[functions.py]: Function fit_kww: Improve Initial Guess

Improve the initial guess of tau in `mdtools.functions.fit_kww`.

src/mdtools/functions.py

@@ -379,8 +379,9 @@ def fit_kww(xdata, ydata, ysd=None, return_valid=False, **kwargs):
         Additional keyword arguments (besides `xdata`, `ydata` and
         `sigma`) to parse to :func:`scipy.optimize.curve_fit`.  See
         there for possible options.  By default, `absolute_sigma` is set
-        to ``True``, `p0` is set to ``[len(ydata) * np.mean(ydata), 1]``
-        and `bounds` is set to ``([0, 0], [np.inf, 1])``.
+        to ``True``, `p0` is set to ``[tau_init, 1]`` and `bounds` is
+        set to ``([0, 0], [np.inf, 1])``.  ``tau_init`` is the point at
+        which `ydata` falls below :math:`1/e`.
 
     Returns
     -------
@@ -422,17 +423,32 @@ def fit_kww(xdata, ydata, ysd=None, return_valid=False, **kwargs):
             return np.full(2, np.nan), np.full(2, np.nan), valid
         else:
             return np.full(2, np.nan), np.full(2, np.nan)
+    xdata = xdata[valid]
+    ydata = ydata[valid]
     if ysd is not None:
         ysd = ysd[valid]
         ysd[ysd == 0] = CURVE_FIT_ZERO_SD
 
+    # Initial guess for tau.
+    # If beta = 1, f(t=tau) = 1/e  =>  tau = f^(-1)(1/e)
+    thresh = 1 / np.e
+    ix = np.argmin(ydata[valid] <= thresh)
+    if ydata[ix] > thresh:
+        # `ydata` never falls below `thresh`.
+        # Linearly extrapolate `ydata` to `thresh`.
+        tau_init = (1 + ydata[-1] - thresh) * xdata[-1]
+    else:
+        tau_init = ydata[ix]
+    # Ensure that `tau_init` is greater than zero.
+    tau_init = max(tau_init, CURVE_FIT_ZERO_SD)
+
     kwargs.setdefault("absolute_sigma", True)
-    kwargs.setdefault("p0", [len(ydata) * np.mean(ydata), 1])
+    kwargs.setdefault("p0", [tau_init, 1])
     kwargs.setdefault("bounds", ([0, 0], [np.inf, 1]))
 
     try:
         popt, pcov = optimize.curve_fit(
-            f=kww, xdata=xdata[valid], ydata=ydata[valid], sigma=ysd, **kwargs
+            f=kww, xdata=xdata, ydata=ydata, sigma=ysd, **kwargs
         )
     except (ValueError, RuntimeError) as err:
         print()