ISSN: 2056-3736 (Online Version) | 2056-3728 (Print Version)

Machine Learning Treasury Yields

Zura Kakushadze and Willie Yu

Correspondence: Zura Kakushadze,

Quantigic Solutions LLC, USA

pdf (3537.63 Kb) | doi:


We give explicit algorithms and source code for extracting factors underlying Treasury yields using (unsupervised) machine learning (ML) techniques, such as nonnegative matrix factorization (NMF) and (statistically deterministic) clustering. NMF is a popular ML algorithm (used in computer vision, bioinformatics/computational biology, document classification, etc.), but is often misconstrued and misused. We discuss how to properly apply NMF to Treasury yields. We analyze the factors based on NMF and clustering and their interpretation. We discuss their implications for forecasting Treasury yields in the context of out-of-sample ML stability issues.


  non-negative matrix factorization, NMF, clustering, k-means, Treasury, yield, machine learning, maturity, time series, out-of-sample, in-sample, weight, factor, exposure, source code, principal component, correlation, forecasting, interest rate, stability, level, slope, steepness, curvature, fixed income, term structure, yield curve.


Almeida, C., Ardison, K., Kubudi, D., Simonsen, A. and Vicente, J. (2018) Forecasting Bond Yields with Segmented Term Structure Models. Journal of Financial Econometrics 16(1): 1-33.Bliss, R.R. (1997) Movements in the Term Structure of Interest Rates. Federal Reserve Bank of Atlanta Economic Review 82(4): 16-33.Bouchaud, J.-P. and Potters, M. (2011) Financial applications of random matrix theory: a short review. In: Akemann, G., Baik, J. and Di Francesco, P. (eds.) The Oxford Handbook of Random Matrix Theory. Oxford, United Kingdom: Oxford University Press.Campbell, L.L. (1960) Minimum coefficient rate for stationary random processes. Information and Control 3(4): 360-371.Diebold, F.X. and Li, C. (2006) Forecasting the Term Structure of Government Bond Yields. Journal of Econometrics 130(2): 337-364.Ding, C., He, X. and Simon, H.D. (2005) On the equivalence of nonnegative matrix factorization and spectral clustering. In: Kargupta, H., Srivastava, J., Kamath, C. and Goodman, A. (eds.) Proceedings of the Fifth SIAM International Conference on Data Mining. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), pp. 606-610.Duffee, G. (2002) Term premia and interest rate forecasts in affine models. Journal of Finance 57(1): 405-443.Duffee, G. (2013) Chapter 7 – Forecasting Interest Rates. In: Elliott, G. and Timmermann, A. (eds.) Handbook of Economic Forecasting. Vol. 2, Part A. Amsterdam, The Netherlands: Elsevier, pp. 385-426.Eckart, C. and Young, G. (1936) The approximation of one matrix by another of lower rank. Psychometrika 1(3): 211-218.Fama, E.F. and MacBeth, J.D. (1973) Risk, Return and Equilibrium: Empirical Tests. Journal of Political Economy 81(3): 607-636.Forgy, E.W. (1965) Cluster analysis of multivariate data: efficiency versus interpretability of classifications. Biometrics 21(3): 768-769.Frobenius, G. (1912) Über Matrizen aus Nicht Negativen Elementen. In: Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, pp. 456-477.Gaujoux, R. and Seoighe, C. (2010). A flexible R package for nonnegative matrix factorization. BMC Bioinformatics 11: 367.Hartigan, J.A. (1975) Clustering Algorithms. New York, NY: John Wiley & Sons, Inc.Hartigan, J.A. and Wong, M.A. (1979) Algorithm AS 136: A K-Means Clustering Algorithm. Journal of the Royal Statistical Society, Series C (Applied Statistics) 28(1): 100-108.Kakushadze, Z. (2015) Heterotic Risk Models. Wilmott Magazine 2015(80): 40-55. Available online:, Z. and Yu, W. (2016a) Factor Models for Cancer Signatures. Physica A 462: 527-559. Available online:, Z. and Yu, W. (2016b) Statistical Industry Classification. Journal of Risk & Control 3(1): 17-65. Available online:, Z. and Yu, W. (2017a) Statistical Risk Models. Journal of Investment Strategies 6(2): 1-40. Available online:, Z. and Yu, W. (2017b) *K-means and Cluster Models for Cancer Signatures. Biomolecular Detection and Quantification 13: 7-31. Available online:, Z. and Yu, W. (2017c) Mutation Clusters from Cancer Exome. Genes 8(8): 201. Available online:, P., Litterman, R.B. and Scheinkman, J. (1994) Explorations into Factors Explaining Money Market Returns. Journal of Finance 49(5): 1861-1882.Lee, D.D. and Seung, H.S. (1999) Learning the parts of objects by non-negative matrix factorization. Nature 401(6755): 788-791. Litterman, R.B. and Scheinkman, J. (1991) Common factors affecting bond returns. Journal of Fixed Income 1(1): 54-61.Lloyd, S.P. (1957) Least square quantization in PCM. Working Paper. Murray Hill, NJ: Bell Telephone Laboratories.Lloyd, S.P. (1982) Least square quantization in PCM. IEEE Transactions on Information Theory 28(2): 129-137.MacQueen, J.B. (1967) Some Methods for classification and Analysis of Multivariate Observations. In: LeCam, L. and Neyman, J. (eds.) Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability. Berkeley, CA: University of California Press, pp. 281-297.Nelson, C. and Siegel, A.F. (1987) Parsimonious modeling of yield curves. Journal of Business 60(4): 473-489.Paatero, P. and Tapper, U. (1994) Positive matrix factorization: A non-negative factor model with optimal utilization of error. Environmetrics 5(1): 111-126. Perron, O. (1907) Zur Theorie der Matrices. Mathematische Annalen 64(2): 248-263.Roy, O. and Vetterli, M. (2007) The effective rank: A measure of effective dimensionality. In: European Signal Processing Conference (EUSIPCO). Poznań, Poland (September 3-7, 2007), pp. 606-610.Shahnaz, F., Berry, M.W., Pauca, V.P. and Plemmons, R.J. (2006) Document clustering using nonnegative matrix factorization. Information Processing and Management 42(2): 373-386.Steinhaus, H. (1957) Sur la division des corps matériels en parties. Bull. Acad. Polon. Sci. 4(12): 801-804.Svensson, L.E.O. (1994) Estimating and interpreting forward interest rates: Sweden 1992-1994. NBER Working Paper No. 4871. Cambridge, MA: National Bureau of Economic Research.Takada, H. H. and Stern, J.M. (2015) Non-negative matrix factorization and term structure of interest rates. AIP Conference Proceedings 1641(1): 369-377.Yang, W., Gibson, J.D. and He, T. (2005) Coefficient rate and lossy source coding. IEEE Transactions on Information Theory 51(1): 381-386.Zass, R. and Shashua, A. (2005) A unifying approach to hard and probabilistic clustering. In: Proceedings of the Tenth IEEE International Conference on Computer Vision (ICCV’05). Washington, DC: IEEE Computer Society, pp. 294-301.