Robust Deep Neural Network Estimation for Multi-dimensional Functional Data

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Functional Data Regression Model

• : fixed vector of length d for the j-th observational point
• : scalar random variable for the i-th subject and j-th observational point
• : error random process with measurement error for the i-th subject and j-th observational point
• : sample size
• : number of observational points
• : true function to estimate

Deep Neural Network Model input and output

• Input: (uniform among all i for the same j)
• Output:

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Deep Neural Network Hyperparameters and Structures

• L: number of layers
• p: neurons per layer (uniform for all layers)
• s: dropout rate (data dependent)
• Loss function: absolute value loss/ square loss/ huber loss/ check loss
• Batch size: data dependent
• Epoch number: data dependent
• Activation function: ReLU
• Optimizer: Adam

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Function descriptions

• "rdnn.R": robust dnn estimation for multi-dimensional funtional data, with dimension no more than 4. More details can be found in the file.

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Examples

• "example.R": 2D and 3D functional data regression examples. Cauchy and Slash distributed measurement errors are added to the observations.