Time Series Analysis of Nutrients on the Lower Mississippi River

Time Series Analysis of Nutrients on the Lower Mississippi River 29 September 2004 @ Northeastern U.

41% of Continental US • Heavily Farmed Regions(N) • 56% of N from 5 states http://www.epa.gov Lower Mississippi Nutrient Time Series

Hypoxia • First noticed in 1972 • Regular checking since 1985 • 1 Million Metric Tons Nitrate/year http://www-personal.umich.edu/~acotel/Webpages/Classwebpages/kirk.ppt Lower Mississippi Nutrient Time Series

Issues & Facts • Anoxic Zone along the Coastal Louisiana waters • Nitrate • Ammonia • Phosphorus • Orthophosphate • Causes • Fertilizer application • Hydrological changes Lower Mississippi Nutrient Time Series

Time Series • What is it? • An ordered sequence of values of a variable at equally spaced time intervals. • Time Series Modeling • accounts for the fact that data points taken over time may have an internal structure (such as autocorrelation, trend or seasonal variation) that should be accounted for. • Yt = Trend + Seasonal + Irregular • Trend: Tt = 5 + 0.1t • Seasonal: St = 2.6 Sin(t/2) • Irregular: It = 0.4 It-1 + Єt Lower Mississippi Nutrient Time Series

Methodology • Summary Statistics • Pretreatment / Missing Values • Trend Analysis • Seasonality • SARIMA • ARCH-GARCH Models • VAR Models • Forecasting Lower Mississippi Nutrient Time Series

Summary Statistics & Data Pretreatment • Summary of Statistics • Number of data, Missing values • Mean, Variance • Maximum, Minimum • Skewness , Kurtosis • Data Pretreatment • Outliers • Transformation • Treatment of Missing Values Lower Mississippi Nutrient Time Series

Trend Analysis • Trend: • A long term movement in a time series. It is the underlying direction (an upward or downward tendency) and rate of change in a time series • Deterministic • Stochastic • Two Tests • Kendell-Tau Test • Dicky-Fuller Test • Trend Fitting / Removal as a part of modeling • Differencing • Detrending Lower Mississippi Nutrient Time Series

Seasonality • Seasonality: • mean periodic fluctuations in the data • Hourly, Daily, Monthly, Annual • Identification • Spectral Analysis [Frequency Domain] • Auto Correlation Function (ACF) & Partial Auto Correlation Function (PACF) [Time Domain] Lower Mississippi Nutrient Time Series

SARIMA Models • SARIMA Models • Seasonal AutoRegressive Integrated Moving Average • SARIMA(p,d,q)(P,D,Q)s • p & q = the nonseasonal coefficients • d = number of nonseasonal differences • P = number of multiplicative autoregressive coefficients • D = number of seasonal differences • Q = number of multiplicative moving average coefficients • s = seasonal period Lower Mississippi Nutrient Time Series

SARIMA Models (Cont.) • ARMA model ARMA(p,0,q) • yt = α0 +Σαi yt-i + Σβi εt-i Where:α0 a constant (intercept) αi the autoregressive model parameters βi the moving average model parameters εt-i white noise (residual OR error) Lower Mississippi Nutrient Time Series

SARIMA Models (Cont.) • Model Selection Criteria • Akaike Information Criterion • AIC = T*ln(residual sum of squares)+2*n • Schwartz Bayesian Criterion • SBC = T*ln(residual sum of squares)+n*ln(T) • Q-Statistic Where T = number of usable observations n = number of parameters estimated Lower Mississippi Nutrient Time Series

ARCH & GARCH • Residuals in SARIMA Models assume • Zero mean • Homoskedasticity: constant variance • GARCH model • Generalized AutoRegressive Conditional Heteroskedastic • extracts information from the residual variance • Modeled after the SARIMA model Lower Mississippi Nutrient Time Series

VAR Models • Models with other variables in the models • Transfer function models • No feedback to Independent variable • VAR (Vector AutoRegression) models • No independent variable (Feedback from one to another) • VAR Analyses & models • Impulse Response Analysis • Structural VARs & Multivariate decompositions • Granger Causality • To decide the “independent” variable components Lower Mississippi Nutrient Time Series

VAR Models (Cont.) • Standard VAR Form • Forecast error variances • in {eyt} • in {ezt} Lower Mississippi Nutrient Time Series

VAR Models (Cont.) • Impulse Response Functions • VAR Form 1 (Analogous to an AR(1) model) • VAR Form 2 (Analogous to a MA model) Lower Mississippi Nutrient Time Series

Study Data • USGS Station @ Tarbert Landing (River Mile 308) • Daily Flow • Jefferson Parish Water Works (River Mile 100) • Daily Nitrate Concentration • Daily Ammonia Concentration • Daily Phosphorus Concentration • Daily Orthophosphate Concentration • Time Travel Model for Flow from RM 308 to RM 100 Lower Mississippi Nutrient Time Series

Data (Cont.) • Summary Table • Data Analysis • Weekly • Monthly • Quarterly Lower Mississippi Nutrient Time Series

Results Lower Mississippi Nutrient Time Series

Results (Cont.) • 40% missing values in Ammonia weekly data • No Significant Trend in • Monthly data & • Quarterly data • Annual seasonality according to • Spectral Analysis • ACF-PACF plot Lower Mississippi Nutrient Time Series

Results (Cont.) Monthly Nitrate/ACF-PACF Plot Lower Mississippi Nutrient Time Series

Results (Cont.) Monthly SAR Model Quarterly SAR Model • No significant improvement with ARCH models Lower Mississippi Nutrient Time Series

Results (Cont.) VAR Models for Monthly data • The numbers in brackets are the specific time lag for that endogenous variable • The other number in the cell is the value of the coefficient for that time lagged variable • Impulse Response Functions (for monthly data) show • A shock to Nitrate had negligible effect on the flow • A shock to Nitrate had a 3 month effect on the nitrate • A shock to flow had a 3 month effect on the flow • A shock to flow had the peak effect on the nitrate at the 3rd month Lower Mississippi Nutrient Time Series

Results (Cont.) Lower Mississippi Nutrient Time Series

Conclusions & Future Research • Conclusions • No significant trend found • Annual seasonality • No improvement with ARCH models • Shocks on Nitrate had negligible effect on flow • Shocks on Flow had significant effect on Nitrate • Future Research • GARCH & VAR models • for Lake data • for other water quality parameters in rivers Lower Mississippi Nutrient Time Series

Time Series Analysis of Nutrients on the Lower Mississippi River