(S, num_records=None, normalize=True, baseline_to_zero=True):
 
     """ 
@@ -23,6 +29,8 @@ def central_difference(S, num_records=None, normalize=True, baseline_to_zero=Tru
     normalize: bool
         If True, normalize the derivative by the scattering signal 
         S(t) to get (1/S) * dS/dt.
+    baseline_to_zero: bool
+        If True, shift each record's minimum to zero before differentiation.
 
     Returns 
     ------- 
@@ -110,15 +118,330 @@ def plot_normalized_derivative(ds, record_no, chn=0):
         spectra['Data_ch' + str(chn)].values[np.newaxis, :],
         dims=['time', 'bins'])
     inp_data = xr.Dataset(inp_data)
-    bins = np.linspace(0, 100, 100)
+
+    bins = np.arange(0, 0.00004-0.3e-6, 0.4e-6)  # 0 to 0.0004 microseconds in steps of 0.4e-6 seconds
+    bins = bins*1e6  # convert to microseconds for plotting
 
     ch_name = f'Data_ch{chn}'
     plt.figure(figsize=(10, 6))
     ax = plt.gca()
-    inp_data[ch_name].plot(ax=ax)
+    # Plot using bins for x-axis
+    ax.plot(bins, spectra['Data_ch' + str(chn)].values, label=ch_name)
+    ax.set_xlim([bins[0], bins[-1]])
     ax.set_title(f'Normalized Derivative of Scattering Signal - Channel {chn} Record {record_no}')
-    ax.set_xlabel('Time (s)')
+    ax.set_xlabel('Time ($\mu$s)')
     ax.set_ylabel('Normalized Derivative')
     plt.grid()
+    ax.legend()
 
     return ax
+
+
+@dataclass(frozen=True)
+class MLEConfig:
+    """
+    Configuration parameters for Maximum Likelihood Estimation (MLE)
+    of the Moteki & Kondo normalized derivative method.
+
+    Parameters
+    ----------
+    h : float
+        Time resolution of the scattering signal (instrument specification).
+    sigma_bar : float
+        Mean value of the noise standard deviation (measured value).
+    delta_sigma : float
+        Increment for the noise standard deviation (measured value).
+    A1 : float
+        Coefficient for the first term in the model (determined experimentally).
+    A2 : float
+        Coefficient for the second term in the model (determined experimentally).
+    A3 : float
+        Coefficient for the third term in the model (determined experimentally).
+    grid_size : int, default=401
+        Number of grid points for tau estimation.
+    grid_margin : float, default=0.5
+        Margin for the tau grid as a fraction of the range.
+    """
+    h: float
+    sigma_bar: float
+    delta_sigma: float
+    A1: float
+    A2: float
+    A3: float
+    grid_size: int = 401
+    grid_margin: float = 0.5
+
+def mle_tau_moteki_kondo(
+    S: Union[xr.DataArray, xr.Dataset],
+    norm_deriv: Union[xr.DataArray, xr.Dataset],
+    p: int,
+    *,
+    ch: Optional[str] = None,
+    event_index: int,
+    event_dim: str = "event_index",
+    S_sample_dim: Optional[str] = None,
+    y_sample_dim: Optional[str] = None,
+    tau_grid: Optional[Union[np.ndarray, xr.DataArray]] = None,
+    k_end: Optional[int] = None,
+    config: Optional[MLEConfig] = None,
+) -> xr.DataArray:
+    """
+    Estimate tau_hat using the Moteki & Kondo grid-search MLE.
+
+    Parameters
+    ----------
+    S : xr.DataArray or xr.Dataset
+        Scattering signal.
+    norm_deriv : xr.DataArray or xr.Dataset
+        Normalized derivative.
+    p : int
+        Number of consecutive points in each k-subset.
+    ch : str, optional
+        Variable to select when S and/or norm_deriv are Datasets.
+        Required if a Dataset contains multiple variables and no unique choice exists.
+    event_index : int
+        Event index to select. This function returns tau_hat(k) for one event only.
+    event_dim : str
+        Name of event dimension.
+    S_sample_dim : str, optional
+        Sample dimension in S.
+    y_sample_dim : str, optional
+        Sample dimension in norm_deriv.
+    tau_grid : 1D array-like, optional
+        Global tau grid for all subsets.
+    k_end : int, optional
+        Largest starting k.
+    config : MLEConfig
+        Calibration / noise / grid settings.
+    """
+    if config is None:
+        raise ValueError("config must be provided.")
+
+    def _to_dataarray(obj: Union[xr.DataArray, xr.Dataset], name: str) -> xr.DataArray:
+        """
+        Accept either a DataArray or Dataset.
+        If a Dataset is provided, select the variable named `ch`.
+        """
+        if isinstance(obj, xr.DataArray):
+            return obj
+        if isinstance(obj, xr.Dataset):
+            if ch is not None:
+                # Use the user input channel.
+                if ch not in obj.data_vars:
+                    raise ValueError(
+                        f"{ch!r} not found in {name}.data_vars={list(obj.data_vars)}"
+                    )
+                return obj[ch]
+            if len(obj.data_vars) == 1:
+                only_var = next(iter(obj.data_vars))
+                return obj[only_var]
+            raise ValueError(
+                f"{name} is a Dataset with multiple variables. "
+                f"Provide ch. Available: {list(obj.data_vars)}"
+            )
+        raise TypeError(f"{name} must be an xarray DataArray or Dataset.")
+
+    # Convert datasets to the selected DataArrays.
+    S = _to_dataarray(S, "S")
+    norm_deriv = _to_dataarray(norm_deriv, "norm_deriv")
+
+    # The method requires one event axis and one sample axis.
+    if event_dim not in S.dims:
+        raise ValueError(f"{event_dim!r} not found in S.dims={S.dims}")
+    if event_dim not in norm_deriv.dims:
+        raise ValueError(f"{event_dim!r} not found in norm_deriv.dims={norm_deriv.dims}")
+
+    # Infer the sample dimension if the user did not specify it.
+    if S_sample_dim is None:
+        s_non_event_dims = [d for d in S.dims if d != event_dim]
+        if len(s_non_event_dims) != 1:
+            raise ValueError(
+                f"Could not infer S sample dim. Non-event dims in S: {s_non_event_dims}"
+            )
+        S_sample_dim = s_non_event_dims[0]
+
+    if y_sample_dim is None:
+        y_non_event_dims = [d for d in norm_deriv.dims if d != event_dim]
+        if len(y_non_event_dims) != 1:
+            raise ValueError(
+                f"Could not infer norm_deriv sample dim. Non-event dims in norm_deriv: {y_non_event_dims}"
+            )
+        y_sample_dim = y_non_event_dims[0]
+
+    # Rename the sample dimensions to a common internal name.
+    S_std = S.rename({S_sample_dim: "sample"})
+    y_std = norm_deriv.rename({y_sample_dim: "sample"})
+
+    # Align the arrays so the same event/sample positions are used in both inputs.
+    S_std, y_std = xr.align(S_std, y_std, join="inner")
+
+    if event_index < 0 or event_index >= S_std.sizes[event_dim]:
+        raise ValueError(
+            f"event_index must be in [0, {S_std.sizes[event_dim] - 1}], got {event_index}"
+        )
+
+    n_samples = S_std.sizes["sample"]
+
+    if p < 2 or p > n_samples:
+        raise ValueError(f"p must be in [2, {n_samples}], got {p}")
+
+    if k_end is None:
+        k_end = n_samples - p
+    if k_end < 0 or k_end > n_samples - p:
+        raise ValueError(f"k_end must be in [0, {n_samples - p}], got {k_end}")
+
+    # Optional tau grid for the 1D grid search in tau.
+    # Moteki & Kondo determine tau numerically by maximizing L_k(tau).
+    if tau_grid is not None:
+        tau_grid_np = np.asarray(
+            tau_grid.data if isinstance(tau_grid, xr.DataArray) else tau_grid,
+            dtype=float,
+        )
+        if tau_grid_np.ndim != 1:
+            raise ValueError("tau_grid must be 1D.")
+    else:
+        tau_grid_np = None
+
+    # Parameters from Appendix A.
+    h = float(config.h)
+    sigma_bar = float(config.sigma_bar)
+    delta_sigma = float(config.delta_sigma)
+    A1, A2, A3 = float(config.A1), float(config.A2), float(config.A3)
+
+    # Time axis used in the fit.
+    # Here we use physical time spacing h so tk is in seconds (or whatever unit h uses).
+    # This must match sigma_bar and delta_sigma units.
+    t = np.arange(n_samples) * h
+
+    if h <= 0:
+        raise ValueError("config.h must be positive.")
+    if sigma_bar <= 0:
+        raise ValueError("config.sigma_bar must be positive.")
+    if delta_sigma < 0:
+        raise ValueError("config.delta_sigma must be >= 0.")
+
+    # Eq. (A.7): finite-difference amplification factor for the derivative noise.
+    Af_d = np.sqrt(130.0) / 12.0
+
+    def _logL_for_tau(yk: np.ndarray, sk: np.ndarray, tk: np.ndarray, tau: float) -> float:
+        """
+        Log-likelihood for one k-subset and one candidate tau.
+
+        Mean model:
+            ybar_i(tau) = -(t_i - tau) / sigma_bar^2      [Eq. (A.4)]
+        where y_i = S'_i / S_i.
+
+        Covariance:
+            Cov[y_i, y_j] = 4 / sigma_bar^6 * (t_i - tau)(t_j - tau) * (delta_sigma)^2   [Eq. (A.10a)]
+            Var[y_i]      = 4 / sigma_bar^6 * (t_i - tau)^2 * (delta_sigma)^2
+                            + (Af_d^2 / h^2) * (1/S_i^2) * (delta S_i)^2                  [Eq. (A.10b)]
+        with
+            delta S_i = sqrt(A1^2 + A2^2 S_i + A3^2 S_i^2)                               [Eq. (A.6)]
+        and
+            (delta y_i)_ran = Af_d * (1/h) * (1/S_i) * delta S_i                          [Eq. (A.7)]
+
+        The full likelihood is the multivariate Gaussian in Eq. (A.9).
+        """
+        # Mean vector of the normalized derivative under the Gaussian beam model.
+        # This is the line I'/I = -(t - tau)/sigma^2 [Eq. (5)] used as the mean [Eq. (A.4)].
+        ybar = -(tk - tau) / (sigma_bar * sigma_bar)
+
+        # Signal-noise amplitude from Appendix A [Eq. (A.6)].
+        deltaS = np.sqrt(A1 * A1 + (A2 * A2) * sk + (A3 * A3) * (sk * sk))
+
+        # Random variance of y = S'/S from finite-difference error propagation [Eq. (A.7)].
+        with np.errstate(divide="ignore", invalid="ignore"):
+            var_rand_k = (Af_d * Af_d) / (h * h) * (deltaS * deltaS) / (sk * sk)
+
+        # If any term is non-finite, this tau candidate is unusable.
+        if not np.all(np.isfinite(var_rand_k)):
+            return -np.inf
+        if np.any(var_rand_k <= 0):
+            return -np.inf
+
+        # Systematic covariance from particle-by-particle fluctuations in sigma [Eq. (A.10a)].
+        dt = (tk - tau).reshape(-1, 1)
+        sys_pref = 4.0 * (delta_sigma * delta_sigma) / (sigma_bar ** 6)
+        Sigma = sys_pref * (dt @ dt.T)
+        # Add the diagonal random variance term [Eq. (A.10b)].
+        Sigma[np.diag_indices_from(Sigma)] += var_rand_k
+
+        # Residual vector y - ybar.
+        r = yk - ybar
+
+        # Use Cholesky factorization for numerical stability when evaluating Eq. (A.9).
+        try:
+            L = np.linalg.cholesky(Sigma)
+        except np.linalg.LinAlgError:
+            return -np.inf
+
+        # Compute statistical distance.
+        # d^2 = (y - ybar)^T Sigma^{-1} (y - ybar) [Eq. (A.11)]
+        z = np.linalg.solve(L, r)
+        d2 = float(z.T @ z)
+        # log |Sigma| from the Cholesky factor.
+        logdet = 2.0 * np.sum(np.log(np.diag(L)))
+
+        # Multivariate normal log-likelihood [Eq. (A.9)].
+        p_local = yk.size
+        return float(-0.5 * (p_local * np.log(2.0 * np.pi) + logdet + d2))
+
+    def _tau_hat_for_one_event(s_event: np.ndarray, y_event: np.ndarray) -> np.ndarray:
+        """
+        For one event, scan all k-subsets of length p and return tau_hat(k).
+        """
+        tau_hat = np.full(k_end + 1, np.nan, dtype=float)
+
+        # Skip events with missing values.
+        if not (np.all(np.isfinite(s_event)) and np.all(np.isfinite(y_event))):
+            return tau_hat
+
+        for k in range(k_end + 1):
+            # Consecutive p-point subset starting at k.
+            # This is the subset over which Moteki & Kondo search for the leading-edge
+            # segment that best matches I'/I [Appendix A.5].
+            yk = y_event[k : k + p]
+            sk = s_event[k : k + p]
+            tk = t[k : k + p]
+
+            if not (np.all(np.isfinite(yk)) and np.all(np.isfinite(sk))):
+                continue
+
+            # If the user did not supply a global tau grid, build a local grid for this k.
+            if tau_grid_np is None:
+                span = float(tk[-1] - tk[0])
+                margin = config.grid_margin * (span + h)
+                grid = np.linspace(tk[0] - margin, tk[-1] + margin, config.grid_size)
+            else:
+                grid = tau_grid_np
+
+            # Grid-search maximization of L_k(tau) [Appendix A.5].
+            best_ll = -np.inf
+            best_tau = np.nan
+            for tau_cand in grid:
+                ll = _logL_for_tau(yk, sk, tk, float(tau_cand))
+                if ll > best_ll:
+                    best_ll = ll
+                    best_tau = float(tau_cand)
+
+            if np.isfinite(best_ll):
+                tau_hat[k] = best_tau
+
+        return tau_hat
+
+    # Select the requested event only, and return tau_hat(k) for that one event.
+    s_event = np.asarray(S_std.sel({event_dim: event_index}).values, dtype=float)
+    y_event = np.asarray(y_std.sel({event_dim: event_index}).values, dtype=float)
+
+    tau_hat_1d = _tau_hat_for_one_event(s_event, y_event)
+
+    return xr.DataArray(
+        tau_hat_1d,
+        dims=("k",),
+        coords={"k": np.arange(k_end + 1)},
+        name="tau_hat",
+        attrs={
+            "long_name": f"MLE tau_hat(k) for {event_dim}={event_index}",
+            "units": "sample_index_or_time_units_of_t",
+        },
+    )