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--- visual3d:documentation:pipeline:expressions:least_squares_fitting_of_data [2024/07/12 13:29] – removed sgranger
+++ visual3d:documentation:pipeline:expressions:least_squares_fitting_of_data [2025/01/24 18:53] (current) – [Best Fit Plane] wikisysop
@@ Line 1: / Line 1: @@
+===== Least Squares Fitting of Data =====
+This page contains a list of all of [[visual3d:documentation:pipeline:expressions:expressions_overview|Evaluate_Expression]]'s functions that are used to find a line or plane of best fit to your data.
+==== Best Fit Plane ====
+**Best_Fit_Plane**(signal, start_event_signal, end_event_signal) - Finds a plane that fits the path of a point from a start event to an end event. [[https://en.wikipedia.org/wiki/Plane_%28geometry%29#Point-normal_form_and_general_form_of_the_equation_of_a_plane|The resulting plane is defined by 4 components]].
+The general form of the equation of a plane in 3D is ax+by+cz+d = 0 where 𝑎, 𝑏, and 𝑐 are the components of the normal vector which is perpendicular to the plane or any vector parallel to the plane.
+If (𝑥0,𝑦0,𝑧0)is a point that lies on the plane, then 𝑑=−(𝑎𝑥0+𝑏𝑦0+𝑐𝑧0) and as 𝑎𝑥+𝑏𝑦+𝑐𝑧−(𝑎𝑥0+𝑏𝑦0+𝑐𝑧0)=0.
+The result of the Best_Fit_Plane function is a signal with the fource components (a,b,c,d)
+<code>
+! Example: Compute a Best_Plane_Fit for one signal across a range of frames
+Evaluate_Expression
+/EXPRESSION=Best_Fit_Plane(CURRENT_SIGNAL,EVENT_LABEL::ORIGINAL::START,EVENT_LABEL::ORIGINAL::END)
+/SIGNAL_TYPES=TARGET
+/SIGNAL_FOLDER=ORIGINAL
+/SIGNAL_NAMES=LSK_1
+/RESULT_TYPES=METRIC
+/RESULT_FOLDERS=PROCESSED
+/RESULT_NAME=LSK1_PLANE
+/APPLY_AS_SUFFIX_TO_SIGNAL_NAME=TRUE
+;
+</code>
+<code>
+! Example: Compute a Best_Plane_Fit for multiple signals at each frame of data
+Evaluate_Expression
+/EXPRESSION=Best_Fit_Plane(CURRENT_SIGNAL)
+/SIGNAL_TYPES=TARGET
+/SIGNAL_FOLDER=ORIGINAL
+/SIGNAL_NAMES=LSK_1+LSK_2+LSK_3
+/RESULT_TYPES=DERIVED
+/RESULT_FOLDERS=PROCESSED
+/RESULT_NAME=LSK_PLANE
+/APPLY_AS_SUFFIX_TO_SIGNAL_NAME=TRUE
+;
+</code>
+==== Best Fit Circle ====
+**Best_Fit_Circle**(signal, start_event_signal, end_event_signal) - Fits a 2D circle to the path of a signal.
+The algorithm computes a best fit plane to the data, rotates this plane into a principal plane, computes the center and radius, the rotates these values back into the original plane. The result is 4 components:
+  - the origin's X-coordinate;
+  - the origin's Y-coordinate;
+  - the origin's Z-coordinate; and
+  - the radius.
+==== Best Fit Sphere ====
+**Best_Fit_Sphere**(signal,start_event_signal,end_event_signal) - Fits a 3D sphere to the path of a signal.
+<code>
+! Example : Fit a sphere to 6 points on a DIAMOND
+! Create 6 TARGETS representing the Vertices
+Create_Target
+/SIGNAL_NAMES=V1
+! /SIGNAL_DESCRIPTION=
+/EXPRESSION=VECTOR((0.5+0*FRAME_NUMBERS::ORIGINAL::FRAMES),0,0)
+! /INCLUDE_CALFILE=FALSE
+;
+Create_Target
+/SIGNAL_NAMES=V2
+! /SIGNAL_DESCRIPTION=
+/EXPRESSION=VECTOR((-0.5+0*FRAME_NUMBERS::ORIGINAL::FRAMES),0,0)
+! /INCLUDE_CALFILE=FALSE
+;
+Create_Target
+/SIGNAL_NAMES=V3
+! /SIGNAL_DESCRIPTION=
+/EXPRESSION=VECTOR((0,0.5+0*FRAME_NUMBERS::ORIGINAL::FRAMES),0)
+! /INCLUDE_CALFILE=FALSE
+;
+Create_Target
+/SIGNAL_NAMES=V4
+! /SIGNAL_DESCRIPTION=
+/EXPRESSION=VECTOR((0,-0.5+0*FRAME_NUMBERS::ORIGINAL::FRAMES),0)
+! /INCLUDE_CALFILE=FALSE
+;
+Create_Target
+/SIGNAL_NAMES=V5
+! /SIGNAL_DESCRIPTION=
+/EXPRESSION=VECTOR((0,0,0.5+0*FRAME_NUMBERS::ORIGINAL::FRAMES))
+! /INCLUDE_CALFILE=FALSE
+;
+Create_Target
+/SIGNAL_NAMES=V6
+! /SIGNAL_DESCRIPTION=
+/EXPRESSION=VECTOR((0,0,-0.5+0*FRAME_NUMBERS::ORIGINAL::FRAMES))
+! /INCLUDE_CALFILE=FALSE
+;
+Evaluate_Expression
+/EXPRESSION=Best_Fit_Sphere(CURRENT_SIGNAL)
+/SIGNAL_TYPES=TARGET
+/SIGNAL_FOLDER=ORIGINAL
+/SIGNAL_NAMES=V1+V2+V3+V4+V5+V6
+/RESULT_TYPES=DERIVED
+! /RESULT_FOLDERS=PROCESSED
+/RESULT_NAME=SPHERE
+! /APPLY_AS_SUFFIX_TO_SIGNAL_NAME=FALSE
+;
+</code>
+Note that TARGETS are created and the sphere computed has each TARGET on its surface.
+{{:Sphere1.jpg}}{{:sphere6view.jpg}}
+<code>
+! Example : Create a TARGET where each frame is on a random location on the surface of a sphere of radius 1
+Evaluate_Expression
+/EXPRESSION=RAND(-PI(),PI(),FRAME_NUMBERS::ORIGINAL::FRAMES)
+/RESULT_TYPES=DERIVED
+/RESULT_FOLDERS=RADIUS
+/RESULT_NAME=VERTICAL
+! /APPLY_AS_SUFFIX_TO_SIGNAL_NAME=FALSE
+;
+Evaluate_Expression
+/EXPRESSION=RAND(0,2*PI(),FRAME_NUMBERS::ORIGINAL::FRAMES)
+/RESULT_TYPES=DERIVED
+/RESULT_FOLDERS=RADIUS
+/RESULT_NAME=HORIZONTAL
+! /APPLY_AS_SUFFIX_TO_SIGNAL_NAME=FALSE
+;
+Create_Target
+/SIGNAL_NAMES=SPHERE
+! /SIGNAL_DESCRIPTION=
+/EXPRESSION=VECTOR(
+COS(DERIVED::RADIUS::VERTICAL)*COS(DERIVED::RADIUS::HORIZONTAL),
+-COS(DERIVED::RADIUS::VERTICAL)*SIN(DERIVED::RADIUS::HORIZONTAL),
+SIN(DERIVED::RADIUS::VERTICAL))
+! /INCLUDE_CALFILE=FALSE
+;
+Evaluate_Expression
+/EXPRESSION=Best_Fit_Sphere(CURRENT_SIGNAL)
+/SIGNAL_TYPES=TARGET
+/SIGNAL_FOLDER=ORIGINAL
+/SIGNAL_NAMES=SPHERE
+/RESULT_TYPES=DERIVED
+! /RESULT_FOLDERS=PROCESSED
+/RESULT_NAME=SPHERE
+! /APPLY_AS_SUFFIX_TO_SIGNAL_NAME=FALSE
+;
+</code>
+{{:random8view.jpg}}
+{{:sphereRandom.jpg}}
+==== Simple Linear Regression ====
+**Simple_Linear_Regression**(signal1, signal2, start_event, end_event); - Fits a signal to a line given by the equation Y = mX + b where
+<code>
+Signal = m Signal1 + b
+</code>
+An explanation of the calculation can be found [[Visual3D:Documentation:Statistics:Compute_Linear_Regression|here]] or [[https://en.wikipedia.org/wiki/Simple_linear_regression|here]].
+Signal1 and Signal2 are both one-component signals and the resulting METRIC contains 6 components:
+  - Slope = m -> Component 1
+  - Intercept = b -> Component 2
+  - Siga = uncertainty in m -> Component 3
+  - Sigb = uncertainty in b -> Component 4
+  - Chi2 = chi square -> Component 5
+  - Q = The R^2 statistic -> Component 6