Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites
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1 Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites Tiago Santos 1 Simon Walk 2 Denis Helic 3 1 Know-Center, Graz, Austria 2 Stanford University 3 Graz University of Technology 3. April 2017 Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
2 Introduction Motivation Success of online collaboration websites depends critically on content contributed by users. For example, StackExchange vs. Google knol. Problem: Key deciding factors of success and failure of online collaboration websites? Goal: Uncover hidden nonlinear behavior in activity dynamics. Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
3 Related Work Nonlinear Time Series Analysis and its Applications Nonlinear time series analysis studies reconstructions of high dimensional dynamical systems from low dimensional ones. Example applications: Small and Tse [1] predicted the outcome of a roulette wheel. Hsieh [2] found nonlinear behavior in stock returns. Strozzi et al. [3] detected events in the stock market. More examples in Marwan et al. [4] and Bradley and Kantz [5]. New application: activity dynamics in online collaboration websites. Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
4 Related Work Dynamical Systems for Networks Dynamical systems provide mathematical formalizations for the evolution of numerical quantities over time [6, 7]. Applications of dynamical system theory to study activity in networks: Ribeiro [8] models daily active users in online communities with behavior of active and inactive users. Walk et al. [9] model activity in collaboration networks with activity decay rate and peer influence growth. Our approach: Characterize activity by its propensity to have originated in a dynamical system Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
5 Methodology Nonlinearity Tests We assess nonlinearity with 9 statistical tests: AR process-based tests: Broock, Dechert and Scheinkman [BDS] test [10] on ARIMA residuals Keenan s one-degree test for nonlinearity [11] McLeod-Li test [12] Tsay s test for nonlinearity [13] Likelihood ratio test for threshold nonlinearity [14] Neural Networks-based tests: Teraesvirta s neural network test [15] White neural network test [16] Other tests: Wald-Wolfowitz runs test [17] on the number of times time series grows Surrogate test - time asymmetry [18] Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
6 Methodology Reconstructing nonlinear dynamical system from univariate time series We reconstruct state space with Takens embedding theorem [19] to get: R t = (x t, x t τ, x t 2τ,..., x t (m 1)τ ) R m. (1) Example: Lorenz dynamical system Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
7 Methodology Reconstructing nonlinear dynamical system from univariate time series We reconstruct state space with Takens embedding theorem [19] to get: R t = (x t, x t τ, x t 2τ,..., x t (m 1)τ ) R m. (1) Example: Lorenz dynamical system Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
8 Methodology Reconstructing nonlinear dynamical system from univariate time series We reconstruct state space with Takens embedding theorem [19] to get: R t = (x t, x t τ, x t 2τ,..., x t (m 1)τ ) R m. (1) Example: Lorenz dynamical system Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
9 Methodology Forecasting univariate time series We employ linear and nonlinear models to forecast time series. Linear models: Linear regression: linear combination of Fourier terms and trend ARIMA models: differenced, linear combination of auto-regressors and lagged moving average error terms Exponential Smoothing (ETS) models: linear combination of lagged terms, such as level, trend, seasonality and error Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
10 Methodology Forecasting univariate time series We employ linear and nonlinear models to forecast time series. Linear models: Linear regression: linear combination of Fourier terms and trend ARIMA models: differenced, linear combination of auto-regressors and lagged moving average error terms Exponential Smoothing (ETS) models: linear combination of lagged terms, such as level, trend, seasonality and error Nonlinear models: Reconstruct dynamical system properties with Takens embedding Forecast univariate time series by following nearby trajectories in reconstructed state space Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
11 Methodology Recurrence Analysis Analyze reconstructed state spaces with Recurrence Plots: Example: Lorenz dynamical system R i,j (ɛ) = Θ(ɛ x i x j ). (2) Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
12 Methodology Recurrence Analysis Analyze reconstructed state spaces with Recurrence Plots: Example: Lorenz dynamical system R i,j (ɛ) = Θ(ɛ x i x j ). (2) Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
13 Experimental Setup Datasets, pre-processing & experiment configuration Datasets and pre-processing: Datasets: 16 randomly picked StackExchange Q&A portals Activity-based time series: number of questions, answers and comments per user per day Pre-processing: weekly aggregation, burn-in, detrend Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
14 Experimental Setup Datasets, pre-processing & experiment configuration Datasets and pre-processing: Datasets: 16 randomly picked StackExchange Q&A portals Activity-based time series: number of questions, answers and comments per user per day Pre-processing: weekly aggregation, burn-in, detrend Test and forecast setup: Significance level of the 9 tests for nonlinearity: 95% Categorize datasets on number of tests indicating nonlinearity Forecast 1 year of activity for all datasets Compare forecast root mean squared error (RMSE) of the 4 models Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
15 Results & Discussion Nonlinearity test results and forecast performance comparison Group datasets on nonlinearity test results and rank forecast RMSE per group with the Friedman test [20]: 10 out of 16 datasets with 4/9 tests indicating nonlinearity. Friedman test rank: 1. ETS, 2. ARIMA, 3. Nonlinear, 4. Linear 6 out of 16 datasets with 5/9 tests indicating nonlinearity. Friedman test rank: 1. Nonlinear, 2. ARIMA, 2. ETS, 4. Linear Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
16 Results & Discussion Nonlinearity test results and forecast performance comparison Group datasets on nonlinearity test results and rank forecast RMSE per group with the Friedman test [20]: 10 out of 16 datasets with 4/9 tests indicating nonlinearity. Friedman test rank: 1. ETS, 2. ARIMA, 3. Nonlinear, 4. Linear 6 out of 16 datasets with 5/9 tests indicating nonlinearity. Friedman test rank: 1. Nonlinear, 2. ARIMA, 2. ETS, 4. Linear Observations: Neural network-based tests are more sensitive to nonlinear dynamics Presence of nonlinear dynamics impacts forecast and modeling efforts Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
17 Results & Discussion Recurrence Plot Analysis Study reconstructed state spaces of 2 datasets deemed nonlinear: The RP empowers activity dynamics modeling efforts: Math: Drift, chaotic dynamics and slowly changing states Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
18 Results & Discussion Recurrence Plot Analysis Study reconstructed state spaces of 2 datasets deemed nonlinear: The RP empowers activity dynamics modeling efforts: Math: Drift, chaotic dynamics and slowly changing states Bitcoin: Periodic dynamics and non-stationary transitions Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
19 Conclusions & Future Work Conclusions & Future Work Conclusions: Group activity-based time series by propensity to originate from dynamical systems Increase accuracy in activity forecast experiments Customize activity models with Recurrence Plots More and longer time series more conclusive results Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
20 Conclusions & Future Work Conclusions & Future Work Conclusions: Group activity-based time series by propensity to originate from dynamical systems Increase accuracy in activity forecast experiments Customize activity models with Recurrence Plots More and longer time series more conclusive results Future work: Understand reason for differences in nonlinear behavior Study underlying collaboration networks Recurrence Quantification Analysis Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
21 Conclusions & Future Work Questions? Thank you very much for your time! Questions? Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 13
22 Backup and References Results table Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 6
23 Backup and References References I Small, M and Tse, CK. Predicting the outcome of roulette. Chaos: an interdisciplinary journal of nonlinear science 2012;22: Hsieh, DA. Chaos and nonlinear dynamics: application to financial markets. The journal of finance 1991;46: Strozzi, F, Zaldívar, JM, and Zbilut, JP. Application of nonlinear time series analysis techniques to high-frequency currency exchange data. Physica A: Statistical Mechanics and its Applications 2002;312: Marwan, N, Romano, MC, Thiel, M, and Kurths, J. Recurrence plots for the analysis of complex systems. Physics reports 2007;438: Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 6
24 Backup and References References II Bradley, E and Kantz, H. Nonlinear time-series analysis revisited. Chaos: An Interdisciplinary Journal of Nonlinear Science 2015;25: Luenberger, DGDG. Introduction to dynamic systems; theory, models, and applications. Tech. rep Guckenheimer, J and Holmes, PJ. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Vol. 42. Springer Science & Business Media, Ribeiro, B. Modeling and predicting the growth and death of membership-based websites. In: Proceedings of the 23rd international conference on World Wide Web. ACM. 2014: Walk, S, Helic, D, Geigl, F, and Strohmaier, M. Activity dynamics in collaboration networks. ACM Transactions on the Web (TWEB) 2016;10:11. Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 6
25 Backup and References References III Broock, W, Scheinkman, JA, Dechert, WD, and LeBaron, B. A test for independence based on the correlation dimension. Econometric reviews 1996;15: Keenan, DM. A Tukey nonadditivity-type test for time series nonlinearity. Biometrika 1985;72: McLeod, AI and Li, WK. Diagnostic checking ARMA time series models using squared-residual autocorrelations. Journal of Time Series Analysis 1983;4: Tsay, RS. Nonlinearity tests for time series. Biometrika 1986;73: Chan, KS. Percentage points of likelihood ratio tests for threshold autoregression. Journal of the Royal Statistical Society. Series B (Methodological) 1991: Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 6
26 Backup and References References IV Teräsvirta, T, Lin, CF, and Granger, CW. Power of the neural network linearity test. Journal of Time Series Analysis 1993;14: Lee, TH, White, H, and Granger, CW. Testing for neglected nonlinearity in time series models: A comparison of neural network methods and alternative tests. Journal of Econometrics 1993;56: Siegel, S. Nonparametric statistics for the behavioral sciences Schreiber, T and Schmitz, A. Surrogate time series. PhysicaD:NonlinearPhenomena 2000;142: Takens, F. Detecting strange attractors in turbulence. In: Dynamical systems and turbulence, Warwick Springer, 1981: Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 6
27 Backup and References References V Demšar, J. Statistical comparisons of classifiers over multiple data sets. Journal of Machine learning research 2006;7:1 30. Tiago Santos Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites 3. April / 6
Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites
Nonlinear Characterization of Activity Dynamics in Online Collaboration Websites Tiago Santos Know-Center Graz, Austria tsantos@know-center.at Simon Walk Stanford University walk@stanford.edu Denis Helic
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