## Research Interests

**NMF+GARCH on multivariate time series**: This is work in progress.**Dynamic PCA on multivariate time series**: we factorize multivariate daily air quality data (at each hour) into latent basis and loadings matrices on a 45 day moving window. All factors are dynamic, yielding 5-dimensional tensors for interpretations. PCA allows extraction of significant factors based on largest eigenvalues (aliasing standard deviation of the orthogonal PCs). Observations from multiple sites are robustly aggregated and missing values are non-linearly interpolated from temporally nearby values. This work revealed heightened dependency between ozone and other pollutants in winter mornings and weakest in summer afternoons. Previous research showed average dependency only. arXiv:1608.03022**Spectra estimation of multivariate time series**: a transformation to frequency domain (via fast Fourier transform) decomposes temporal observations into sum of sinusoids at natural frequencies. Noise can be ignored by thresholding frequencies with low amplitudes or by smoothing a cross-periodogram (yielding a spectral estimate and equivalent of cross-covariance). Thus, we built cross-coherence for a portfolio of daily ETF trade prices to find similarly-behaving assets, regardless of their conventional classification. This can further be used to diversify a portfolio risk or provide hedge by equivalently behaving assets.**Spectra estimation for irregularly spaced multivariate time series**: this problem is similar to the usual, equi-spaced, series, but many challenges arise such as selecting candidate frequencies, deducing inference on periodogram, etc.**Multivariate time series**: ARIMA, GARCH, dynamic PCA, online NMF, cross-covariance and cross-coherence in spectral domain, inference and forecasting**Machine learning**: PCA, NMF, dimension reduction, pattern learning, regressions, classification, missing data imputation

**Applications to finance**: volatility modeling, valuation of derivative (bonds, futures, options, swaps, …). **Software Packages**: QFRM package (Python, R): pricing and visualization of 20+ exotic (financial) options using 4 key algorithms: Black Scholes model, lattice tree, Monte Carlo simulations, and finite differences. Additionally, some pricing of fixed income instruments.

** Applications to air quality: **multivariate spatio-temporal modeling and forecasting of air pollutants, accounting for missing values and spatial dispersion; identifying latent sources.