Bayesian function registration with random truncation

Question

1. What are the advantages of incorporating random truncation in Bayesian models for function registration?

2. What transformations are applied to the observation (data) space and parameter space in function registration?

3. What is the main parameter of interest in the model?

4. What are the two mechanisms for the prior distribution of random truncation T?

Accepted Answer

The advantages of incorporating random truncation in Bayesian models for function registration are threefold. First, the level of smoothness of the functional phase parameter is informed by the data, avoiding potential mis-specifications of the number of basis functions. Second, it allows flexibly incorporating prior beliefs or desired constraints on the shape of the functional phase parameter, such as controlling the amount of shape-alteration in the observed functions. Third, the model allows inference on the random truncation parameter, providing additional information about the shapes of the functions in the data, such as keeping a larger number of basis functions for the phase parameter when the observed functions exhibit a lot of local features that must be registered.

Accepted Answer

For the observation (data) space, the square-root velocity transformation Qdf ThdtTh 1/4 signdf 0 dtThTh ffi ffi ffi ffi ffi ffi ffi ffi ffi jf 0 dtThj p dq; g is used to transform the functions. The resulting function, denoted by q, belongs to the transformed observation space Q, which is a subset of L 2 d 1/20; 1Th [23]. For the parameter space, two transformations are applied: the square-root velocity transformation QdgThdtTh 1/4 signdg 0 dtThTh ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi jg 0 dtThj p 1/4 ffi ffi ffi ffi ffi ffi ffi ffi ffi g 0 dtTh p cdtTh; EQUATION The first transformation is the square-root velocity transformation in Eq (1) (note that g 0 (t) > 0 8 t). The second transformation is the inverse exponential map for a unit sphere that allows us to linearize the space C, which is a transformed representation space of the warping functions. The resulting function, denoted by g, belongs to a subset of a linear space, defined as A {g 2 T 1 (C)|exp 1 (g) > 0} (the notation T 1 (C) refers to the tangent space of C at the function 1; see S1 File for details).

Accepted Answer

The main parameter of interest in the model is the warping function g2G, represented via g2A. It is crucial in modeling the difference between q1([t]) and (q2, g)([t]) by a zero-mean N-dimensional Gaussian distribution. The warping function g2G is fully specified by the pair (g, T), where g is a function of g and T, and T is a prior distribution for T. The model uses a Gaussian process to model g and a general distribution tT that does not depend on g to model T. The pair (g, T) results in a valid warping function by restricting the joint prior to the domain Bfdg; T: exp1dGTh > 0g. The full model is given by Model 1, which includes the likelihood function Ldg; T; EQUATION. This likelihood function is identical to that of [11, 19], except that the truncated parameter g replaces the parameter g.

Accepted Answer

The two mechanisms for the prior distribution of random truncation T are: (1) Random Number of Basis Functions, where TM represents the number of basis functions used to represent the parameter g, and (2) Random Indicators, where a sequence {kh i } i = 1, . .., 1, kh i 2 {0, 1} controls which basis functions are kept in the basis expansion of g. The first mechanism involves a prior distribution on the set of positive integers, while the second mechanism uses a random sequence to switch basis functions on and off, resulting in a prior distribution on the collection of vectors X.

Accepted Answer

The posterior distribution is sampled via Markov chain Monte Carlo by iteratively drawing from the full conditional distributions of (g, T) and s 2 1. The algorithm updates the posterior at each step using a Metropolis-within-Gibbs approach. For sampling (g, T), a Z-mixture proposal is used, where g is updated by drawing from a Gaussian process prior and a tuning parameter b z is used to determine the proposal distribution. The Karhunen-Loeve expansion is used to sample a new function x from Gaussiand0; C g Th. The proposal for T is generated based on the density Q T djTh, and the acceptance ratio is calculated using the dominating prior Gaussian measure. For sampling s 2 1, a conjugate inverse-gamma distribution is used with shape parameter N 2 th a and scale parameter 1 2 SSEdg Th th b.

Accepted Answer

The multiple function Bayesian registration model with random basis truncation is an extension of Model (1) that aims to make inference on g 1; . . .; g C, determined by the pairs (g 1 , T 1 ). .., (g C , T C ). The truncated template function q* is treated as a parameter, and the same truncation mechanisms as described in the pairwise case are considered. A Gaussian process prior is assigned to q*, and a prior t*T to the random truncation T q. The likelihood function is given by EQUATION, and sampling from the posterior distribution is performed using a Metropolis-within-Gibbs algorithm. The algorithm updates each of the truncated warping parameters sequentially and updates the template function q* with an acceptance ratio. The model uses priors and proposals for T i and T q, and updates s 2 1 by drawing from an inverse-gamma distribution.

Accepted Answer

In the simulation study, the proposed pairwise Bayesian registration model is assessed by simulating two observed functions, f1 and f2, both warped versions of a template function. The true warping g_true is randomly generated, and f1 is constructed as f2 multiplied by g_true. The model is then applied to register f2 to f1, obtaining g_est, an estimate of the true warping. The performance is compared to five sets of true warping functions, each with varying degrees of local features. The proposed model is compared to the model in [11] using 20 basis functions (M20). The results show how well the proposed model performs in registering the observed functions and estimating the true warping.

Accepted Answer

The inverse-gamma distribution with parameters a=0.1 and b=0.1 is used for s21. This prior distribution is chosen to reflect beliefs about the smoothness of the warping functions. A smaller mean in the Poisson priors (pois20, pois50, pois80) indicates a belief in smoother warping functions, while a larger mean suggests less smoothness. The choice of prior distributions allows for flexibility in modeling the warping functions based on prior knowledge or assumptions about their smoothness.

Accepted Answer

The deterministic Dynamic Programming (DP) algorithm is used for pairwise registration in the implementation stage. This algorithm is implemented in the R package fdasrvf [27]. Pairwise registration involves aligning and comparing sets of warping functions to establish correspondences between them. The DP algorithm provides a good starting point for the MCMC sampling algorithm, allowing the chain to mix faster and improving the overall performance of the Bayesian model. The S1 File includes examples with different starting points to demonstrate the algorithm's effectiveness. Overall, the DP algorithm plays a crucial role in the implementation stage by facilitating accurate registration of warping functions.

Accepted Answer

The proposed Bayesian model offers several advantages over the M20 model in terms of registration performance and flexibility. Firstly, the Bayesian model allows for registration of more local features, as seen in Fig 4, where the model caplow generates estimated warping functions based on at least 30 basis elements even when the true warping function is very smooth. In contrast, the M20 model struggles to produce a good template for functions with different shapes, such as neural spike trains and handwriting curves. Secondly, the Bayesian model provides a more principled exploration of the parameter space, allowing for a more informed inference on the level of smoothness of the warping function. This is evident in Fig 3, where the posterior means of the number of active basis functions are influenced by the underlying data, regardless of the prior chosen for the random truncation T. Additionally, the Bayesian model offers flexibility in incorporating constraints on the level of smoothness via the prior distribution. For example, the model pois20 generates fewer active basis functions than the model pois80, regardless of the level of smoothness in the observed functions. Overall, the proposed Bayesian model outperforms the M20 model in terms of registration performance and provides a more adaptable approach to modeling and registration.

Accepted Answer

The five application domains for the proposed multiple function Bayesian registration model are: 1. Right knee flexion and pelvis right roll, 2. Growth rates, 3. Pinch force, and two unspecified domains. The first domain involves gait variables for 12 participants, linearly scaled to observe on the same time interval. The second domain uses growth rate functions from the Berkeley growth study, widely used to assess registration performance. The third domain records pinch force exerted by thumb and forefingers, collected for 20 test subjects, with varying starting times and force durations, necessitating registration for temporal variation prior to analysis.

Accepted Answer

The Fourier basis is used to represent the template function in the Gaussian process prior. It is especially suitable when the observed functions are periodic on [0, 1], such as gait cycle variables. For other datasets, the Fourier period is set to 2. The Fourier basis with corresponding eigenvalues l2i1/4s2qiA1:2 is specified, where s2q is fixed based on the scale of the observed functions. This choice of prior yields good registration results across different shapes of observed functions and is robust to alternative prior choices for model parameters. The Fourier basis allows for a discrete uniform prior on [5, M max = 200] for the number of basis functions M, ensuring a minimum level of complexity while allowing for as many local features as possible. This approach is suitable when there is no strong prior information for the warping parameter and a desire to explore the parameter space thoroughly.

Accepted Answer

The main contribution of the proposed method lies in the randomization of the truncation process. This is achieved by introducing a new random truncation parameter that controls how many or which basis functions are used to represent the functional parameters. The resulting Bayesian models can explore the full parameter space instead of a small subset of a truncated parameter space. This allows for more flexibility and better inference on the amount of truncation that has occurred, helping to detect significant shape changes during registration. The proposed method also avoids potential mis-specifications of the truncation parameter by informing the posterior distribution on the truncation parameter with both the data and the prior. Additionally, it enables fine-tuning of the models based on the application of interest, making it a versatile and effective approach for pairwise and multiple function registration.

Bayesian function registration with random truncation

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Most frequently asked questions

1. What are the advantages of incorporating random truncation in Bayesian models for function registration?

2. What transformations are applied to the observation (data) space and parameter space in function registration?

3. What is the main parameter of interest in the model?

4. What are the two mechanisms for the prior distribution of random truncation T?

5. How is the posterior distribution sampled via Markov chain Monte Carlo in the given section?

6. What is the multiple function Bayesian registration model with random basis truncation?

7. How does the proposed pairwise Bayesian registration model perform in a simulation study?

8. What are the prior distributions for s21?

9. What algorithm is used for pairwise registration in implementation stage?

10. What are the advantages of the proposed Bayesian model over the M20 model in terms of registration performance and flexibility?

11. What are the five application domains for the proposed multiple function Bayesian registration model?

12. What is the Fourier basis used for?

13. What is the main contribution of the proposed method?

References

Functional Data Analysis

Inverse problems: A Bayesian perspective

MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster

Shape Analysis of Elastic Curves in Euclidean Spaces

Functional and Shape Data Analysis

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