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Farnell DJJ, Claes P. Initial Steps towards a Multilevel Functional Principal Components Analysis Model of Dynamical Shape Changes. J Imaging 2023; 9:jimaging9040086. [PMID: 37103237 PMCID: PMC10144090 DOI: 10.3390/jimaging9040086] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/21/2023] [Revised: 04/12/2023] [Accepted: 04/14/2023] [Indexed: 04/28/2023] Open
Abstract
In this article, multilevel principal components analysis (mPCA) is used to treat dynamical changes in shape. Results of standard (single-level) PCA are also presented here as a comparison. Monte Carlo (MC) simulation is used to create univariate data (i.e., a single "outcome" variable) that contain two distinct classes of trajectory with time. MC simulation is also used to create multivariate data of sixteen 2D points that (broadly) represent an eye; these data also have two distinct classes of trajectory (an eye blinking and an eye widening in surprise). This is followed by an application of mPCA and single-level PCA to "real" data consisting of twelve 3D landmarks outlining the mouth that are tracked over all phases of a smile. By consideration of eigenvalues, results for the MC datasets find correctly that variation due to differences in groups between the two classes of trajectories are larger than variation within each group. In both cases, differences in standardized component scores between the two groups are observed as expected. Modes of variation are shown to model the univariate MC data correctly, and good model fits are found for both the "blinking" and "surprised" trajectories for the MC "eye" data. Results for the "smile" data show that the smile trajectory is modelled correctly; that is, the corners of the mouth are drawn backwards and wider during a smile. Furthermore, the first mode of variation at level 1 of the mPCA model shows only subtle and minor changes in mouth shape due to sex; whereas the first mode of variation at level 2 of the mPCA model governs whether the mouth is upturned or downturned. These results are all an excellent test of mPCA, showing that mPCA presents a viable method of modeling dynamical changes in shape.
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Affiliation(s)
| | - Peter Claes
- Medical Imaging Research Center, UZ Leuven, 3000 Leuven, Belgium
- Department of Electrical Engineering, Processing of Speech and Images (ESAT-PSI), KU Leuven, 3000 Leuven, Belgium
- Department of Human Genetics, KU Leuven, 3000 Leuven, Belgium
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Farnell DJJ. An Exploration of Pathologies of Multilevel Principal Components Analysis in Statistical Models of Shape. J Imaging 2022; 8:jimaging8030063. [PMID: 35324618 PMCID: PMC8950128 DOI: 10.3390/jimaging8030063] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2022] [Revised: 02/22/2022] [Accepted: 03/01/2022] [Indexed: 12/10/2022] Open
Abstract
3D facial surface imaging is a useful tool in dentistry and in terms of diagnostics and treatment planning. Between-group PCA (bgPCA) is a method that has been used to analyse shapes in biological morphometrics, although various “pathologies” of bgPCA have recently been proposed. Monte Carlo (MC) simulated datasets were created here in order to explore “pathologies” of multilevel PCA (mPCA), where mPCA with two levels is equivalent to bgPCA. The first set of MC experiments involved 300 uncorrelated normally distributed variables, whereas the second set of MC experiments used correlated multivariate MC data describing 3D facial shape. We confirmed results of numerical experiments from other researchers that indicated that bgPCA (and so also mPCA) can give a false impression of strong differences in component scores between groups when there is none in reality. These spurious differences in component scores via mPCA decreased significantly as the sample sizes per group were increased. Eigenvalues via mPCA were also found to be strongly affected by imbalances in sample sizes per group, although this problem was removed by using weighted forms of covariance matrices suggested by the maximum likelihood solution of the two-level model. However, this did not solve problems of spurious differences between groups in these simulations, which was driven by very small sample sizes in one group. As a “rule of thumb” only, all of our experiments indicate that reasonable results are obtained when sample sizes per group in all groups are at least equal to the number of variables. Interestingly, the sum of all eigenvalues over both levels via mPCA scaled approximately linearly with the inverse of the sample size per group in all experiments. Finally, between-group variation was added explicitly to the MC data generation model in two experiments considered here. Results for the sum of all eigenvalues via mPCA predicted the asymptotic amount for the total amount of variance correctly in this case, whereas standard “single-level” PCA underestimated this quantity.
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Affiliation(s)
- Damian J J Farnell
- School of Dentistry, Cardiff University, Heath Park, Cardiff CF14 4XY, UK
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Farnell DJJ, Richmond S, Galloway J, Zhurov AI, Pirttiniemi P, Heikkinen T, Harila V, Matthews H, Claes P. An exploration of adolescent facial shape changes with age via multilevel partial least squares regression. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2021; 200:105935. [PMID: 33485077 PMCID: PMC7920996 DOI: 10.1016/j.cmpb.2021.105935] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2020] [Accepted: 01/05/2021] [Indexed: 05/24/2023]
Abstract
BACKGROUND AND OBJECTIVES Multilevel statistical models represent the existence of hierarchies or clustering within populations of subjects (or shapes in this work). This is a distinct advantage over single-level methods that do not. Multilevel partial-least squares regression (mPLSR) is used here to study facial shape changes with age during adolescence in Welsh and Finnish samples comprising males and females. METHODS 3D facial images were obtained for Welsh and Finnish male and female subjects at multiple ages from 12 to 17 years old. 1000 3D points were defined regularly for each shape by using "meshmonk" software. A three-level model was used here, including level 1 (sex/ethnicity); level 2, all "subject" variations excluding sex, ethnicity, and age; and level 3, age. The mathematical formalism of mPLSR is given in an Appendix. RESULTS Differences in facial shape between the ages of 12 and 17 predicted by mPLSR agree well with previous results of multilevel principal components analysis (mPCA); buccal fat is reduced with increasing age and features such as the nose, brow, and chin become larger and more distinct. Differences due to ethnicity and sex are also observed. Plausible simulated faces are predicted from the model for different ages, sexes and ethnicities. Our models provide good representations of the shape data by consideration of appropriate measures of model fit (RMSE and R2). CONCLUSIONS Repeat measures in our dataset for the same subject at different ages can only be modelled indirectly at the lowest level of the model at discrete ages via mPCA. By contrast, mPLSR models age explicitly as a continuous covariate, which is a strong advantage of mPLSR over mPCA. These investigations demonstrate that multivariate multilevel methods such as mPLSR can be used to describe such age-related changes for dense 3D point data. mPLSR might be of much use in future for the prediction of facial shapes for missing persons at specific ages or for simulating shapes for syndromes that affect facial shape in new subject populations.
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Affiliation(s)
- D J J Farnell
- School of Dentistry, Cardiff University, Heath Park, Cardiff CF14 4XY, United Kingdom.
| | - S Richmond
- School of Dentistry, Cardiff University, Heath Park, Cardiff CF14 4XY, United Kingdom
| | - J Galloway
- School of Dentistry, Cardiff University, Heath Park, Cardiff CF14 4XY, United Kingdom
| | - A I Zhurov
- School of Dentistry, Cardiff University, Heath Park, Cardiff CF14 4XY, United Kingdom
| | - P Pirttiniemi
- Research Unit of Oral Health Sciences, Faculty of Medicine, University of Oulu, Oulu, Finland; Medical Research Center Oulu (MRC Oulu), Oulu University Hospital, Oulu, Finland
| | - T Heikkinen
- Research Unit of Oral Health Sciences, Faculty of Medicine, University of Oulu, Oulu, Finland; Medical Research Center Oulu (MRC Oulu), Oulu University Hospital, Oulu, Finland
| | - V Harila
- Research Unit of Oral Health Sciences, Faculty of Medicine, University of Oulu, Oulu, Finland; Medical Research Center Oulu (MRC Oulu), Oulu University Hospital, Oulu, Finland
| | - H Matthews
- Medical Imaging Research Center, UZ Leuven, 3000 Leuven, Belgium; Department of Human Genetics, KU Leuven, 3000 Leuven, Belgium; Facial Sciences Research Group, Murdoch Children's Research Institute, Melbourne; Department of Paediatrics, University of Melbourne, Melbourne, Australia
| | - P Claes
- Medical Imaging Research Center, UZ Leuven, 3000 Leuven, Belgium; Department of Human Genetics, KU Leuven, 3000 Leuven, Belgium; Department of Electrical Engineering, ESAT/PSI, KU Leuven, 3000 Leuven, Belgium
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Farnell DJJ, Khor C, Ayre WN, Doyle Z, Chadwick EA. Initial Investigations of the Cranial Size and Shape of Adult Eurasian Otters ( Lutra lutra) in Great Britain. J Imaging 2020; 6:106. [PMID: 34460547 PMCID: PMC8321200 DOI: 10.3390/jimaging6100106] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2020] [Revised: 09/30/2020] [Accepted: 10/04/2020] [Indexed: 11/24/2022] Open
Abstract
Three-dimensional (3D) surface scans were carried out in order to determine the shapes of the upper sections of (skeletal) crania of adult Eurasian otters (Lutra lutra) from Great Britain. Landmark points were placed on these shapes using a graphical user interface (GUI) and distance measurements (i.e., the length, height, and width of the crania) were found by using the landmark points. Male otters had significantly larger skulls than females (P < 0.001). Differences in size also occurred by geographical area in Great Britain (P < 0.05). Multilevel Principal Components Analysis (mPCA) indicated that sex and geographical area explained 31.1% and 9.6% of shape variation in "unscaled" shape data and that they explained 17.2% and 9.7% of variation in "scaled" data. The first mode of variation at level 1 (sex) correctly reflected size changes between males and females for "unscaled" shape data. Modes at level 2 (geographical area) also showed possible changes in size and shape. Clustering by sex and geographical area was observed in standardized component scores. Such clustering in a cranial shape by geographical area might reflect genetic differences in otter populations in Great Britain, although other potentially confounding factors (e.g., population age-structure, diet, etc.) might also drive regional differences. This work provides a successful first test of the effectiveness of 3D surface scans and multivariate methods, such as mPCA, to study the cranial morphology of otters.
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Affiliation(s)
- Damian J. J. Farnell
- School of Dentistry, Cardiff University, Heath Park, Cardiff CF14 4XY, UK; (C.K.); (W.N.A.); (E.A.C.)
| | - Chern Khor
- School of Dentistry, Cardiff University, Heath Park, Cardiff CF14 4XY, UK; (C.K.); (W.N.A.); (E.A.C.)
| | - Wayne Nishio Ayre
- School of Dentistry, Cardiff University, Heath Park, Cardiff CF14 4XY, UK; (C.K.); (W.N.A.); (E.A.C.)
| | - Zoe Doyle
- School of Biosciences, Cardiff University, Heath Park, Cardiff CF10 3AX, UK;
| | - Elizabeth A. Chadwick
- School of Dentistry, Cardiff University, Heath Park, Cardiff CF14 4XY, UK; (C.K.); (W.N.A.); (E.A.C.)
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