1.1 The registr model and algorithm

The main model for exponential family registration is

\[ \begin{eqnarray*} E\left[Y_i\left(h_i^{-1}(t_i^*)\right) | c_i, h_i^{-1} \right] &=& \mu_i(t) \\ g\left[\mu_i(t)\right]&=& \alpha(t) + \sum_{k = 1}^K c_{ik}\psi_k(t). \end{eqnarray*} \] For subject \(i\), inverse warping function \(h_i^{-1}\) maps unregistered time \(t_i^*\) to registered time \(t\) such that \(h_i^{-1}(t_i^*) = t\). \(Y_i\left(t_i^*\right)\) and \(Y_i\left(h_i^{-1}(t_i^*)\right)\) are the unregistered and registered response curves, respectively. The subject-specific means \(\mu_i(t)\) are related to the population-level mean \(\alpha(t)\) and a linear combination of population-level basis functions \(\psi(t)\) and subject-specific scores \(c_i\) through a known link function \(g\).

The registr algorithm is based on this model and iterates between the following steps:

1. Estimate subject-specific means \(\mu_i(t)\) using GFPCA, conditional on current estimate of \(h_i^{-1}(t_i^*)\).
2. Estimate inverse warping functions \(h_i^{-1}(t_i^*)\), conditional on current estimate of \(\mu_i(t)\).

The methods implemented in registr are described in more detail in this paper.

2.1 A note on data formatting

Use of this package requires that data be in a specific format: a long-form data frame with variables id, index, and value, where the value column contains functional observations for all subjects, the id column identifies which observations belong to which subject, and index provides the grid (domain) over which the values are observed.

The variable id should be a unique identifier in that each id identifies a single subject. Since we assume there is only one curve per subject for this package, id uniquely identifies each curve as well. Other covariates can be included in the data as long as the variables id, index, and value are present.

3.1 Simulate data for registration

simulate_unregistered_curves() generates curves with both unregistered and registered time grids.The code below generates data with \(I = 10\) subjects and \(D = 200\) using this function:

registration_data = simulate_unregistered_curves(I = 50, D = 200, seed = 2018)

head(registration_data)
#>   id       index value latent_mean           t
#> 1  1 0.000000000     0   -1.408706 0.000000000
#> 2  1 0.005025126     0   -1.447980 0.004501377
#> 3  1 0.010050251     0   -1.486200 0.009015300
#> 4  1 0.015075377     0   -1.523325 0.013541660
#> 5  1 0.020100503     0   -1.559314 0.018080346
#> 6  1 0.025125628     0   -1.594127 0.022631248

The resulting object,registration_data, is a data frame with variables id, value, index, latent_mean, and t, which is consistent with the format our registr software requires. id is the identifier for a particular subject, the value variable contains binary observations, and latent_mean contains continuous observations used to generate the binary observations for the value variable. Note that when family = "binomial" we will use the binary value variable as the observations for each subject and when family = "gaussian" we use the latent_mean variable as the outcome.

The variables index and t are both time grids. Evaluated on the grid index the data is unregistered, and on the grid t the data is registered. Registered and unregistered curves are plotted below.

Each curve has one main peak, but the location of that peak is shifted. When curves are registered the peaks are aligned.

3.2 Simulate data for GFPCA

simulate_functional_data() simulates data with a population-level mean and two orthogonal principal components based on sine and cosine functions. The code below generates data with \(I = 100\) subjects and \(D = 200\) time points per subject using this function:

fpca_data = simulate_functional_data(I = 100, D = 200, seed = 2018)

ls(fpca_data)
#> [1] "Y"     "alpha" "psi1"  "psi2"

head(fpca_data$Y)
#>   id value       index latent_mean
#> 1  1     0 0.000000000   -1.549878
#> 2  1     0 0.005025126   -1.540848
#> 3  1     0 0.010050251   -1.528362
#> 4  1     0 0.015075377   -1.512449
#> 5  1     1 0.020100503   -1.493144
#> 6  1     0 0.025125628   -1.470490

The resulting object,fpca_data, is a list that contains the true population-level mean (alpha) and principal components (psi1 and psi2), and a dataframe (Y). The dataframe Y contains variables id, value, index and latent_mean. This data is plotted below.

The left panel of the figure above shows the latent means for each subject, along with the population-level mean, \(\alpha(t)\), in red. The middle and right panels show the first and second principal components, \(\psi_1(t)\) and \(\psi_2(t)\), respectively. Using the \(logit^{-1}(\cdot)\) function we can convert the subject-specific means to probabilities; these probabilities are used to generate the binary values. Binary values and latent probability curve for one subject in the dataset is shown below.

We can alter the score variance for the principal components using the arguments lambda1 and lambda2. The default setting is for all subjects to have the same number of time points. However, by specifying vary_D = TRUE, we can generate data with uneven grid lengths for each subject.

4.1 Analyzing binary data

To register binary data use the following code:

registr_bin = register_fpca(Y = registration_data, family = "binomial", Kt = 8, Kh = 4, npc = 1, verbose = 2)

The argument Y specifies the input dataset; this code uses the simulated registration_data. Kt and Kh specify number of B-spline basis functions for the subject-specific means and warping functions, respectively, and npc indicates the number of functional principal components to use. The latter can also be chosen based on an explained share of variance, see argument npc_varExplained.

Underlying probabilities of the binary data are plotted above. At left probabilities on unregistered domain \(t^*\), center are probabilities on true registered domain \(t\), and at right are probabilities on estimated registered domain \(\widehat{t}\). After registration the underlying probabilities are aligned – though it is important to note that the algorithm registers based on the underlying binary observations, not the true probabilities.

The true and estimated warping functions are plotted above.

4.2 Analyzing gaussian data

To register continuous data use the following code:

registration_data$value = registration_data$latent_mean
registr_gauss = register_fpca(Y = registration_data, family = "gaussian", npc = 1, Kt = 10)

5.1 Binomial FPCA using bfpca()

The registr package includes a novel variational EM algorithm for binary functional principal component analysis (bfpca), derived from methods for binary probabilistic PCA (Tipping 1999).

This bfpca() function works for data that is sparse and irregular (subjects do not have to be observed on the same grid and do not have to have the same number of grid points), as well as dense, regularly observed data. The following code runs bfpca on the fpca_data dataset.

bfpca_object = bfpca(fpca_data$Y, npc = 2, Kt = 8, print.iter = TRUE)

The argument print.iter = TRUE prints the error after each iteration. The true and estimated population-level mean and FPCs are plotted below.

The algorithm runs quickly and does a good job recovering the true FPCs. Note that while the truth and estimation are not perfectly aligned, this is to be expected – the data used to estimate these functions are binary observations that are generated for the truth with some variability, so results are not expected to perfectly align. One would expect results to get better with increasing number of time points per subject.

In registr, the estimated FPCs can be easily visualized using the internal function plot.fpca. This function is automatically called when calling the general plot function on an object of class fpca.

if (have_ggplot2 && requireNamespace("cowplot", quietly = TRUE)) {
  registr:::plot.fpca(bfpca_object)
}

5.2 Generalized FPCA using gfpca_twoStep()

If register_fpca() is called with fpca_type = "two-step", the GFPCA step is performed with function gfpca_twoStep(). As Gertheiss, Goldsmith, and Staicu (2017) outline, in comparison to purely marginal and thus biased GFPCA approaches, this two-step approach can be seen as a “quick-fix” for the marginal approach of Hall, Müller, and Yao (2008) that works well in practice.

Our implementation is based on the codebase accompanying their paper which can be found at github.com/jeff-goldsmith/gfpca. We further adapted the functions to work more efficiently both regarding the need for computation time and RAM, especially for large data settings with thousands of curves.

7.1 Registering incomplete curves

Incomplete curves arise in many applications. Incompleteness refers to functional data where (some) curves were not observed from the very beginning and/or until the very end of the common domain. Such a data structure is e.g. observed in the presence of drop-out in panel studies.

The registr package offers the possibility to flexibly account for different types of incompleteness structures in the registration using registr() as well as in the joint approach using register_fpca(). All incomplete curve functionalities are outlined in the separate vignette "incomplete_curves".

7.2 Parametric inverse warping functions

The registr package currently supports two types of inverse warping functions: nonparmetric B-spline basis functions (default), or parametric 2-knot piecewise linear functions. With warping = "piecewise_linear2", the registration step estimates the \(x\) and \(y\) (or \(t_i^*\) and \(t\)) coordinates of each of the two knots to construct an inverse warping function that consists of 3 line segments.

To register data with parametric inverse warping functions, using the following code:

registration_data = simulate_unregistered_curves(I = 10, D = 50, seed = 2018)

registr_parametric = register_fpca(Y = registration_data, family = "binomial", 
                                   Kt = 8, Kh = 4, npc = 1, gradient = FALSE,
                                   warping = "piecewise_linear2")

This argument works for all families. Note that the gradient option is currently unavailable for parametric inverse warping functions. Below are the resulting inverse warping functions from both the nonparametric (left) and piecewise_linear2 (right) specifications. The slopes of the 3 line segments that make up one’s parametric function can provide some interpretation about how that particular subject’s data was warped to align with the population mean.

Beyond interpretability, another advantage of parametric inverse warping functions is the ability to specify prior information about how they should look. The register_fpca() function can include normally-distributed priors which pull warping functions toward the identity line. Specifically, the priors pull the knot coordinates toward (0.25, 0.25) and (0.75, 0.75).

To activate the priors, one must specify priors = TRUE and choose a value for the argument prior_sd, the standard deviation that will be applied to all 4 prior distributions.

registr_par_priors = register_fpca(Y = registration_data, family = "binomial", 
                                   Kt = 8, Kh = 4, npc = 1, gradient = FALSE,
                                   warping = "piecewise_linear2",
                                   priors = TRUE, prior_sd = 0.1)

As expected, a smaller variance lead to stronger tendency toward the prior means. This is demonstrated in the warping function plots below.

The ability to specify priors is currently only available for binary curve registration, not Gaussian.

7.3 Periodic B-spline basis functions for GFPCA

In some cases it may be of interest to place periodic boundary conditions on the B-spline basis functions for the population-level and subject-specific mean templates. The periodic conditions ensure that the resulting function starts and ends with the same value. This may be useful when modeling cyclical data, such as daily physical activity patterns. To use periodic B-spline basis functions during registration, use the option periodic = TRUE:

registr_periodic = register_fpca(Y = registration_data, family = "binomial", 
                                 Kt = 8, Kh = 4, npc = 1, gradient = FALSE,
                                 periodic = TRUE)

Note that the gradient option is currently unavailable for periodic = TRUE.

The resulting population mean and principal component from registering the continuous data using periodic = FALSE (default) and periodic = TRUE are plotted below, with dotted lines to demonstrate the unification of the start and end of the periodic functions.

7.4 Choosing the template function

By default registr() estimates the overall mean of all curves and uses this mean curve as the template function to which all curves are registered. In the joint approach (register_fpca()) this mean curve is used as template for an initial registration step, before the main iteration between registration and GFPCA starts.

In some situations the overall mean is not the most reasonable choice for the template. See for example the following curves with a trailing incompleteness structure where some processes were not observed until their very end. The shape of the mean curve does not match any of the observed curve shapes well.

Performing the registration with this mean curve as template would lead to the following result:

reg1 = registr(Y = temp_dat, family = "gaussian", Kh = 4,
               incompleteness = "trailing", lambda_inc = 0)
if (have_ggplot2) {
  ggplot(reg1$Y, aes(x = index, y = value, col = id)) + 
    geom_line() +
    ggtitle("Registration with overall mean (black) as template function")
}

Alternatively, the template function can be manually defined using one of the following options:

1. Define the template as the mean curve only based on some subset of the observed curves Y
2. Define the template as one observed curve
3. Define the template curve independently from the observed curves

The subset of curves (for option 1) or one specific curve (for options 2 and 3) can be specified using the argument Y_template. In the following example, the red curve with id 1 is used as template:

Y_template = temp_dat %>% filter(id == 1)
reg2 = registr(Y = temp_dat, family = "gaussian", Kh = 4, Y_template = Y_template,
               incompleteness = "trailing", lambda_inc = 0)
if (have_ggplot2) {
  ggplot(reg2$Y, aes(x = index, y = value, col = id)) +
    geom_line() +
    ggtitle("Registration with red curve as template")
}