Repository: ssciwr/vstt (language: python)

Issue:
document statistics
should document how each statistic is calculated, what the units are, etc


Discussion / hints:
* Reaction time: How is he reaction
    time calculated? Time between target appearance and
        movement onset?
* Movement time: how is movement time calculated? It seems that this is the time only when the
        cursor is moved (e.g. when the cursor is not moved, 0 is displayed)? Would it be possible to
        add an additional movement time parameter (e.g. overall movement time) defined as the time
        between target appearance until the target is reached?
* Distance & RMSE: how are the distance and RMSE calculated? For example, the distance and
        RMSE seems to be very small when the cursor is moved in the opposite direction. What is the
        unit of the parameters (e.g. cm)?

Fix patch:
diff --git a/docs/reference/images/positions.svg b/docs/reference/images/positions.svg
new file mode 100644
index 00000000..c8f14a37
--- /dev/null
+++ b/docs/reference/images/positions.svg
@@ -0,0 +1,196 @@
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diff --git a/docs/reference/images/times.svg b/docs/reference/images/times.svg
new file mode 100644
index 00000000..b7825155
--- /dev/null
+++ b/docs/reference/images/times.svg
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diff --git a/docs/reference/statistics.rst b/docs/reference/statistics.rst
index e919183c..4378d3f1 100644
--- a/docs/reference/statistics.rst
+++ b/docs/reference/statistics.rst
@@ -1,20 +1,61 @@
 Statistics
 ==========
 
+VSTT calculates a number of statistics for each trial.
+These are displayed in the application, are provided in the statistics Pandas DataFrame,
+and are also contained in the excel data export.
+Here we provide the definitions used to calculate them.
+
 Time
 ----
 
+.. figure:: images/times.svg
+   :alt: target timepoints
+
+   The timepoints of events for a target.
+
+The following timepoints are used to calculate timing statistics:
+
+   * :math:`t_{start}` is when timestamps begin for this target
+   * :math:`t_{display}` is when the target is first displayed
+   * :math:`t_{move}` is when the cursor is first moved
+   * :math:`t_{final}` is when the cursor reaches the target or timeout occurs
+
+The following timing statistics are calculated, all in units of seconds:
+
    * Time
+      * :math:`t_{final} - t_{display}`
       * Time from target being displayed to the target being reached by the cursor or timeout
    * Reaction time
+      * :math:`t_{move} - t_{display}`
       * Time from target being displayed to first cursor movement
    * Movement time
+      * :math:`t_{final} - t_{move}`
       * Time from first cursor movement to the target being reached by the cursor or timeout
 
 Distance
 --------
 
+.. figure:: images/positions.svg
+   :alt: target cursor positions
+
+   The cursor positions for a target.
+
+The cursor location at a timestamp is given by a pair of :math:`(x, y)` coordinates,
+where :math:`(0, 0)` corresponds to the center of the screen,
+and 1 in these units is equal to the height of the screen.
+
+Given :math:`n` pairs of :math:`(x, y)` cursor locations, labelled in order from :math:`1` to :math:`n`,
+the following statistics are calculated, all in units of screen height:
+
    * Distance
+      * :math:`\sum_{i=1}^{n-1}\sqrt{(x_{i+1}-x_i)^2+(y_{i+1}-y_i)^2}`
       * Euclidean point-to-point distance travelled by the cursor
+
    * RMSE
-      * The Root Mean Square Error (RMSE) of the perpendicular distance from each cursor point to the ideal path to the target
+      * :math:`\sqrt{\frac{\sum_{i=2}^{n}\left[(x_t-x_1)(y_1-y_i)-(x_1-x_i)(y_t-y_1)\right]^2}{(n-1)\left[(x_t-x_1)^2+(y_t-y_1)^2\right]}}`
+      * where :math:`(x_t, y_t)` is the location of the target
+      * Root Mean Square Error (RMSE) from the ideal path to the target
+      * this is the RMS of the perpendicular distance of each cursor point from the ideal path
+      * where the ideal path is the straight line from the first cursor point to the target
+      * based on `Distance from a point to a line: Line defined by two points <https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line#Line_defined_by_two_points>`_


Test patch:
diff --git a/tests/test_task.py b/tests/test_task.py
index b6150e6d..e129a861 100644
--- a/tests/test_task.py
+++ b/tests/test_task.py
@@ -197,12 +197,21 @@ def test_task_fixed_intervals_no_user_input(window: Window) -> None:
     # at ~target_duration the first target is displayed for target_duration secs,
     # subsequent ones should be displayed every ~target_duration starting from ~2*target_duration
     all_to_target_timestamps = data["to_target_timestamps"][0][0]
-    # require timestamps to be accurate within 0.1s
-    # On CI the fps / dropped frames can be erratic so this is a conservative value.
-    # If running on real hardware these should actually be within 2/fps
-    delta = 0.1
-    assert abs(all_to_target_timestamps[0][0]) < delta
-    assert abs(all_to_target_timestamps[0][-1] - 2 * target_duration) < delta
+    # require timestamps to be accurate within 0.5s
+    allowed_error_on_timestamp = 0.5
+    # this is a weak requirement to avoid tests failing on CI where many frames can get dropped
+    # if running tests locally allowed_error_on_timestamp should be ~2/fps
+    assert abs(all_to_target_timestamps[0][0]) < allowed_error_on_timestamp
+    assert (
+        abs(all_to_target_timestamps[0][-1] - 2 * target_duration)
+        < allowed_error_on_timestamp
+    )
     for count, to_target_timestamps in enumerate(all_to_target_timestamps[1:]):
-        assert abs(to_target_timestamps[0] - (count + 2) * target_duration) < delta
-        assert abs(to_target_timestamps[-1] - (count + 3) * target_duration) < delta
+        assert (
+            abs(to_target_timestamps[0] - (count + 2) * target_duration)
+            < allowed_error_on_timestamp
+        )
+        assert (
+            abs(to_target_timestamps[-1] - (count + 3) * target_duration)
+            < allowed_error_on_timestamp
+        )