autoFRK 1.1.1

autoFRK: Automatic Fixed Rank Kriging. The Python version with PyTorch

autoFRK-python

autoFRK-python is a Python implementation of the R package autoFRK v1.4.3 (Tzeng S et al., 2021). autoFRK provides a Resolution Adaptive Fixed Rank Kriging (FRK) approach for handling regular and irregular spatial data, reducing computational cost through multi-resolution basis functions.

Features

• Spatial modeling based on multi-resolution basis functions
• Supports single or multiple time points
• Offers approximate or EM-based model estimation
• Suitable for global latitude-longitude data
• Implemented in PyTorch, supporting CPU and GPU (requires PyTorch with CUDA support for GPU)

## Main Functions

• AutoFRK

  Automatic Fixed Rank Kriging.

• MRTS

  Multi-Resolution Thin-Plate Spline basis function.

Installation

Install via pip:

pip install autoFRK

Install directly from GitHub:

pip install git+https://github.com/Josh-test-lab/autoFRK-python.git

Or clone and install manually:

git clone https://github.com/Josh-test-lab/autoFRK-python.git
cd autoFRK-python
pip install .

Usage

1. Import and Initialize

import torch
from autoFRK import AutoFRK

# Initialize the autoFRK model
model = AutoFRK(dtype=torch.float64, device="cpu")

2. Model Fitting

# Assume `data` is (n, T) observations (NA allowed) and `loc` is (n, d) spatial coordinates  corresponding to n locations
data = torch.randn(100, 1)  # Example data
loc = torch.rand(100, 2)    # Example 2D coordinates

model_object = model.forward(
    data=data,
    loc=loc,
    maxit=50,
    tolerance=1e-6,
    method="fast",          # "fast" or "EM"
    n_neighbor=3
)

print(result.keys())
# ['M', 's', 'negloglik', 'w', 'V', 'G', 'LKobj']

forward() returns a dictionary including:

• M: ML estimate of M.
• s: Estimate for the scale parameter of measurement errors.
• negloglik: Negative log-likelihood.
• w: K by T matrix with w[t] as the t-th column.
• V: K by K matrix of the prediction error covariance matrix of w[t].
• G: User specified basis function matrix or an automatically generated object from MRTS.
• LKobj: Not used yet.

3. Predicting New Data

# Assume `newloc` contains new spatial coordinates
newloc = torch.rand(20, 2)

pred = model.predict(
    obj=result,
    newloc=newloc,
    se_report=True
)

print(pred['pred.value'])  # Predicted values
print(pred.get('se'))            # Standard errors

predict() can optionally return standard errors (se_report=True). If obj is not provided, the most recent forward() result is used.

Arguments

• AutoFRK

AutoFRK.forward() supports various parameters:

Parameter	Description	Type	Default
data	n by T data matrix (NA allowed) with z[t] as the t-th column.	torch.Tensor	(Required)
loc	n by d matrix of coordinates corresponding to n locations.	torch.Tensor	(Required)
mu	n-vector or scalar for µ.	float | torch.Tensor	0
D	n by n matrix (preferably sparse) for the covariance matrix of the measurement errors up to a constant scale.	torch.Tensor	Identity matrix
G	n by K matrix of basis function values with each column being a basis function taken values at loc. Automatically determined if None.	torch.Tensor	None
maxK	Maximum number of basis functions considered. Default is None, which means 10 · √n (for n > 100) or n (for n ≤ 100).	int	None
Kseq	User-specified vector of numbers of basis functions considered. Default is None, which is determined from maxK.
maxknot	Maximum number of knots used in generating basis functions.	torch.Tensor	None
method	"fast" or "EM"; "fast" fills missing data using k-nearest-neighbor imputation, while "EM" handles missing data via the EM algorithm.	str	"fast"
n_neighbor	Number of neighbors used in the "fast" imputation method.	int	3
maxit	Maximum number of iterations used in the "EM" imputation method.	int	50
tolerance	Precision tolerance for convergence check used in the "EM" imputation method.	float	1e-6
requires_grad	If True, enables gradient computation for data tensor.	bool	False
calculate_with_spherical	If True, calculates thin-plate spline distances using spherical coordinates.	bool	False
finescale	Logical; if True, an (approximate) stationary finer-scale process η[t] will be included based on the LatticeKrig package. Only the diagonals of D are used. (Not used yet)	bool	FALSE
dtype	Data type used in computations (e.g.,torch.float64).	torch.dtype	torch.float64
device	Target computation device ("cpu", "cuda", "mps", etc.). If None, automatically selected.	torch.device | str	None

AutoFRK.predict() supports various parameters:

Parameter	Description	Type	Default
obj	A model object obtained from AutoFRK. If None, the model object produced by the forward method will be used.	dict | None	None
obsData	A vector with observed data used for prediction. Default is None, which uses the data input from obj.	torch.Tensor | None	None
obsloc	A matrix with rows being coordinates of observation locations for obsData. Only objects using mrts basis functions can have obsloc different from the loc input of object. Default is None.	torch.Tensor | None	None
mu_obs	A vector or scalar for the deterministic mean values at obsloc.	float | torch.Tensor	0
newloc	A matrix with rows being coordinates of new locations for prediction. Default is None, which gives prediction at the locations of the observed data.	torch.Tensor | None	None
basis	A matrix with each column being a basis function taken values at newloc. Can be omitted if object was fitted using default MRTS basis functions.	torch.Tensor | None	None
mu_new	A vector or scalar for the deterministic mean values at newloc.	float | torch.Tensor	0
se_report	Logical; if True, the standard error of prediction is reported.	bool	False
dtype	Data type used in computations (e.g., torch.float64). Defaults to the dtype of the model obj if available	torch.dtype	torch.float64
device	Target device for computations (e.g., 'cpu', 'cuda', 'mps'). If None, it will be selected automatically, with the device of the model obj used first if available.	torch.device | str	None

• MRTS

MRTS.forward() supports various parameters:

Parameter	Description	Type	Default
knot	m by d matrix (d ≤ 3) for m locations of d-dimensional knots as in ordinary splines. Missing values are not allowed.	torch.Tensor	(Required)
k	The number (≤m) of basis functions.	int	None
x	n by d matrix of coordinates corresponding to n locations where the values of basis functions are to be evaluated. Default is None, which uses the m by d matrix in knot.	torch.Tensor | None	None
maxknot	Maximum number of knots to be used in generating basis functions. If maxknot <m, a deterministic subset selection of knots will be used. To use all knots, set maxknot ≥ m.	int	5000
calculate_with_spherical	If True, calculates thin-plate spline distances using spherical coordinates.	bool	False
dtype	Data type used in computations (e.g.,torch.float64).	torch.dtype	torch.float64
device	Target computation device ("cpu", "cuda", "mps", etc.). If None, automatically selected.	torch.device | str	None

MRTS.predict() supports various parameters:

Parameter	Description	Type	Default
obj	A model object obtained from MRTS. If None, the model object produced by the forward method will be used.	dict | None	None
newx	n by d matrix of coordinates corresponding to n locations where prediction is desired.	torch.Tensor | None	None
calculate_with_spherical	If True, calculates thin-plate spline distances using spherical coordinates. The default None uses the method specified in forward.	bool | None	None
dtype	Data type used in computations (e.g., torch.float64). Defaults to the dtype of the model obj if available	torch.dtype	torch.float64
device	Target device for computations (e.g., 'cpu', 'cuda', 'mps'). If None, it will be selected automatically, with the device of the model obj used first if available.	torch.device | str	None

Example Code

• AutoFRK

import torch
from autoFRK import AutoFRK

# Generate fake data
n, T = 200, 1
data = torch.randn(n, T)
loc = torch.rand(n, 2)

# Initialize model
model = AutoFRK(device="cpu")

# Fit model
res = model.forward(
    data=data,
    loc=loc
)

# Predict new data
newloc = torch.rand(10, 2)
pred = model.predict(
    newloc=newloc
)

print("Predicted values:", pred['pred.value'])

• MRTS

import torch
from autoFRK import MRTS

Experimental Features

• Spherical coordinate basis function computation
• Gradient tracking (using torch's requires_grad_())

Authors

• ShengLi Tzeng — Original Paper Author
• Hsin-Cheng Huang — Original Paper Author
• Wen-Ting Wang — R Package Author
• Yao-Chih Hsu — Python Package Author

Contributors

• Hao-Yun Huang — Spherical Coordinate Thin Plate Spline Provider
• Yi-Xuan Xie — Python Package Tester
• Xuan-Chun Wang — Python Package Tester

License

• GPL (>= 3)

Development and Contribution

• Report bugs or request features on GitHub issues

References

• Tzeng S, Huang H, Wang W, Nychka D, Gillespie C (2021). autoFRK: Automatic Fixed Rank Kriging. R package version 1.4.3, https://CRAN.R-project.org/package=autoFRK
• Wang, J. W.-T. (n.d.). autoFRK. GitHub. Retrieved January 7, 2023, from https://egpivo.github.io/autoFRK/
• Tzeng, S. & Huang, H.-C. (2018). Resolution Adaptive Fixed Rank Kriging. Technometrics. https://doi.org/10.1080/00401706.2017.1345701
• Nychka, D., Hammerling, D., Sain, S., & Lenssen, N. (2016). LatticeKrig: Multiresolution Kriging Based on Markov Random Fields

Citation

• To cite the Python package autoFRK-python in publications use:

  Tzeng S, Huang H, Wang W, Hsu Y (2025). _autoFRK-python: Automatic Fixed Rank Kriging. The Python version with PyTorch_. Python package version 1.1.1, 
  <https://pypi.org/project/autoFRK/>.

• A BibTeX entry for LaTeX users to cite the Python package is:

  @Manual{,
    title = {autoFRK-python: Automatic Fixed Rank Kriging. The Python version with PyTorch},
    author = {ShengLi Tzeng and Hsin-Cheng Huang and Wen-Ting Wang and Yao-Chih Hsu},
    year = {2025},
    note = {Python package version 1.1.1},
    url = {https://pypi.org/project/autoFRK/},
  }

• To cite the R package autoFRK in publications use:

  Tzeng S, Huang H, Wang W, Nychka D, Gillespie C (2021). _autoFRK: Automatic Fixed Rank Kriging_. R package version 1.4.3,
  <https://CRAN.R-project.org/package=autoFRK>.

• A BibTeX entry for LaTeX users to cite the R package is:

  @Manual{,
    title = {autoFRK: Automatic Fixed Rank Kriging},
    author = {ShengLi Tzeng and Hsin-Cheng Huang and Wen-Ting Wang and Douglas Nychka and Colin Gillespie},
    year = {2021},
    note = {R package version 1.4.3},
    url = {https://CRAN.R-project.org/package=autoFRK},
  }

Release Notes

v1.1.1

• Fixed a ValueError caused by a missing v in the model object when using the "EM" method.
• Fixed an issue with absent indices in the EM0miss function when using the "EM" method with missing data.
• Fixed a bug in the EM0miss function where some variables could not be found when handling missing data with the "EM" method.
• Improved the handling of device selection to reduce redundant checks and repeated triggers.
• Added input validation for the mu and mu_new variable.
• Updated additional functions to fully support requires_grad.
• Update README.

v1.1.0

• Added dtype and device parameters to AutoFRK.predict() and MRTS.predict().
• Added logger_level parameter to AutoFRK.__init__() and MRTS.__init__() (default: 20). Options include NOTSET(0), DEBUG(10), INFO(20), WARNING(30), ERROR(40), CRITICAL(50).
• Enhanced automatic device selection, including MPS support.
• Fixed device assignment issue when device is not specified, preventing redundant parameter transfers.

v1.0.0

• Ported R package autoFRK to Python.

Repositories

• Python Repository: https://github.com/Josh-test-lab/autoFRK-python
• R Repository: https://github.com/egpivo/autoFRK

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