hlab-analysis-methods

Code, slides, and figs on various analysis techniques

This repository contains a variety of instructional code and slides for Hlab analysis mini-lectures. The purpose is to provide examples of applied mathematical analyses to typical neural data recorded in the Hlab.

Structure

Each week we will discuss a specific topic and dive into real data. The repository will continuously grow with new folders added for each topic. Expected topics to be covered are:

1. Linear algebra & matrix/vector data representations
   • inner / outer products
   • matrices as affine functions
   • geometrical perspective

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2. Statistical signal/image processing
   • fourier analysis, filtering, spectrograms
   • wavelets & convolution
   • auto-correlations / auto-regression
   • time-series decomposition (singular spectrum)
   • object identification / edge & ridge detection

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3. Matrix/tensor factorization
   • general framework of matrix factorization
   • eigenvectors/eigenvalues & PCA
   • NMF, sparse PCA, ICA
   • CANDECOMP/PARAFAC
   • smoothing/denoising & dimensionality reduction

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4. Unsupervised clustering
   • k-means, k-medians
   • mixture models, EM
   • density-based clustering

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5. Graph Theory
   • representation & construction
   • decomposition (laplacian eigmaps, spectral clustering)
   • inference & applications to time series data

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6. Supervised learning (regression, classification)
   • linear, multiple regression
   • logistic regression
   • regularization

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7. curve fitting
   • linear, polynomial, exponential
   • splines
   • spline regression

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8. Parametric and non-parametric statistics
   • bootstrapping / sampling
   • density estimation
   • quantitative comparisons of distributions
   • self-created statistical tests

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9. Probability theory & bayesian analysis
   • stochastic vs. deterministic signals
   • bayes rule & chain rule
   • conditional probability and expectation/conditional expectation
   • graphical models & inference

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10. Data visualization
    • dimensionality reduction and visualization
    • multi-variate distributions
    • time-series & movies

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11. Deep learning
    • single-layer MLPs <--> logistic regression
    • Naive Bayes
    • multi-layer MLPs and non-linear activations
    • RNNs for time-series prediction