still messing with docstring, sorry

Author: pavelkomarov

pynumdiff/kalman_smooth/_kalman_smooth.py

@@ -260,7 +260,7 @@ def constant_jerk(x, dt, params=None, options=None, r=None, q=None, forwardbackw
 
 def robustdiff(x, dt, order, log_q, log_r, proc_huberM=6, meas_huberM=0):
     """Perform outlier-robust differentiation by solving the Maximum A Priori optimization problem:
-    :math:`\\min_{\\{x_n\\}} \\sum_{n=0}^{N-1} V(R^{-1/2}(y_n - C x_n)) + \\sum_{n=1}^{N-1} J(Q^{-1/2}(x_n - A x_{n-1}))`,
+    :math:`\\argmin_{\\{x_n\\}} \\sum_{n=0}^{N-1} V(R^{-1/2}(y_n - C x_n)) + \\sum_{n=1}^{N-1} J(Q^{-1/2}(x_n - A x_{n-1}))`,
     where :math:`A,Q,C,R` come from an assumed constant derivative model and :math:`V,J` are the :math:`\\ell_1` norm or Huber
     loss rather than the :math:`\\ell_2` norm optimized by RTS smoothing. This problem is convex, so this method calls
     :code:`convex_smooth`.
@@ -270,11 +270,12 @@ def robustdiff(x, dt, order, log_q, log_r, proc_huberM=6, meas_huberM=0):
     deviation. In other words, this choice affects which portion of inputs are treated as outliers. For example, assuming
     Gaussian inliers, the portion beyond :math:`M\\sigma` is :code:`outlier_portion = 2*(1 - scipy.stats.norm.cdf(M))`. The
     inverse of this is :code:`M = scipy.stats.norm.ppf(1 - outlier_portion/2)`. As :math:`M \\to \\infty`, Huber becomes the
-    1/2-sum-of-squares case, :math:`\\frac{1}{2}\\|\\cdot\\|_2^2`, and the normalization constant of the Huber loss (See
-    :math:`c_2` `in section 6 <https://jmlr.org/papers/volume14/aravkin13a/aravkin13a.pdf>`_, missing a :math:`\\sqrt{\\cdot}`
-    term there, see p2700) approaches 1 as :math:`M` increases. Similarly, as :code:`M` approaches 0, Huber reduces to the
-    :math:`\\ell_1` norm case, because the normalization constant approaches :math:`\\frac{\\sqrt{2}}{M}`, cancelling the
-    :math:`M` multiplying :math:`|\\cdot|` and leaving behind :math:`\\sqrt{2}`, the proper :math:`\\ell_1` normalization.
+    1/2-sum-of-squares case, :math:`\\frac{1}{2}\\|\\cdot\\|_2^2`, because the normalization constant of the Huber loss (See
+    :math:`c_2` in `section 6 of this paper <https://jmlr.org/papers/volume14/aravkin13a/aravkin13a.pdf>`_, missing a
+    :math:`\\sqrt{\\cdot}` term there, see p2700) approaches 1 as :math:`M` increases. Similarly, as :code:`M` approaches 0,
+    Huber reduces to the :math:`\\ell_1` norm case, because the normalization constant approaches :math:`\\frac{\\sqrt{2}}{M}`,
+    cancelling the :math:`M` multiplying :math:`|\\cdot|` in the Huber function, and leaving behind :math:`\\sqrt{2}`, the
+    proper :math:`\\ell_1` normalization.
 
     Note that :code:`log_q` and :code:`proc_huberM` are coupled, as are :code:`log_r` and :code:`meas_huberM`, via the relation
     :math:`\\text{Huber}(q^{-1/2}v, M) = q^{-1}\\text{Huber}(v, Mq^{-1/2})`, but these are still independent enough that for
@@ -283,8 +284,8 @@ def robustdiff(x, dt, order, log_q, log_r, proc_huberM=6, meas_huberM=0):
     :param np.array[float] x: data series to differentiate
     :param float dt: step size
     :param int order: which derivative to stabilize in the constant-derivative model (1=velocity, 2=acceleration, 3=jerk)
-    :param float log_q: base 10 logarithm of the process noise variance, so :code:`q = 10**log_q`
-    :param float log_r: base 10 logarithm of the measurement noise variance, so :code:`r = 10**log_r`
+    :param float log_q: base 10 logarithm of process noise variance, so :code:`q = 10**log_q`
+    :param float log_r: base 10 logarithm of measurement noise variance, so :code:`r = 10**log_r`
     :param float proc_huberM: quadratic-to-linear transition point for process loss
     :param float meas_huberM: quadratic-to-linear transition point for measurement loss