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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. Hypoxia. First noticed in 1972 Regular checking since 1985 1 Million Metric Tons Nitrate/year.

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Time Series Analysis of Nutrients on the Lower Mississippi River

29 September 2004

@ Northeastern U.


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http://www.epa.gov

Lower Mississippi Nutrient Time Series


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Hypoxia River

  • 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


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Issues & Facts River

  • Anoxic Zone along the Coastal Louisiana waters

    • Nitrate

    • Ammonia

    • Phosphorus

    • Orthophosphate

  • Causes

    • Fertilizer application

    • Hydrological changes

Lower Mississippi Nutrient Time Series


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Time Series River

  • 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


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Methodology River

  • Summary Statistics

  • Pretreatment / Missing Values

  • Trend Analysis

  • Seasonality

  • SARIMA

  • ARCH-GARCH Models

  • VAR Models

  • Forecasting

Lower Mississippi Nutrient Time Series


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Summary Statistics & Data Pretreatment River

  • 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


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Trend Analysis River

  • 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


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Seasonality River

  • 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


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SARIMA Models River

  • 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


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SARIMA Models (Cont.) River

  • 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


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SARIMA Models (Cont.) River

  • 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


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ARCH & GARCH River

  • 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


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VAR Models River

  • 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


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VAR Models (Cont.) River

  • Standard VAR Form

  • Forecast error variances

    • in {eyt}

    • in {ezt}

Lower Mississippi Nutrient Time Series


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VAR Models (Cont.) River

  • 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


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Study Data River

  • 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


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Data (Cont.) River

  • Summary Table

  • Data Analysis

    • Weekly

    • Monthly

    • Quarterly

Lower Mississippi Nutrient Time Series


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Results River

Lower Mississippi Nutrient Time Series


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Results (Cont.) River

  • 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


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Results (Cont.) River

Monthly Nitrate/ACF-PACF Plot

Lower Mississippi Nutrient Time Series


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Results (Cont.) River

Monthly SAR Model

Quarterly SAR Model

  • No significant improvement with ARCH models

Lower Mississippi Nutrient Time Series


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Results (Cont.) River

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


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Results (Cont.) River

Lower Mississippi Nutrient Time Series


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Results (Cont.) River

Lower Mississippi Nutrient Time Series


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Results (Cont.) River

Lower Mississippi Nutrient Time Series


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Conclusions & Future Research River

  • 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


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