用于行为和神经数据分析的非线性回归模型
Non-linear regression models for behavioral and neural data analysis
摘要：
回归模型是经验科学中的流行工具，用于推断给定实验数据集的一组变量对因变量的影响。在神经科学和认知心理学中，广义线性模型（GLM）-包括线性回归，逻辑斯谛回归和泊松GLM-是研究参与者选择，反应时间和神经激活的因素的选择回归模型。然而，这些方法是有限的，因为它们仅捕获每个回归变量的线性贡献。在这里，我们介绍了一个扩展的GLM称为广义无限制模型（GUM），它允许推断出更丰富的一组贡献的回归变量的因变量，包括回归变量之间可能的相互作用。在GUM中，每个回归量通过线性或非线性函数传递，并且不同的所得变换回归量的贡献可以求和或相乘以生成因变量的预测值。我们提出了一个贝叶斯处理这些模型中，我们赋予功能与高斯过程先验，我们提出了两种方法来计算后验的功能给定的数据集：拉普拉斯方法和稀疏变分方法，它更好地为大型数据集的规模。对于每种方法，我们评估模型估计的质量，并详细说明如何拟合超参数（例如定义函数的预期平滑度）。最后，我们说明了权力的行为数据集，受试者报告的一系列光栅的平均感知方向的方法。该方法允许恢复光栅角度到每个主体的感知证据上的映射，以及基于其位置的光栅的影响。总的来说，GUM提供了一个非常丰富和灵活的框架来运行神经科学、心理学等领域的非线性回归分析。
英文摘要：
Regression models are popular tools in empirical sciences to infer the influence of a set of variables onto a dependent variable given an experimental dataset. In neuroscience and cognitive psychology, Generalized Linear Models (GLMs) -including linear regression, logistic regression, and Poisson GLM- is the regression model of choice to study the factors that drive participant's choices, reaction times and neural activations. These methods are however limited as they only capture linear contributions of each regressors. Here, we introduce an extension of GLMs called Generalized Unrestricted Models (GUMs), which allows to infer a much richer set of contributions of the regressors to the dependent variable, including possible interactions between the regressors. In a GUM, each regressor is passed through a linear or nonlinear function, and the contribution of the different resulting transformed regressors can be summed or multiplied to generate a predictor for the dependent variable. We propose a Bayesian treatment of these models in which we endow functions with Gaussian Process priors, and we present two methods to compute a posterior over the functions given a dataset: the Laplace method and a sparse variational approach, which scales better for large dataset. For each method, we assess the quality of the model estimation and we detail how the hyperparameters (defining for example the expected smoothness of the function) can be fitted. Finally, we illustrate the power of the method on a behavioral dataset where subjects reported the average perceived orientation of a series of gratings. The method allows to recover the mapping of the grating angle onto perceptual evidence for each subject, as well as the impact of the grating based on its position. Overall, GUMs provides a very rich and flexible framework to run nonlinear regression analysis in neuroscience, psychology, and beyond.
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