lme.rst

Mixed Effects / Multilevel Models

TODO: description...

Linear Mixed Effects Model via Design Matrices

TODO: one way to do it is to manually encode the variables into design matrices...

y ∼ Normal(Xw + Zu, σ)

where

• X is the fixed variables' design matrix,
• Z is the random variables' design matrix,
• w is the vector of fixed-effects coefficients,
• u is the vector of random effects coefficients, and
• σ is the noise standard deviation.

TensorFlow

import probflow as pf

class LinearMixedEffectsModel(pf.Model):

    def __init__(self, Nf, Nr):
        self.Nf = Nf
        self.w = pf.Parameter([Nf, 1])
        self.u = pf.Parameter([Nr, 1])
        self.sigma = pf.ScaleParameter([1, 1])

    def __call__(self, x):
        X = x[:, :self.Nf]
        Z = x[:, self.Nf:]
        return pf.Normal(X @ self.w() + Z @ self.u(), self.sigma())

PyTorch

import probflow as pf
import torch

class LinearMixedEffectsModel(pf.Model):

    def __init__(self, Nf, Nr):
        self.Nf = Nf
        self.w = pf.Parameter([Nf, 1])
        self.u = pf.Parameter([Nr, 1])
        self.sigma = pf.ScaleParameter([1, 1])

    def __call__(self, x):
        x = torch.tensor(x)
        X = x[:, :self.Nf]
        Z = x[:, self.Nf:]
        return pf.Normal(X @ self.w() + Z @ self.u(), self.sigma())