From 7239b6f343e8c784f00b4c159761b15c12baa3af Mon Sep 17 00:00:00 2001
From: JuleAdrian <j.adrian@fz-juelich.de>
Date: Wed, 4 Nov 2026 11:20:51 +0100
Subject: [PATCH 1/7] solve errors in acceleration computation and adjust user
 guide

---
 notebooks/user_guide.ipynb               | 42 ++++++++++-------
 pedpy/methods/acceleration_calculator.py | 57 +++++++++++++++---------
 pedpy/methods/method_utils.py            | 18 ++++----
 3 files changed, 70 insertions(+), 47 deletions(-)

diff --git a/notebooks/user_guide.ipynb b/notebooks/user_guide.ipynb
index c2d77b5d..75453f7b 100644
--- a/notebooks/user_guide.ipynb
+++ b/notebooks/user_guide.ipynb
@@ -2683,37 +2683,45 @@
     "a specific frame step ($n$) is needed.\n",
     "Together with the {class}`trajectory data<trajectory_data.TrajectoryData>` $fps$ of\n",
     "the trajectory data $fps$ the time frame $\\Delta t$ for\n",
-    "computing the speed becomes:\n",
+    "computing the speed and the acceleration becomes:\n",
     "\n",
     "$$\n",
     "\\Delta t = 2 n / fps\n",
     "$$\n",
     "\n",
-    "This time step describes how many frames before and after the current\n",
-    "position $X(t_k)$ are used to compute the movement.\n",
-    "These positions are called $X(t_{k+n})$, $X(t_{k-n})$\n",
-    "respectively.\n",
+    "The acceleration at time $t_k$ can be calculated from the speeds\n",
+    "$v(t_{k+n})$ and $v(t_{k-n})$:\n",
     "\n",
-    "In order to compute the acceleration at time $t_k$, we first calculate the\n",
-    "displacements $\\bar{X}$ around $t_{k+n}$ and $t_{k-n}$:\n",
-    "\n",
-    "$$\n",
-    "\\bar{X}(t_{k+n}) = X(t_{k+2n}) - X(t_{k})\n",
     "$$\n",
+    "v(t_{k+n}) = \bar{X}(t_{k+n}) / \Delta t\n",
+    "$$\n",    
     "$$\n",
-    "\\bar{X}(t_{k-n}) = X(t_{k}) - X(t_{k-2n})\n",
+    "v(t_{k-n}) = \bar{X}(t_{k-n}) / \Delta t\n",
     "$$\n",
-    "\n",
-    "The acceleration is then calculated from the difference of the displacements\n",
+    "using the displacements $\bar{X}(t_{k+n})$ and $\bar{X}(t_{k-n})$\n",
+    "during $\Delta t$ around $t_{k+n}$ and $t_{k-n}$:\n",
     "\n",
     "$$\n",
-    "\\Delta\\bar{X}(t_k) = \\bar{X}(t_{k+n}) - \\bar{X}(t_{k-n})\n",
+    "\bar{X}(t_{k+n}) = X(t_{k+2n}) - X(t_{k})\n",
+    "$$\n",    
+    "$$\n",    
+    "\bar{X}(t_{k-n}) = X(t_{k}) - X(t_{k-2n})\n",
     "$$\n",
+    "The acceleration $a_i(t_k)$ is then calculated from the difference of\n",
+    "the speeds divided by the time interval $\Delta t$:\n",
     "\n",
-    "divided by the square of the time interval $\\Delta t$:\n",
-    "\n",
     "$$\n",
-    "a_i(t_k) = \\Delta\\bar{X}(t_k) / \\Delta t^{2}\n",
+    "a_i(t_k) = (v(t_{k+n}) - v(t_{k-n})) / \Delta t\n",
+    "which is simplified to\n",
+    "$$\n",
+    "a_i(t_k) = \Delta\bar{X}(t_k) / (\Delta t^{2})\n",
+    "$$\n",
+    "using\n",
+    "$$\n",
+    "\Delta\bar{X}(t_k) = \bar{X}(t_{k+n}) - \bar{X}(t_{k-n})\n",
+    "$$\n",
+    "$$\n",
+    "\Delta\bar{X}(t_k) = X(t_{k+2n}) - 2*X(t_{k}) + X(t_{k-2n})\n",
     "$$\n",
     "\n",
     "When getting closer to the start, or end of the trajectory data, it is not\n",
diff --git a/pedpy/methods/acceleration_calculator.py b/pedpy/methods/acceleration_calculator.py
index fee9ed47..cc802c90 100644
--- a/pedpy/methods/acceleration_calculator.py
+++ b/pedpy/methods/acceleration_calculator.py
@@ -30,23 +30,27 @@ def compute_individual_acceleration(
 
     For computing the individuals' acceleration at a specific frame
     :math:`a_i(t_k)`, a specific frame step (:math:`n`) is needed.
-    Together with the
-    :attr:`~trajectory_data.TrajectoryData.frame_rate` of the trajectory data
-    :math:`fps` the time frame :math:`\Delta t` for computing the speed
-    becomes:
+    Together with the :attr:`~trajectory_data.TrajectoryData.frame_rate` 
+    of the trajectory data :math:`fps` the time frame :math:`\Delta t` 
+    for computing the speed and the acceleration becomes:
 
     .. math::
 
         \Delta t = 2 n / fps
 
-    This time step describes how many frames before and after the current
-    position :math:`X(t_k)` are used to compute the movement.
-    These positions are called :math:`X(t_{k+n})`, :math:`X(t_{k-n})`
-    respectively.
+    The acceleration at time :math:`t_k` can be calculated from the speeds 
+    :math:`v(t_{k+n})` and :math:`v(t_{k-n})`:
 
-    In order to compute the acceleration at time :math:`t_k`, we first
-    calculate the displacements :math:`\bar{X}` around :math:`t_{k+n}` and
-    :math:`t_{k-n}`:
+    .. math::
+
+        v(t_{k+n}) = \bar{X}(t_{k+n}) / \Delta t
+
+    .. math::
+
+        v(t_{k-n}) = \bar{X}(t_{k-n}) / \Delta t
+
+    using the displacements :math:`\bar{X}(t_{k+n})` and :math:`\bar{X}(t_{k-n})`
+    during :math:`\Delta t` around :math:`t_{k+n}` and :math:`t_{k-n}`:
 
     .. math::
 
@@ -56,18 +60,29 @@ def compute_individual_acceleration(
 
         \bar{X}(t_{k-n}) = X(t_{k}) - X(t_{k-2n})
 
-    The acceleration is then calculated from the difference of the
-    displacements
+    
+    The acceleration :math:`a_i(t_k)` is then calculated from the difference of 
+    the speeds divided by the time interval :math:`\Delta t`:
 
     .. math::
 
-        \Delta\bar{X}(t_k) = \bar{X}(t_{k+n}) - \bar{X}(t_{k-n})
+        a_i(t_k) = (v(t_{k+n}) - v(t_{k-n})) / \Delta t
 
-    divided by the square of the time interval :math:`\Delta t`:
+    which is simplified to
 
     .. math::
 
-        a_i(t_k) = \Delta\bar{X}(t_k) / \Delta t^{2}
+        a_i(t_k) = \Delta\bar{X}(t_k) / (\Delta t^{2})
+
+    using 
+
+    .. math::
+
+        \Delta\bar{X}(t_k) = \bar{X}(t_{k+n}) - \bar{X}(t_{k-n})
+    
+    .. math::
+
+        \Delta\bar{X}(t_k) = X(t_{k+2n}) - 2*X(t_{k}) + X(t_{k-2n})
 
 
     When getting closer to the start, or end of the trajectory data, it is not
@@ -298,14 +313,14 @@ def _compute_individual_acceleration(
         components in x and y direction if compute_acceleration is True
     """
     columns = [ID_COL, FRAME_COL, ACC_COL]
-    time_interval = movement_data.window_size / frame_rate
+    time_interval = 0.5 * movement_data.window_size / frame_rate
 
     # Compute displacements in x and y direction
     movement_data[["dd_x", "dd_y"]] = (
         shapely.get_coordinates(movement_data.end_position) - shapely.get_coordinates(movement_data.mid_position)
     ) - (shapely.get_coordinates(movement_data.mid_position) - shapely.get_coordinates(movement_data.start_position))
 
-    movement_data[ACC_COL] = np.linalg.norm(movement_data[["dd_x", "dd_y"]], axis=1) / time_interval**2
+    movement_data[ACC_COL] = np.linalg.norm(movement_data[["dd_x", "dd_y"]], axis=1) / (time_interval**2)
 
     if movement_direction is not None:
         # Projection of the displacement onto the movement direction
@@ -318,12 +333,12 @@ def _compute_individual_acceleration(
         movement_data[ACC_COL] = (
             np.dot(movement_data[["dd_x", "dd_y"]].to_numpy(), movement_direction)
             / np.linalg.norm(movement_direction)
-            / time_interval**2
+            / (time_interval**2)
         )
 
     if compute_acceleration_components:
-        movement_data[A_X_COL] = movement_data["dd_x"].to_numpy() / time_interval**2
-        movement_data[A_Y_COL] = movement_data["dd_y"].to_numpy() / time_interval**2
+        movement_data[A_X_COL] = movement_data["dd_x"].to_numpy() / (time_interval**2)
+        movement_data[A_Y_COL] = movement_data["dd_y"].to_numpy() / (time_interval**2)
         columns.append(A_X_COL)
         columns.append(A_Y_COL)
 
diff --git a/pedpy/methods/method_utils.py b/pedpy/methods/method_utils.py
index 76bb0464..c006942b 100644
--- a/pedpy/methods/method_utils.py
+++ b/pedpy/methods/method_utils.py
@@ -1002,8 +1002,8 @@ def _compute_movement_acceleration_exclude_border(
 ) -> pd.DataFrame:
     """Compute the individual movement in the time interval frame_step.
 
-    The movement is computed for the interval [frame - frame_step: frame +
-    frame_step], if one of the boundaries is not contained in the trajectory
+    The movement is computed for the interval [frame - 2 * frame_step: frame +
+    2 * frame_step], if one of the boundaries is not contained in the trajectory
     frame these points will not be considered.
 
     Args:
@@ -1022,16 +1022,16 @@ def _compute_movement_acceleration_exclude_border(
     """
     df_movement = traj_data.data.copy(deep=True)
 
-    df_movement[START_POSITION_COL] = df_movement.groupby(by=ID_COL).point.shift(2 * frame_step)
-    df_movement["start_frame"] = df_movement.groupby(by=ID_COL).frame.shift(2 * frame_step)
+    df_movement[START_POSITION_COL] = df_movement.groupby(by=ID_COL).point.shift(-2 * frame_step)
+    df_movement["start_frame"] = df_movement.groupby(by=ID_COL).frame.shift(-2 * frame_step)
 
-    df_movement[MID_POSITION_COL] = df_movement.groupby(by=ID_COL).point.shift(frame_step)
-    df_movement["mid_frame"] = df_movement.groupby(by=ID_COL).frame.shift(frame_step)
+    df_movement[MID_POSITION_COL] = df_movement.point
+    df_movement["mid_frame"] = df_movement.frame
 
-    df_movement[END_POSITION_COL] = df_movement.point
-    df_movement["end_frame"] = df_movement.frame
+    df_movement[END_POSITION_COL] = df_movement.groupby(by=ID_COL).point.shift(2 * frame_step)
+    df_movement["end_frame"] = df_movement.groupby(by=ID_COL).frame.shift(2 * frame_step)
 
-    df_movement[WINDOW_SIZE_COL] = df_movement.end_frame - df_movement.mid_frame
+    df_movement[WINDOW_SIZE_COL] = df_movement.end_frame - df_movement.start_frame
     return df_movement[
         [
             ID_COL,

From 7db436c074b91fcd20e30b88ec9ef53e705a213e Mon Sep 17 00:00:00 2001
From: JuleAdrian <j.adrian@fz-juelich.de>
Date: Wed, 4 Nov 2026 11:45:17 +0100
Subject: [PATCH 2/7] adjust user guide

---
 notebooks/user_guide.ipynb | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/notebooks/user_guide.ipynb b/notebooks/user_guide.ipynb
index 75453f7b..339467af 100644
--- a/notebooks/user_guide.ipynb
+++ b/notebooks/user_guide.ipynb
@@ -2694,7 +2694,7 @@
     "\n",
     "$$\n",
     "v(t_{k+n}) = \bar{X}(t_{k+n}) / \Delta t\n",
-    "$$\n",    
+    "$$\n",
     "$$\n",
     "v(t_{k-n}) = \bar{X}(t_{k-n}) / \Delta t\n",
     "$$\n",
@@ -2703,8 +2703,8 @@
     "\n",
     "$$\n",
     "\bar{X}(t_{k+n}) = X(t_{k+2n}) - X(t_{k})\n",
-    "$$\n",    
-    "$$\n",    
+    "$$\n",
+    "$$\n",
     "\bar{X}(t_{k-n}) = X(t_{k}) - X(t_{k-2n})\n",
     "$$\n",
     "The acceleration $a_i(t_k)$ is then calculated from the difference of\n",

From 1e19032fbe2daae8009c3cdd45f08a89cfa1d300 Mon Sep 17 00:00:00 2001
From: JuleAdrian <j.adrian@fz-juelich.de>
Date: Wed, 4 Nov 2026 12:03:10 +0100
Subject: [PATCH 3/7] adjust user guide

---
 notebooks/user_guide.ipynb | 156 +++++++++++++++++++++++++++++++++----
 1 file changed, 142 insertions(+), 14 deletions(-)

diff --git a/notebooks/user_guide.ipynb b/notebooks/user_guide.ipynb
index 339467af..cc75d7a0 100644
--- a/notebooks/user_guide.ipynb
+++ b/notebooks/user_guide.ipynb
@@ -4,6 +4,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "is_executing": true,
      "outputs_hidden": false
@@ -52,6 +53,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -66,6 +68,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -92,6 +95,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -143,6 +147,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -163,6 +168,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -183,6 +189,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -231,6 +238,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -275,6 +283,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -286,6 +295,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -337,6 +347,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -355,6 +366,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -367,6 +379,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -387,6 +400,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -404,6 +418,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -697,6 +712,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -711,6 +727,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -743,6 +760,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -759,6 +777,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -774,6 +793,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -789,6 +809,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -880,6 +901,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -973,6 +995,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -988,6 +1011,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1008,6 +1032,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1030,6 +1055,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -1077,6 +1103,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1120,6 +1147,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1134,6 +1162,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1158,6 +1187,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1176,6 +1206,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1190,6 +1221,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -1258,6 +1290,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1282,6 +1315,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1299,6 +1333,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -1319,6 +1354,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1333,6 +1369,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1350,6 +1387,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -1374,6 +1412,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1468,6 +1507,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -1521,6 +1561,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1545,6 +1586,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -1642,6 +1684,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1691,6 +1734,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1713,6 +1757,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -1744,6 +1789,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1771,6 +1817,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1791,6 +1838,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -1815,6 +1863,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1846,6 +1895,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -1866,6 +1916,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -1894,6 +1945,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2012,6 +2064,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2031,6 +2084,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -2064,6 +2118,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2094,6 +2149,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2113,6 +2169,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -2145,6 +2202,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2162,6 +2220,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -2186,6 +2245,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2223,6 +2283,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2243,6 +2304,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -2275,6 +2337,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2293,6 +2356,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -2317,6 +2381,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2371,6 +2436,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2448,6 +2514,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2469,6 +2536,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -2489,6 +2557,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2504,6 +2573,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2518,6 +2588,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2536,6 +2607,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -2556,6 +2628,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2613,6 +2686,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2635,6 +2709,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -2658,6 +2733,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2670,6 +2746,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2679,10 +2756,10 @@
     "\n",
     "Compute the individual acceleration for each pedestrian.\n",
     "\n",
-    "For computing the individuals' acceleration at a specific frame $a_i(t_k)$,\n",
+    "For computing the individuals' acceleration at a specific frame $a_i(t_k)$\n",
     "a specific frame step ($n$) is needed.\n",
     "Together with the {class}`trajectory data<trajectory_data.TrajectoryData>` $fps$ of\n",
-    "the trajectory data $fps$ the time frame $\\Delta t$ for\n",
+    "the trajectory data, the time frame $\\Delta t$ for\n",
     "computing the speed and the acceleration becomes:\n",
     "\n",
     "$$\n",
@@ -2693,35 +2770,36 @@
     "$v(t_{k+n})$ and $v(t_{k-n})$:\n",
     "\n",
     "$$\n",
-    "v(t_{k+n}) = \bar{X}(t_{k+n}) / \Delta t\n",
+    "v(t_{k+n}) = \\bar{X}(t_{k+n}) / \\Delta t\n",
     "$$\n",
     "$$\n",
-    "v(t_{k-n}) = \bar{X}(t_{k-n}) / \Delta t\n",
+    "v(t_{k-n}) = \\bar{X}(t_{k-n}) / \\Delta t\n",
     "$$\n",
-    "using the displacements $\bar{X}(t_{k+n})$ and $\bar{X}(t_{k-n})$\n",
-    "during $\Delta t$ around $t_{k+n}$ and $t_{k-n}$:\n",
+    "using the displacements $\\bar{X}(t_{k+n})$ and $\\bar{X}(t_{k-n})$\n",
+    "during $\\Delta t$ around $t_{k+n}$ and $t_{k-n}$:\n",
     "\n",
     "$$\n",
-    "\bar{X}(t_{k+n}) = X(t_{k+2n}) - X(t_{k})\n",
+    "\\bar{X}(t_{k+n}) = X(t_{k+2n}) - X(t_{k})\n",
     "$$\n",
     "$$\n",
-    "\bar{X}(t_{k-n}) = X(t_{k}) - X(t_{k-2n})\n",
+    "\\bar{X}(t_{k-n}) = X(t_{k}) - X(t_{k-2n})\n",
     "$$\n",
     "The acceleration $a_i(t_k)$ is then calculated from the difference of\n",
-    "the speeds divided by the time interval $\Delta t$:\n",
+    "the speeds divided by the time interval $\\Delta t$:\n",
     "\n",
     "$$\n",
-    "a_i(t_k) = (v(t_{k+n}) - v(t_{k-n})) / \Delta t\n",
+    "a_i(t_k) = (v(t_{k+n}) - v(t_{k-n})) / \\Delta t\n",
+    "$$\n",
     "which is simplified to\n",
     "$$\n",
-    "a_i(t_k) = \Delta\bar{X}(t_k) / (\Delta t^{2})\n",
+    "a_i(t_k) = \\Delta\\bar{X}(t_k) / (\\Delta t^{2})\n",
     "$$\n",
     "using\n",
     "$$\n",
-    "\Delta\bar{X}(t_k) = \bar{X}(t_{k+n}) - \bar{X}(t_{k-n})\n",
+    "\\Delta\\bar{X}(t_k) = \\bar{X}(t_{k+n}) - \\bar{X}(t_{k-n})\n",
     "$$\n",
     "$$\n",
-    "\Delta\bar{X}(t_k) = X(t_{k+2n}) - 2*X(t_{k}) + X(t_{k-2n})\n",
+    "\\Delta\\bar{X}(t_k) = X(t_{k+2n}) - 2*X(t_{k}) + X(t_{k-2n})\n",
     "$$\n",
     "\n",
     "When getting closer to the start, or end of the trajectory data, it is not\n",
@@ -2741,6 +2819,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -2763,6 +2842,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -3182,6 +3262,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -3264,6 +3345,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -3280,6 +3362,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -3434,6 +3517,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -3474,6 +3558,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -4072,6 +4157,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -4101,6 +4187,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -4506,6 +4593,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -4542,6 +4630,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -4775,6 +4864,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -4789,6 +4879,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -4802,6 +4893,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -4827,6 +4919,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -4839,6 +4932,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -4853,6 +4947,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -4879,6 +4974,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -4929,6 +5025,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -4943,6 +5040,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -4955,6 +5053,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -4967,6 +5066,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -4979,6 +5079,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -4992,6 +5093,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -5006,6 +5108,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -5023,6 +5126,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -5037,6 +5141,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -5052,6 +5157,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -5067,6 +5173,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -5081,6 +5188,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -5124,6 +5232,7 @@
   {
    "cell_type": "markdown",
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -5138,6 +5247,7 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
+    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -5160,7 +5270,25 @@
    ]
   }
  ],
- "metadata": {},
+ "metadata": {
+  "kernelspec": {
+   "display_name": "pedpy_venv_2026_10_30",
+   "language": "python",
+   "name": "pedpy_venv_2026_10_30"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.15.1"
+  }
+ },
  "nbformat": 4,
  "nbformat_minor": 4
 }

From b6f3d09015c7d2b4463f246182d431cc87fda794 Mon Sep 17 00:00:00 2001
From: JuleAdrian <j.adrian@fz-juelich.de>
Date: Wed, 4 Nov 2026 12:28:51 +0100
Subject: [PATCH 4/7] adjust formatting

---
 pedpy/methods/acceleration_calculator.py | 16 ++++++++--------
 1 file changed, 8 insertions(+), 8 deletions(-)

diff --git a/pedpy/methods/acceleration_calculator.py b/pedpy/methods/acceleration_calculator.py
index cc802c90..508244ef 100644
--- a/pedpy/methods/acceleration_calculator.py
+++ b/pedpy/methods/acceleration_calculator.py
@@ -30,15 +30,15 @@ def compute_individual_acceleration(
 
     For computing the individuals' acceleration at a specific frame
     :math:`a_i(t_k)`, a specific frame step (:math:`n`) is needed.
-    Together with the :attr:`~trajectory_data.TrajectoryData.frame_rate` 
-    of the trajectory data :math:`fps` the time frame :math:`\Delta t` 
+    Together with the :attr:`~trajectory_data.TrajectoryData.frame_rate`
+    of the trajectory data :math:`fps` the time frame :math:`\Delta t`
     for computing the speed and the acceleration becomes:
 
     .. math::
 
         \Delta t = 2 n / fps
 
-    The acceleration at time :math:`t_k` can be calculated from the speeds 
+    The acceleration at time :math:`t_k` can be calculated from the speeds
     :math:`v(t_{k+n})` and :math:`v(t_{k-n})`:
 
     .. math::
@@ -61,7 +61,7 @@ def compute_individual_acceleration(
         \bar{X}(t_{k-n}) = X(t_{k}) - X(t_{k-2n})
 
     
-    The acceleration :math:`a_i(t_k)` is then calculated from the difference of 
+    The acceleration :math:`a_i(t_k)` is then calculated from the difference of
     the speeds divided by the time interval :math:`\Delta t`:
 
     .. math::
@@ -74,7 +74,7 @@ def compute_individual_acceleration(
 
         a_i(t_k) = \Delta\bar{X}(t_k) / (\Delta t^{2})
 
-    using 
+    using
 
     .. math::
 
@@ -320,7 +320,7 @@ def _compute_individual_acceleration(
         shapely.get_coordinates(movement_data.end_position) - shapely.get_coordinates(movement_data.mid_position)
     ) - (shapely.get_coordinates(movement_data.mid_position) - shapely.get_coordinates(movement_data.start_position))
 
-    movement_data[ACC_COL] = np.linalg.norm(movement_data[["dd_x", "dd_y"]], axis=1) / (time_interval**2)
+    movement_data[ACC_COL] = np.linalg.norm(movement_data[["dd_x", "dd_y"]], axis=1) / time_interval**2
 
     if movement_direction is not None:
         # Projection of the displacement onto the movement direction
@@ -337,8 +337,8 @@ def _compute_individual_acceleration(
         )
 
     if compute_acceleration_components:
-        movement_data[A_X_COL] = movement_data["dd_x"].to_numpy() / (time_interval**2)
-        movement_data[A_Y_COL] = movement_data["dd_y"].to_numpy() / (time_interval**2)
+        movement_data[A_X_COL] = movement_data["dd_x"].to_numpy() / time_interval**2
+        movement_data[A_Y_COL] = movement_data["dd_y"].to_numpy() / time_interval**2
         columns.append(A_X_COL)
         columns.append(A_Y_COL)
 

From 7587a81c6c11a13943fa0ff2c3251f21e8fc8291 Mon Sep 17 00:00:00 2001
From: JuleAdrian <j.adrian@fz-juelich.de>
Date: Wed, 4 Nov 2026 12:39:22 +0100
Subject: [PATCH 5/7] adjust formatting

---
 pedpy/methods/acceleration_calculator.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/pedpy/methods/acceleration_calculator.py b/pedpy/methods/acceleration_calculator.py
index 508244ef..31c4c1fa 100644
--- a/pedpy/methods/acceleration_calculator.py
+++ b/pedpy/methods/acceleration_calculator.py
@@ -60,7 +60,7 @@ def compute_individual_acceleration(
 
         \bar{X}(t_{k-n}) = X(t_{k}) - X(t_{k-2n})
 
-    
+
     The acceleration :math:`a_i(t_k)` is then calculated from the difference of
     the speeds divided by the time interval :math:`\Delta t`:
 
@@ -79,7 +79,7 @@ def compute_individual_acceleration(
     .. math::
 
         \Delta\bar{X}(t_k) = \bar{X}(t_{k+n}) - \bar{X}(t_{k-n})
-    
+
     .. math::
 
         \Delta\bar{X}(t_k) = X(t_{k+2n}) - 2*X(t_{k}) + X(t_{k-2n})

From ced1d542f3e4c87b7a844ac32efa8ded36ce80c4 Mon Sep 17 00:00:00 2001
From: JuleAdrian <j.adrian@fz-juelich.de>
Date: Wed, 4 Nov 2026 12:46:36 +0100
Subject: [PATCH 6/7] adjust formatting

---
 notebooks/user_guide.ipynb | 128 +++----------------------------------
 1 file changed, 10 insertions(+), 118 deletions(-)

diff --git a/notebooks/user_guide.ipynb b/notebooks/user_guide.ipynb
index cc75d7a0..64d47eb5 100644
--- a/notebooks/user_guide.ipynb
+++ b/notebooks/user_guide.ipynb
@@ -4,7 +4,6 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
-    "collapsed": false,
     "jupyter": {
      "is_executing": true,
      "outputs_hidden": false
@@ -53,7 +52,6 @@
   {
    "cell_type": "markdown",
    "metadata": {
-    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -68,7 +66,6 @@
   {
    "cell_type": "markdown",
    "metadata": {
-    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -95,7 +92,6 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
-    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -147,7 +143,6 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
-    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     },
@@ -168,7 +163,6 @@
   {
    "cell_type": "markdown",
    "metadata": {
-    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -189,7 +183,6 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
-    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -238,7 +231,6 @@
   {
    "cell_type": "markdown",
    "metadata": {
-    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -283,7 +275,6 @@
   {
    "cell_type": "markdown",
    "metadata": {
-    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }
@@ -295,7 +286,6 @@
   {
    "cell_type": "markdown",
    "metadata": {
-    "collapsed": false,
     "jupyter": {
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@@ -347,7 +337,6 @@
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@@ -366,7 +355,6 @@
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@@ -379,7 +367,6 @@
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@@ -400,7 +387,6 @@
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@@ -418,7 +404,6 @@
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@@ -712,7 +697,6 @@
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@@ -727,7 +711,6 @@
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@@ -760,7 +743,6 @@
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@@ -777,7 +759,6 @@
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@@ -793,7 +774,6 @@
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@@ -809,7 +789,6 @@
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@@ -901,7 +880,6 @@
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@@ -995,7 +973,6 @@
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@@ -1011,7 +988,6 @@
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@@ -1032,7 +1008,6 @@
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@@ -1055,7 +1030,6 @@
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@@ -1103,7 +1077,6 @@
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@@ -1147,7 +1120,6 @@
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@@ -1162,7 +1134,6 @@
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@@ -1187,7 +1158,6 @@
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@@ -1206,7 +1176,6 @@
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@@ -1221,7 +1190,6 @@
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@@ -1290,7 +1258,6 @@
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@@ -1315,7 +1282,6 @@
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@@ -1333,7 +1299,6 @@
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@@ -1354,7 +1319,6 @@
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@@ -1369,7 +1333,6 @@
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@@ -1387,7 +1350,6 @@
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@@ -1412,7 +1374,6 @@
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@@ -1507,7 +1468,6 @@
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@@ -1561,7 +1521,6 @@
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@@ -1586,7 +1545,6 @@
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@@ -1684,7 +1642,6 @@
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@@ -1734,7 +1691,6 @@
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@@ -1757,7 +1713,6 @@
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@@ -1789,7 +1744,6 @@
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@@ -1817,7 +1771,6 @@
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@@ -1838,7 +1791,6 @@
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@@ -1916,7 +1866,6 @@
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@@ -1945,7 +1894,6 @@
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@@ -2064,7 +2012,6 @@
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@@ -2118,7 +2064,6 @@
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@@ -2245,7 +2186,6 @@
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@@ -2709,7 +2635,6 @@
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@@ -2746,7 +2670,7 @@
   {
    "cell_type": "markdown",
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+    "jp-MarkdownHeadingCollapsed": true,
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@@ -2763,43 +2687,43 @@
     "computing the speed and the acceleration becomes:\n",
     "\n",
     "$$\n",
-    "\\Delta t = 2 n / fps\n",
+    "    \\Delta t = 2 n / fps\n",
     "$$\n",
     "\n",
     "The acceleration at time $t_k$ can be calculated from the speeds\n",
     "$v(t_{k+n})$ and $v(t_{k-n})$:\n",
     "\n",
     "$$\n",
-    "v(t_{k+n}) = \\bar{X}(t_{k+n}) / \\Delta t\n",
+    "    v(t_{k+n}) = \\bar{X}(t_{k+n}) / \\Delta t\n",
     "$$\n",
     "$$\n",
-    "v(t_{k-n}) = \\bar{X}(t_{k-n}) / \\Delta t\n",
+    "    v(t_{k-n}) = \\bar{X}(t_{k-n}) / \\Delta t\n",
     "$$\n",
     "using the displacements $\\bar{X}(t_{k+n})$ and $\\bar{X}(t_{k-n})$\n",
     "during $\\Delta t$ around $t_{k+n}$ and $t_{k-n}$:\n",
     "\n",
     "$$\n",
-    "\\bar{X}(t_{k+n}) = X(t_{k+2n}) - X(t_{k})\n",
+    "    \\bar{X}(t_{k+n}) = X(t_{k+2n}) - X(t_{k})\n",
     "$$\n",
     "$$\n",
-    "\\bar{X}(t_{k-n}) = X(t_{k}) - X(t_{k-2n})\n",
+    "    \\bar{X}(t_{k-n}) = X(t_{k}) - X(t_{k-2n})\n",
     "$$\n",
     "The acceleration $a_i(t_k)$ is then calculated from the difference of\n",
     "the speeds divided by the time interval $\\Delta t$:\n",
     "\n",
     "$$\n",
-    "a_i(t_k) = (v(t_{k+n}) - v(t_{k-n})) / \\Delta t\n",
+    "    a_i(t_k) = (v(t_{k+n}) - v(t_{k-n})) / \\Delta t\n",
     "$$\n",
     "which is simplified to\n",
     "$$\n",
-    "a_i(t_k) = \\Delta\\bar{X}(t_k) / (\\Delta t^{2})\n",
+    "    a_i(t_k) = \\Delta\\bar{X}(t_k) / (\\Delta t^{2})\n",
     "$$\n",
     "using\n",
     "$$\n",
-    "\\Delta\\bar{X}(t_k) = \\bar{X}(t_{k+n}) - \\bar{X}(t_{k-n})\n",
+    "    \\Delta\\bar{X}(t_k) = \\bar{X}(t_{k+n}) - \\bar{X}(t_{k-n})\n",
     "$$\n",
     "$$\n",
-    "\\Delta\\bar{X}(t_k) = X(t_{k+2n}) - 2*X(t_{k}) + X(t_{k-2n})\n",
+    "    \\Delta\\bar{X}(t_k) = X(t_{k+2n}) - 2*X(t_{k}) + X(t_{k-2n})\n",
     "$$\n",
     "\n",
     "When getting closer to the start, or end of the trajectory data, it is not\n",
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@@ -4879,7 +4791,6 @@
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@@ -4919,7 +4829,6 @@
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@@ -4932,7 +4841,6 @@
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@@ -5025,7 +4931,6 @@
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@@ -5040,7 +4945,6 @@
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@@ -5066,7 +4969,6 @@
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@@ -5079,7 +4981,6 @@
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@@ -5093,7 +4994,6 @@
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@@ -5108,7 +5008,6 @@
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@@ -5126,7 +5025,6 @@
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@@ -5141,7 +5039,6 @@
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@@ -5157,7 +5054,6 @@
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@@ -5173,7 +5069,6 @@
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@@ -5188,7 +5083,6 @@
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@@ -5232,7 +5126,6 @@
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@@ -5247,7 +5140,6 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
-    "collapsed": false,
     "jupyter": {
      "outputs_hidden": false
     }

From de57a9701747bde049d747acf876d7f8f581fdee Mon Sep 17 00:00:00 2001
From: JuleAdrian <j.adrian@fz-juelich.de>
Date: Wed, 4 Nov 2026 13:20:38 +0100
Subject: [PATCH 7/7] adjust formatting

---
 notebooks/user_guide.ipynb | 20 +-------------------
 1 file changed, 1 insertion(+), 19 deletions(-)

diff --git a/notebooks/user_guide.ipynb b/notebooks/user_guide.ipynb
index 64d47eb5..b9bedddb 100644
--- a/notebooks/user_guide.ipynb
+++ b/notebooks/user_guide.ipynb
@@ -5162,25 +5162,7 @@
    ]
   }
  ],
- "metadata": {
-  "kernelspec": {
-   "display_name": "pedpy_venv_2026_10_30",
-   "language": "python",
-   "name": "pedpy_venv_2026_10_30"
-  },
-  "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
-    "version": 3
-   },
-   "file_extension": ".py",
-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
-   "version": "3.15.1"
-  }
- },
+ "metadata": {},
  "nbformat": 4,
  "nbformat_minor": 4
 }