2.3 Resource distribution mapping

Annual shrimp catch distribution mapping at 5-year interval is presented using ArcGIS version 9 software.

## 2.3.1 SARIMA and SARIMAX model development

ARIMA model is developed on stationary data, but very few data sets in the fisheries field are found to be stationary in nature. It becomes mandatory to test the time series data sets for stationarity before using for modelling. Augmented Dickey-Fuller [20, 21] (ADF) test used to test stationary of the original time series data. If data were found to be nonstationary, then it is made stationary by transformation procedure. Stationary time series data were further used for SARIMA model development. SARIMA model was developed by Box-Jenkins [7] Seasonal Auto Regressive Integrated Moving Average (SARIMA) model for seasonal quarterly time series data following four steps as model identification, parameter estimation and model validation (diagnostic checking) and finally forecasting by the following methodology:

SARIMA model defined as ARIMA with seasonal parameters denoted as ARIMA (p,d,q) (P,D,Q) s,

where p = auto regression (AR) order, d = differencing order, q = moving average order, P = seasonal AR order, D = seasonal differencing order, Q = seasonal MA order and s is seasonality using the back shift operator B (the operator B is such that B zt ¼ zt�1) is expressed as,

$$
\rho(B^s)\phi(B)\nabla^d\nabla\_s^D\mathbf{z}\_t = \Theta(B^s)\theta(B)\boldsymbol{\varepsilon}\_t \tag{1}
$$

where,

$$\begin{aligned} \nabla &= \mathbf{1} - B \\ \nabla' &= \mathbf{1} - B^{\circ} \\ \phi(B) &= \mathbf{1} - \phi\_1 B - \dots - \phi\_p B^p \\ \theta(B) &= \mathbf{1} - \theta\_1 B - \dots - \theta\_q B^q \\ \varphi(B^{\circ}) &= \mathbf{1} - \varrho\_1 B^{\circ} - \dots - \varrho\_P (B^{\circ})^P \\ \Theta(B^{\circ}) &= \mathbf{1} - \Theta\_1 B^{\circ} - \dots - \Theta\_Q (B^{\circ})^Q \end{aligned}$$

Forecasting Shrimp and Fish Catch in Chilika Lake over Time Series Analysis DOI: http://dx.doi.org/10.5772/intechopen.85458

Parameters of the model are

2.2 Sample data collection

Time Series Analysis - Data, Methods, and Applications

2.3 Resource distribution mapping

2.3.1 SARIMA and SARIMAX model development

ArcGIS version 9 software.

(p,d,q) (P,D,Q) s,

where,

86

B zt ¼ zt�1) is expressed as,

The quarter wise estimated total shrimp catch (MT) data and monthly total fish catch (MT) data along with physicochemical parameters of water quality [14] of the Chilika lagoon for the period April 2001 to March 2015 were collected from Chilika Development Authority (CDA). The systematic random sampling methods with landing centre approach [17–19] modified with site-specific conditions followed on monthly basis were used to catch estimation in Chilika lagoon. In this study, quarter wise (seasons) catch of the shrimp was taken; such as first quarter (Q1) consists of March, April and May months together (summer season); second quarter (Q2) consists of June, July and August months (monsoon season); third quarter (Q3) consists of September, October and November months (post monsoon season) and fourth quarter (Q4) consists of December, January and February months (winter season). This quarter wise shrimp catch and total fish catch data for the period 2001 to 2015 of the Chilika lagoon were used for the development of SARIMA time series prediction modelling, and SARIMAX model has been described using monthly total fish catch data with the physicochemical parameters of water quality of the lagoon [14].

Annual shrimp catch distribution mapping at 5-year interval is presented using

ARIMA model is developed on stationary data, but very few data sets in the fisheries field are found to be stationary in nature. It becomes mandatory to test the time series data sets for stationarity before using for modelling. Augmented Dickey-Fuller [20, 21] (ADF) test used to test stationary of the original time series data. If data were found to be nonstationary, then it is made stationary by transformation procedure. Stationary time series data were further used for SARIMA model development. SARIMA model was developed by Box-Jenkins [7] Seasonal Auto Regressive Integrated Moving Average (SARIMA) model for seasonal quarterly time series data following four steps as model identification, parameter estimation and model validation (diagnostic checking) and finally forecasting by the following methodology:

SARIMA model defined as ARIMA with seasonal parameters denoted as ARIMA

where p = auto regression (AR) order, d = differencing order, q = moving average order, P = seasonal AR order, D = seasonal differencing order, Q = seasonal MA order and s is seasonality using the back shift operator B (the operator B is such that

<sup>s</sup> zt <sup>¼</sup> <sup>Θ</sup> Bs ð Þθð Þ <sup>B</sup> <sup>ε</sup><sup>t</sup> (1)

<sup>φ</sup> <sup>B</sup><sup>s</sup> ð Þϕð Þ <sup>B</sup> <sup>∇</sup><sup>d</sup>∇<sup>D</sup>

<sup>ϕ</sup>ð Þ¼ <sup>B</sup> <sup>1</sup> � <sup>ϕ</sup>1<sup>B</sup> � <sup>⋯</sup> � <sup>ϕ</sup>pBp <sup>θ</sup>ð Þ¼ <sup>B</sup> <sup>1</sup> � <sup>θ</sup>1<sup>B</sup> � <sup>⋯</sup> � <sup>θ</sup>qBq <sup>φ</sup> Bs ð Þ¼ <sup>1</sup> � <sup>φ</sup>1Bs � <sup>⋯</sup> � <sup>φ</sup><sup>P</sup> Bs ð Þ<sup>P</sup> <sup>Θ</sup> Bs ð Þ¼ <sup>1</sup> � <sup>Θ</sup>1Bs � <sup>⋯</sup> � <sup>Θ</sup><sup>Q</sup> Bs ð Þ<sup>Q</sup>

∇ ¼ 1 � B <sup>∇</sup><sup>s</sup> <sup>¼</sup> <sup>1</sup> � <sup>B</sup><sup>s</sup> ϕ1, ⋯, ϕp, θ1, ⋯, θq, φ1, ⋯, φP, Θ1, ⋯, Θ<sup>Q</sup> and σ2. Now, SARIMA model for time series data yt using Eq. (1) can be defined as;

$$\mathcal{Y}\_t = \varrho(B^\circ)\phi(B)\nabla^d\nabla\_\varepsilon^D\mathcal{Y}\_t + \Theta(B^\circ)\theta(B)\varepsilon\_t \tag{2}$$

Moreover, the SARIMAX model (seasonal ARIMA with explanatory variable) can be represented as a time-series forecasting model using the multiple regressions with seasonal ARIMA model that takes care of the residual's serial correlations.

Further, SARIMAX model can be defined using Eq. (2) and regressors as follows;

$$\mathcal{Y}\_t = \varrho(B^\prime)\phi(B)\nabla^d\nabla\_\varepsilon^D\mathcal{Y}\_t + \Theta(B^\prime)\theta(B)\varepsilon\_t + \beta\mathfrak{x}\_t \tag{3}$$

Where, xt is the input series at time t and β is the regression coefficients of the input series considering the assumptions of ARIMA and regression model together [22].

Here, the sequence ε 0 t S is independently and identically distributed (i.i.d) random variable with zero mean and constant variance σ2, which represents the error term in the model.

Model parameters estimation and its significance were performed using statistical software such as SAS, R, MATLAB, SPSS, etc. Generally 70% data sets were taken for model development and 30% data for model validation for long time series data, but one can change the percentage based on the data availability. Model development approach (a) model identification: model parameters are identified by investigating autocorrelation function (ACF) and partial auto correlation function (PACF) of the time series data (see [23] for more details); (b) parameter estimation: identified model parameters are estimated using various recursive statistical algorithms computation [23]; (c) model validation: inspection of residuals for no significant lags using ACF. The best model was selected based on model selection criteria and finally forecast is done data using best model for the given period of time. The known model selection criteria (minimum is better), such as Akaike [24], known as Akaike Information criterion (AIC), Bayesian Information Criterion proposed by Schwartz [25], known as SBC and R square fit statistics, RMSE (root mean square error), are used for identification of the best fitted model. The minimum is preferable criteria except R square (maximum) for identifying best selection model. Finally, the Ljung and Box [26] statistics Q based on the autocorrelations of the residuals were used for testing adequacy of estimated models. Finally, forecasting or prediction for the short term period is done based on best fitted model. For shrimp catch forecasting modelling, Quarter wise shrimp catch data for the period April, 2001 to March, 2013 was used for best model selection and estimation and the data April, 2014 and March, 2015 were used for validation of the model for Chilika lagoon. Further based on developed model, quarterly shrimp catch was forecasted up to 2018. Further SARIMAX model is described using monthly total fish catch data with the physicochemical parameters of water quality as a factor of regressor of the lagoon as regressors [14]. The total fish data sets for the period April 2001–- March 2011 were used for training data sets, and the period April 2011–March 2014 was used as testing and validation data sets for the model.
