Abstract
Assume we would like to predict variables y from variables x through a function f(x) such that the squared deviations between actual and predicted values are minimized (a so-called squared error loss function, see Eq. 1.11). Then the regression function which optimally achieves this is given by f(x) = E(y|x) (Winer 1971; Bishop 2006; Hastie et al. 2009), that is the goal in regression is to model the conditional expectancy of y (the “outputs” or “responses”) given x (the “predictors” or “regressors”). For instance, we may have recorded in vivo the average firing rate of p neurons on N independent trials i, arranged in a set of row vectors X = {x 1,…, x i ,…, x N }, and would like to see whether with these we can predict the movement direction (angle) y i of the animal on each trial (a “decoding” problem). This is a typical multiple regression problem (where “multiple” indicates that we have more than one predictor). Had we also measured more than one output variable, e.g., several movement parameters like angle, velocity, and acceleration, which we would like to set in relation to the firing rates of the p recorded neurons, we would get into the domain of multivariate regression.
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References
Aarts, E., Verhage, M., Veenvliet, J.V., Dolan, C.V., van der Sluis, S.: A solution to dependency: using multilevel analysis to accommodate nested data. Nat. Neurosci. 17, 491–496 (2014)
Balaguer-Ballester, E., Lapish, C.C., Seamans, J.K., Daniel Durstewitz, D.: Attractor dynamics of cortical populations during memory-guided decision-making. PLoS Comput. Biol. 7, e1002057 (2011)
Bishop, C.M.: Pattern Recognition and Machine Learning. Springer, New York (2006)
Brette, R., Gerstner, W.: Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. J. Neurophysiol. 94, 3637–3642 (2005)
Buzsaki, G., Draguhn, A.: Neuronal oscillations in cortical networks. Science. 304, 1926–1929 (2004)
Cleveland, W.S.: Robust locally weighted regression and smoothing scatterplots. J. Am. Stat. Assoc. 74, 829–836 (1979)
Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst. 2, 303–314 (1989)
Demanuele, C., Kirsch, P., Esslinger, C., Zink, M., Meyer-Lindenberg, A., Durstewitz, D.: Area-specific information processing in prefrontal cortex during a probabilistic inference task: a multivariate fMRI BOLD time series analysis. PLoS One. 10, e0135424 (2015b)
Duchi, J., Hazan, E., Singer, Y.: Adaptive subgradient methods for online learning and stochastic optimization. J. Mach. Learn. Res. 12, 2121–2159 (2011)
Duda, R.O., Hart, P.E.: Pattern Classification and Scene Analysis. Wiley, New York (1973)
Fahrmeir, L., Tutz, G.: Multivariate Statistical Modelling Based on Generalized Linear Models. Springer, New York (2010)
Fan, J., Yao, Q.: Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York (2003)
Friston, K.J., Harrison, L., Penny, W.: Dynamic causal modelling. Neuroimage. 19, 1273–1302 (2003)
Graves, A., Wayne, G., Reynolds, M., Harley, T., Danihelka, I., et al.: Hybrid computing using a neural network with dynamic external memory. Nature. 538, 471–476 (2016)
Haase, R.F.: Multivariate General Linear Models. SAGE, Thousand Oaks, CA (2011)
Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning (Vol. 2, No. 1) Springer, New York (2009)
Hertz, J., Krogh, A.S., Palmer, R.G.: Introduction to the theory of neural computation. Addison-Wesley, Reading, MA (1991)
Hoerl, A.E., Kennard, R.W.: Ridge regression: biased estimation for nonorthogonal problems. Technometrics. 12, 55–67 (1970)
Hotelling, H.: Relations between two sets of variants. Biometrika. 28, 321–377 (1936)
Kim, J., Calhoun, V.D., Shim, E., Lee, J.H.: Deep neural network with weight sparsity control and pre-training extracts hierarchical features and enhances classification performance: evidence from whole-brain resting-state functional connectivity patterns of schizophrenia. NeuroImage. 124, 127–146 (2016)
Kohonen, T.: Self-Organising and Associative Memory. Springer, Berlin (1989)
Kriegeskorte, N.: Deep neural networks: a new framework for modeling biological vision and brain information processing. Annu. Rev. Vis. Sci. 1, 417–446 (2015)
Krzanowski, W.J.: Principles of Multivariate Analysis. A User’s Perspective, Rev. edn. Oxford Statistical Science Series. OUP, Oxford (2000)
Lapish, C.C., Balaguer-Ballester, E., Seamans, J.K., Phillips, A.G., Durstewitz, D.: Amphetamine exerts dose-dependent changes in prefrontal cortex attractor dynamics during working memory. J. Neurosci. 35, 10172–10187 (2015)
LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature. 521, 436–444 (2015)
Liu, W., Wang, Z., Liu, X., Zeng, N., Liu, Y., Alsaadi, F.E.: A survey of deep neural network architectures and their applications. Neurocomputing. 234, 11–26 (2017)
McDonald, G.C.: Ridge regression. WIREs Comp. Stat. 1, 93–100 (2009)
Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A.A., Veness, J., Bellemare, M.G., Graves, A., Riedmiller, M., Fidjeland, A.K., Ostrovski, G., Petersen, S., Beattie, C., Sadik, A., Antonoglou, I., King, H., Kumaran, D., Wierstra, D., Legg, S., Hassabis, D.: Human-level control through deep reinforcement learning. Nature. 518, 529–533 (2015)
Murayama, Y., Biessmann, F., Meinecke, F.C., Müller, K.R., Augath, M., Oeltermann, A., Logothetis, N.K.: Relationship between neural and hemodynamic signals during spontaneous activity studied with temporal kernel CCA. Magn. Reson. Imaging. 28, 1095–1103 (2010)
Naundorf, B., Wolf, F., Volgushev, M.: Unique features of action potential initiation in cortical neurons. Nature. 20, 1060–1063 (2006)
Obenchain, R.L.: Classical F-tests and confidence regions for ridge regression. Technometrics. 19, 429–439 (1977)
Ohiorhenuan, I.E., Mechler, F., Purpura, K.P., Schmid, A.M., Hu, Q., Victor, J.D.: Sparse coding and high-order correlations in fine-scale cortical networks. Nature. 466, 617–621 (2010)
Petersen, K.B., Pedersen, M.S.: The Matrix Cookbook. www2.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf/imm3274.pdf (2012)
Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science. 290, 2323–2326 (2000)
Rumelhart, D.E., Hinton, G.E., Williams, R.J.: Learning representations by back-propagating errors. Nature. 323, 533–536 (1986)
Rumelhart, D.E., McClelland, J.E.: Parallel Distributed Processing. MIT Press, Cambridge, MA (1986)
Schmidhuber, J.: Deep learning in neural networks: an overview. Neural Netw. 61, 85–117 (2015)
Schneidman, E., Berry, M.J., Segev, R., Bialek, W.: Weak pairwise correlations imply strongly correlated network states in a neural population. Nature. 440, 1007–1012 (2006)
Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B. 58, 267–288 (1996)
Ruder, S.: An overview of gradient descent optimization algorithms. arXiv:1609.04747 (2016)
West, B.T., Welch, K.B., Galecki, A.T.: Linear Mixed Models: A Practical Guide Using Statistical Software. Chapman & Hall, London (2006)
Winer, B.J.: Statistical Principles in Experimental Design. McGraw-Hill, New York (1971)
Yamins, D.L.K., DiCarlo, J.J.: Using goal-driven deep learning models to understand sensory cortex. Nat. Neurosci. 19, 356–365 (2016)
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Durstewitz, D. (2017). Regression Problems. In: Advanced Data Analysis in Neuroscience. Bernstein Series in Computational Neuroscience. Springer, Cham. https://doi.org/10.1007/978-3-319-59976-2_2
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