I have posted a similar question about the same problem, having been suggested to use a polynomial Robust Linear Model, which worked fine for most cases, as can be seen here:
Non-algebric curve-fitting along weighted pointcloud (if possible using python)
But since then I have done some research, and I think a Non-Parametric regression model might be a better choice, since the model must fit locally and the result should respect some conditions which are not parametric.
The problem statement is this:
"Given a data set consisting of a set of coordinates in the form (positionX, positionY, weight), representing candidate locations of simmetry points of a human back surface, 'weight' being a representation of fitness (bilateral symmetry according to some symmetry function) of each point, find the curve that most likely represent the midline of the back, possibly misaligned due to postural inadequacies, considering that:
- The line must run along the full height of the dataset;
- For each Y (independent variable) there should correspond X (dependent variable) representing the most probable position of the midline;
- There is coupling between successive X positions, that is, X is not random, there are not discontinuities, for the back midline "in the real life" doesn't have discontinuities, sharp corners or points of high curvature. Therefore, the resulting curve should be reasonably "smooth and well-behaved".
- The method should "bridge" two regions with good fit separated by a region with more locally scattered points.
- "when in doubt", the midline tends to run along the vertical middle of the bounding box of the dataset.
I have found two articles in Wikipedia that seem to be applyable to this problem (Kernel Regression and RANSAC), but my current knowledge is not enough (mathematical and statistical notation, programming) in not enough to solve the problem on my own. Here are two representative images found in those articles, that resemble the conditions of my problem:
RANSAC algorithm with a linear model:
Kernel regression:
Also, I want to provide my own data:
Sample unordered set of (x,y,weight) coordinates:
[[ -0.7898176 -3.35201728 4.36142086]
[ 2.99221402 -3.35201728 1.11907575]
[ 6.97475149 -3.35201728 2.4320322 ]
[ -4.82443609 -2.35201728 0.6479064 ]
[ -1.32418909 -2.35201728 1.88004944]
[ 0.07067882 -2.35201728 1.10982834]
[ 3.09169448 -2.35201728 1.8557436 ]
[ 7.10399403 -2.35201728 2.03906224]
[ -3.07207606 -1.35201728 0.35500973]
[ 2.63202993 -1.35201728 5.32397834]
[ 5.19884868 -1.35201728 1.63816326]
[ 7.65721835 -1.35201728 1.13843392]
[ 2.48172754 -0.35201728 6.65584512]
[ 6.0905911 -0.35201728 1.15552652]
[ 8.62497546 -0.35201728 0.30407144]
[ -4.7300089 0.64798272 0.31481496]
[ -3.03274093 0.64798272 0.95337568]
[ 2.19653614 0.64798272 10.3675204 ]
[ 6.20384058 0.64798272 1.42106077]
[ -4.08636605 1.64798272 0.28875288]
[ 2.03344989 1.64798272 13.04648211]
[ -4.11717795 2.64798272 0.39713141]
[ 1.93304283 2.64798272 10.41313242]
[ -4.37994815 3.64798272 0.84588643]
[ 1.66081408 3.64798272 14.96380955]
[ -4.19024027 4.64798272 0.73216113]
[ 1.60252433 4.64798272 14.72419286]
[ 6.77837359 4.64798272 0.6186005 ]
[ -4.14362668 5.64798272 0.93673165]
[ 1.55372968 5.64798272 12.9421123 ]
[ -4.62223541 6.64798272 0.6510101 ]
[ 1.527865 6.64798272 10.80209351]
[ 6.86820685 6.64798272 0.82550801]
[ -4.68259732 7.64798272 0.45321369]
[ 1.36167494 7.64798272 6.45338514]
[ -5.19205787 8.64798272 0.23935013]
[ 1.21003466 8.64798272 10.13528877]
[ 7.6689546 8.64798272 0.32421776]
[ -5.36436818 9.64798272 0.79809416]
[ 1.26248534 9.64798272 7.67036253]
[ 7.35472418 9.64798272 0.92555691]
[ -5.61723652 10.64798272 0.4741007 ]
[ 1.23101086 10.64798272 7.97064105]
[ -7.83024735 11.64798272 0.47557318]
[ 1.20348982 11.64798272 8.20694816]
[ 1.14422758 12.64798272 9.26244889]
[ 9.18164464 12.64798272 0.72428381]
[ 1.0827069 13.64798272 10.08599118]
[ 6.80116007 13.64798272 0.4571425 ]
[ 9.384236 13.64798272 0.42399893]
[ 1.04053491 14.64798272 10.48370805]
[ 9.16197972 14.64798272 0.39930227]
[ -9.85958581 15.64798272 0.39524976]
[ 0.9942501 15.64798272 8.39992264]
[ 8.07642416 15.64798272 0.61480371]
[ 9.55088151 15.64798272 0.54076473]
[ -7.13657331 16.64798272 0.32929172]
[ 0.92606211 16.64798272 7.83597033]
[ 8.74291069 16.64798272 0.74246827]
[ -7.20022443 17.64798272 0.52555351]
[ 0.81344517 17.64798272 6.81654834]
[ 8.52844624 17.64798272 0.70543711]
[ -6.97465178 18.64798272 1.04527813]
[ 0.61959631 18.64798272 10.33529022]
[ 5.733054 18.64798272 1.2309691 ]
[ 8.14818453 18.64798272 1.37532423]
[ -6.82823664 19.64798272 2.0314052 ]
[ 0.56391636 19.64798272 13.61447357]
[ 5.79971126 19.64798272 0.30148347]
[ 8.01499476 19.64798272 1.72465327]
[ -6.78504689 20.64798272 2.88657804]
[ -4.79580634 20.64798272 0.36201975]
[ 0.548376 20.64798272 7.8414544 ]
[ 7.62258506 20.64798272 1.52817905]
[-10.50328534 21.64798272 0.90358671]
[ -6.59976138 21.64798272 2.62980169]
[ -3.71180255 21.64798272 1.27094175]
[ 0.5060743 21.64798272 11.06117677]
[ 4.51983105 21.64798272 1.74626435]
[ 7.50948795 21.64798272 3.46497629]
[ 11.10199877 21.64798272 1.78047269]
[-10.15444935 22.64798272 1.47486166]
[ -6.26274479 22.64798272 4.73707852]
[ -3.45440904 22.64798272 1.72516012]
[ 0.52759064 22.64798272 12.58470433]
[ 4.22258017 22.64798272 2.63827535]
[ 7.03480033 22.64798272 3.506412 ]
[ 10.63560314 22.64798272 3.56076386]
[ -5.95693623 23.64798272 2.97403863]
[ -3.66261423 23.64798272 2.31667236]
[ 0.52051366 23.64798272 12.5526344 ]
[ 4.21083787 23.64798272 1.95794387]
[ 6.82438636 23.64798272 4.77995659]
[ 10.18138299 23.64798272 5.21836205]
[ -9.94629932 24.64798272 0.4074823 ]
[ -5.74101948 24.64798272 2.60992238]
[ 0.52987226 24.64798272 10.68846987]
[ 6.29981921 24.64798272 3.56204471]
[ 9.96431168 24.64798272 2.85079129]
[ -9.64229717 25.64798272 0.4503241 ]
[ -5.579063 25.64798272 0.64475469]
[ 0.52053534 25.64798272 10.05046667]
[ 5.79167815 25.64798272 0.92797027]
[ 10.05116919 25.64798272 2.52194933]
[ -8.55286247 26.64798272 0.94447148]
[ 0.45065604 26.64798272 10.97432823]
[ 5.50068393 26.64798272 2.39645232]
[ 10.08992273 26.64798272 2.77716257]
[-16.62381217 27.64798272 0.2021621 ]
[ -9.62146213 27.64798272 0.62245778]
[ -7.66905507 27.64798272 2.84466396]
[ 0.38656111 27.64798272 10.74369366]
[ 5.76925402 27.64798272 1.13362978]
[ 9.83525197 27.64798272 1.18241147]
[-15.64874512 28.64798272 0.18279302]
[ -7.52932494 28.64798272 2.94012191]
[ 0.32171219 28.64798272 10.73770466]
[ 9.4062684 28.64798272 1.41714298]
[-12.71287717 29.64798272 0.70268073]
[ -7.59473877 29.64798272 2.16183026]
[ 0.20748772 29.64798272 12.97312987]
[ 3.92952496 29.64798272 1.54987681]
[ 9.05148017 29.64798272 2.40563748]
[ 14.96021523 29.64798272 0.55258241]
[-12.14428813 30.64798272 0.36365363]
[ -7.12360666 30.64798272 2.54312163]
[ 0.40594038 30.64798272 12.64839117]
[ 4.59465757 30.64798272 1.23496581]
[ 8.54333134 30.64798272 2.18912857]
[-10.6296531 31.64798272 1.4839259 ]
[ -7.09532763 31.64798272 2.0113838 ]
[ 0.37037733 31.64798272 12.2071139 ]
[ 3.01253349 31.64798272 3.01591777]
[ 4.64523695 31.64798272 3.50267541]
[ 8.39369696 31.64798272 2.53195817]
[ -7.07947026 32.64798272 1.01324147]
[ 0.39269437 32.64798272 9.67368625]
[ 8.58669997 32.64798272 1.00475646]
[ 12.02329114 32.64798272 0.50782399]
[-10.13060786 33.64798272 0.31475653]
[ -7.30360407 33.64798272 0.35065243]
[ 0.49556923 33.64798272 9.66608818]
[ -5.37822311 34.64798272 0.38727401]
[ 0.4958055 34.64798272 7.5415026 ]
[ 6.07719006 34.64798272 0.63012453]
[ -4.64579055 35.64798272 0.39990249]
[ 0.46323666 35.64798272 4.60449213]
[ 4.72819312 35.64798272 0.98050594]
[ -4.62418372 36.64798272 0.64160709]
[ 0.48866236 36.64798272 4.29331656]
[ 5.06493722 36.64798272 0.59888608]
[ 0.49730481 37.64798272 1.32828464]
[ -1.31849217 38.64798272 0.70780886]
[ 1.70966455 38.64798272 0.88052135]
[ 0.06305774 39.64798272 0.47366487]
[ 2.13639356 39.64798272 0.67971461]
[ -0.84726354 40.64798272 0.63787522]
[ 0.55723562 40.64798272 0.62855097]
[ 2.22359779 40.64798272 0.33884894]
[ 0.77309816 41.64798272 0.4605534 ]
[ 0.56144565 42.64798272 0.43678788]]
An image showing one very good fit (leftmost), and two fits that I consider not good, all from different subjects, obtained by fitting a Polynomial Robust Linear Model, acoording to the previously linked question: (the middle fit "jumps" out of the main midline, while the rightmost runs a long length parallel to the "true" midline instead of right through it)
It seems to me that, being parametric and global, polynomial models end up being sensitive to interfering conditions present in the current problem domain.
I thank for any help and in special to the fellow who answered the previous question, which took me out of a dead-end and presented me the amazing statsmodels Python module, which I would like to keep using if possible with a non-parametric regression approach.
