Effect of regime switching on behavior of albacore under the influence of phytoplankton concentration

Abstract

The influence of the sea environment on fish behaviors is an important ecological and environmental factor. In this article, we analyze the relationship between phytoplankton distribution in the sea and the behavioral process of a diadromous fish, albacore (Thunnus alalunga), by development of statistical models. It is well known that animals change their behavioral patterns, such as traveling and foraging, according to the conditions of their encountered habitat. To account for these characteristics in fish behavior, we develop a regime switching model with a hidden latent process, whose output is influenced by phytoplankton concentration. Geolocation prediction experiments suggested that the model can improve the prediction accuracies compared with the general regime switching model with no covariates, or traditional linear nonstationary time series models. Furthermore, robust geolocation predictions are yielded by the model even when the fish switches its migratory behavior.

Appendix

1.1 Applying EM algorithm for parameter estimation of regime switching model

The estimation of ${\varvec{\theta}}^{*}$ by applying EM algorithm is carried out by repeating the following procedures. We first give an initial estimate of ${\varvec{\theta}}^{*}_0$ and then find the estimate of ${\varvec{\theta}}^{*}$ which maximizes $Q({\varvec{\theta}}^{*}|{\varvec{\theta}}^{*}_0).$ Next, we substitute the value of ${\varvec{\theta}}^{*}_0$ with the estimate ${\varvec{\theta}}^{*}$ obtained above, and then find optimal estimate of ${\varvec{\theta}}^{*}$ in the same way.It can be shown that under certain general conditions, the sequence of ${\varvec{\theta}}^{*}$ obtained by using this procedure converges to the maximum likelihood estimator which maximizes $\log(L({\varvec{\theta}}^{*})),$ where $L({\varvec{\theta}}^{*})=\sum_{\mathbf{S}^{*}}f(\user2{\varPhi}^{*}, {\mathbf{S}^{*}}).$ In computation, we update the value of ${\varvec{\theta}}^{*}$ by repeating the procedure above until the percentage decrease in $Q({\varvec{\theta}}^{*}|{\varvec{\theta}}^{*}_0)$ becomes sufficiently small. The value of ${\varvec{\theta}}^{*}$ tends to reach convergence by iterations of about five times in average, although the number of iterations depends on statistical structure of measured data.

For numerical optimization of $Q({\varvec{\theta}}^{*}|{\varvec{\theta}}^{*}_0),$ the initial values of ${\varvec{\theta}}^{*}$ and P ^* _jk are necessary.Analytic solutions for ${\varvec{\theta}}^{*}$ which maximizes $Q({\varvec{\theta}}^{*}|{\varvec{\theta}}^{*}_0)$ exists only in certain situations. One of such situations is when ρ^* = 0 and all other parameters of ${\varvec{\theta}}^{*}$ are distinct. In this case, the solutions which maximizes $Q({\varvec{\theta}}^{*}|{\varvec{\theta}}^{*}_0)$ can be given by the estimators

$$ \hat{\mu}_{0,j}=\frac{\sum_{t=1}^{T}\gamma^{*}_t(j)v_t}{\sum_{t=1}^T\gamma^{*}_t(j)}, $$

(25)

$$ \hat{\sigma}^2_{0,j}=\frac{\sum_{t=1}^{T}\gamma^{*}_t(j)(v_t-\hat{\mu}_j)^2}{\sum_{t=1}^T\gamma^{*}_t(j)}, $$

(26)

and

$$ P^{*}_{0,jk}= \frac{\sum_{t=1}^{T}\gamma^{*}_t(j,k)}{\sum_{t=1}^T\gamma^{*}_t(j)}, $$

(27)

where $\gamma^{*}_{t}(j)=P(S^{*}_t=j|{\mathbf{V}},{\varvec{\theta}^{*}}_0)$ and $\gamma^{*}_t(j,k)=P(S^{*}_t=j,S^{*}_{t+1}=k|{\mathbf{V}},{\varvec{\theta}^{*}}_0).$ We use the above estimators as well as ρ^* = 0 as the initial values for maximization of $Q({\varvec{\theta}}^{*}|{\varvec{\theta}}^{*}_0).$

1.2 The maximum likelihood estimation of the regime switching model based on Wrapped Normal distribution

The consideration in Sect. 3.1 shows that the conditional probability distribution of L _t , under (S _t , S _t−1, L _t−1) are given, follows a normal distribution with ${\mathbb{E}(L_t|S_t, S_{t-1}, L_{t-1})=\mu_{S_t}+\rho_{t-1}\sigma_{S_t}X_{t-1}}$ and ${\mathbb{V}(L_t|S_t, S_{t-1}, L_{t-1})=\sigma^2_{S_t}, }$ where $X_{t-1}=(L_{t-1}-\mu_{S_{t-1}})/\sigma_{S_{t-1}}$. Therefore the conditional pdf of the circular random variable ϕ _t under (S _t , S _t−1, L _t−1) is given by

$$ f(\phi_t|S_t, S_{t-1}, \phi_{t-1})= \frac{1}{2\pi} \left( 1+2\sum_{p=1}^{\infty}\nu^{p^2}_{S_t} \cos p\left(\phi_t-\mu^{\prime}_{t}\right) \right), $$

where $\mu^{\prime}_{t}=(\mu_{S_t}+\rho_{t-1}\sigma_{S_t}X_{t-1})$(mod2π), $\nu_{S_t}=\exp(-\sigma^2_{S_t}/2)$ and X _t-1 = (ϕ _t-1 − μ _{S t-1})/σ _{S t-1}. Furthermore, the conditional pdf of $\varvec{\phi}=(\phi_1,\ldots,\phi_T)$ under ${\mathbf{S}}=(S_1,\ldots,S_T)$ is

$$ f(\varvec{\phi}|{\mathbf{S}})= \left(\frac{1}{2\pi}\right)^T \prod_{t=1}^{T} \left( 1+2\sum_{p=1}^{\infty}\nu^{p^2}_{S_t} \cos p\left(\phi_t-\mu^{\prime}_{t}\right) \right). $$

The estimation of the parameter, ${\varvec{\theta}}=(\{\mu_j\}, \{\sigma_j\}, \rho),$ requires maximization of the conditional expectation,

$$ {\mathbb{E}}_{\varvec{\theta}_0} (\log(L_{F}({\varvec{\theta}}))|\user2{\phi}) \equiv R({\varvec{\theta}}|{\varvec{\theta}}_0), $$

which is analogous to (6), where $L_{F}({\varvec{\theta}})$ is the simultaneous likelihood function of ϕ and S such that

$$ \log(L_{F}({\varvec{\theta}}))= \log(f(\user2{\varvec{\phi}}|{\varvec{S}}))+ \log(f({\varvec{S}})). $$

Here the first term of the right hand side can be approximated by Taylor expansion,

$$ \begin{aligned} \log(f(\user2{\varvec{\phi}}|{\mathbf{S}})) =\, &-T\log2\pi+\sum_{t=1}^{T} \log \left( 1+2\sum_{p=1}^{\infty}\nu^{p^2}_{S_t}\cos p(\phi_t-\mu^{\prime}_{t}) \right)\\ \,\approx &-T\log2\pi+ 2\sum_{t=1}^{T}\sum_{p=1}^{\infty}e^{-{p^2}\sigma^2_{S_t}/2} \cos p\left(\phi_t-\mu^{\prime}_{t}\right)\\ & -2\sum_{t=1}^{T}\sum_{p,q=1}^{\infty} e^{-({p^2+q^2})\sigma^2_{S_t}/{2}} \cos p\left(\phi_t-\mu^{\prime}_{t}\right)\\ & \times \cos q\left(\phi_t-\mu^{\prime}_{t}\right), \end{aligned} $$

where $\mu^{\prime}_{t}= (\mu_{S_t}+\rho C_{t-1}\sigma_{S_t}X_{t-1})({\text{mod}}2\pi).$Also the second term is

$$ \log(f({\mathbf{S}}))=\log(\pi_{S_1})+\sum_{t=2}^{T}\log(P_{S_{t-1}S_{t}}). $$

By analogy of construction of $Q({\varvec{\theta}}|{\varvec{\theta}}_0), R({\varvec{\theta}}|{\varvec{\theta}}_0)$ can be constructed by

$$ \begin{aligned} R({\varvec{\theta}}|{\varvec{\theta}}_0) \approx &-T\log2\pi+ 2\sum_{t=1}^{T-1}\sum_{p=1}^{\infty}\sum_{j,k=1}^{N} e^{{-{p^2}\sigma^2_{k}}/{2}}\left({{\text{cos}}\; p}\phi_{t,j,k}^{(D)}\right) \gamma_t(j,k)\\ & -2\sum_{t=1}^{T-1}\sum_{p,q=1}^{\infty}\sum_{j,k=1}^{N} e^{{-({p^2}+{q^2})}\sigma^2_{k}/{2}}\left({{\text{cos}}\; p}\phi_{t,j,k}^{(D)}{{\text{cos}}\; q}\phi_{t,j,k}^{(D)}\right)\\ &\times \gamma_t(j,k)\\ & +\sum_{j=1}^{N}(\log\pi_j)\gamma_1(j) +\sum_{t=1}^{T-1}\sum_{j,k=1}^{N}(\log P_{jk})\gamma_t(j,k),\\ \end{aligned} $$

(28)

where

$$ \begin{aligned}\phi^{(D)}_{t,j,k}=\phi_{t+1}-\left(\mu_k+\rho C_{t}\sigma_{k} \left(\frac{\phi_{t}-\mu_{j}}{\sigma_{j}}\right)\right)\left({\text{mod}}2\pi\right),&\\ \gamma_t(j,k)=\,P(S_t=j, S_{t+1}=k|\user2{\varPhi}, {\varvec{\theta}}_0), &\\ \gamma_t(j)=\,P(S_t=j|\user2{\varPhi}, {\varvec{\theta}}_0)=\sum_{k=1}^{N}\gamma_{t}(j,k). \end{aligned} $$

The values of γ _t (j, k) and $\gamma_t(j)\,(j,k=1,\ldots,N$) can be obtained by using the algorithm given in Appendix. Maximization of $R({\varvec{\theta}}|{\varvec{\theta}}_0)$ is carried out by means of a numerical method. This maximization requires initial values of parameters. The estimates obtained by (25)–(27) with ρ = 0 are used for the initial values of μ _j ’s, σ _j ’s and ρ in (28), respectively.

1.3 An extension of Baum and Welch algorithm

To estimate γ _t (j) and γ _t (j, k), we apply the modified algorithm proposed by Buckle et al. (2002). Let α _t (j) and $\beta_t(j)\,(j=1,\ldots,N$) be time-dependent sequences satisfying

$$ \begin{aligned} \alpha_t(j)=&\,P(S_t=j|{\mathbf{V}}_t), \quad t=1,\ldots,T, \\ \beta_t(j)=&\, P(v_{t+1},\ldots,v_{T}|S_t=j,{\mathbf{V}}_t) f({\mathbf{V}}_t)/f({\mathbf{V}}_T), \quad t=1,\ldots,T-1, \\ \beta_{T}(j)=&\,1, \end{aligned} $$

where v _t denotes the observation at time t and ${{\mathbf{V}}}_t=(v_{1},\ldots,v_{t}). \{\alpha_t(j)\}$ and {β _t (j)} can be estimated by the following algorithms,

• (i) Forward probability

  $$ \begin{aligned} \alpha_t(k)=&\sum_{j=1}^{N}P(S_{t-1}=j, S_t=k|{\mathbf{V}}_t) \\ =&\sum_{j=1}^{N}f(v_t|S_t=k, S_{t-1}=j, {\mathbf{V}}_{t-1}) P_{jk}\alpha_{t-1}(j)/\kappa_t, \end{aligned} $$

  where

  $$ \begin{aligned} \kappa_t=&\sum_{j=1}^N\sum_{k=1}^N f(v_t|S_t=k, S_{t-1}=j, {\mathbf{V}}_{t-1})P_{jk}\alpha_{t-1}(j), \\ \alpha_1(j)=&f(v_1|S_1=j)\pi_j/\kappa_1, \quad j=1,\ldots,N. \end{aligned} $$

  Here, $f(v_t|S_t=k, S_{t-1}=j, {\mathbf{V}}_{t-1})$ is the pdf of normal distribution with

  $$ {\mathbb{E}}(v_t|S_t=k, S_{t-1}=j, {\mathbf{V}}_{t-1})= \mu_k+\rho_{t-1}\frac{\sigma_k}{\sigma_j}(v_{t-1}-\mu_j). $$

• (ii) Backward probability

  $$ \begin{aligned} \beta_t(j)=&\, \sum_{k=1}^{N}\beta_{t+1}(k)f(v_{t+1}|S_{t+1}=k, S_{t}=j, {\mathbf{V}}_{t})P_{jk}/\kappa_{t+1}, \quad t=T-1,\ldots,1, \\ \beta_T(j)=&\, 1 \end{aligned} $$

  Having obtained the values of {α _t (j)} and $\{\beta_{t}(j)\}, \gamma_t(j)=P(S_t=j|{\mathbf{V}}_t, {\varvec{\theta}}_0)$ and $\gamma_t(j,k)=P(S_t=j, S_{t+1}=k|{\mathbf{V}}_t, {\varvec{\theta}}_0)$ are determined from

  $$ \gamma_t(j)=\alpha_t(j)\beta_t(j), $$

  and

  $$ \gamma_t(j,k)=\beta_{t+1}(k) f(v_{t+1}|S_{t+1}=k, S_{t}=j, {\mathbf{V}}_{t})P_{jk} \alpha_t(j)/\kappa_{t+1}, $$

  respectively.

1.4 Conditional variance of M _T+h

From the fact that {M ^* _t } is sufficiently approximated by {Y _t } and the assumption of boundedness on {Y _t }, {M ^* _t } also have similar boundedness (i.e., ${\mathbb{E}(M^{*2}_t) < K^{'} < \infty}$). Under this property and the assumption |ρ ^* _t | < 1, {M ^* _t } also have the infinite moving average expression,

$$ \widetilde{M}^{*}_{T+h}=\psi_0\zeta_{T+h}+\psi_{T+h-1}\zeta_{T+h-1}+\cdots, $$

where ψ₀ = 1 and $\psi_{T+h-i}=\beta^i_1\prod_{j=1}^iC_{T+h-j}\;(i=1,2,\ldots$). Now we assume an h-step ahead predictor,

$$ \widetilde{M}^{*}_{T}(h)=\psi^{*}_T\zeta_{T}+\psi^{*}_{T-1}\zeta_{T-1}+\cdots. $$

Then the mean squared error of prediction is

$$ E(M^{*}_{T+h}-\widetilde{M}^{*}_{T}(h))^2 =(\psi^2_0+\psi^2_{T+h-1}\cdots+\psi^2_{T+1})\sigma^2_{\zeta}+ \sum_{j=0}^{\infty}(\psi_{T+j}-\psi^{*}_{T+j})^2\sigma^2_{\zeta}, $$

which is minimized by ψ ^* _T+j  = ψ _T+j . Therefore $\widetilde{M}^{*}_{T}(h)$ can be expressed by

$$ \widetilde{M}^{*}_{T}(h)=\psi_T\zeta_{T}+\psi_{T+1}\zeta_{T-1}+\cdots, $$

This means that M ^* _T+h can be written by

$$ \begin{aligned} M^{*}_{T+h} &= \psi_0\zeta_{T+h}+\psi_{T+h-1}\zeta_{T+h-1}+\cdots+\psi_{T+1}\zeta_{T+1}+ \widetilde{M}^{*}_{T}(h) \\ & \equiv e_{T}(h)+\widetilde{M}^{*}_{T}(h), \end{aligned} $$

where e _T (h) means prediction errors. Thus the conditional variance of M ^* _T+h is

$$ {\mathbb{V}}(M^{*}_{T+h}|{\mathbf{V}}) = E(e^2_{T}(h)) =\left( 1+\sum_{j=1}^{h-1}\psi^2_{T+h-j}\right) \sigma^2_{\zeta}, $$

where the values of $\sigma^2_{\zeta}$ and ψ _T+h−j can be obtained by the residual variance based on the fitting of (12) and $\tilde{\psi}_{T+h-j}=\hat{\beta}^j_1\prod_{k=1}^{j}C_{T+h-k},$ respectively.

1.5 Definitions of nonstationary time series models for comparison

• (a) Autoregressive integrated moving average model (ARIMA) (Box and Jenkins 1976)

  $$ \nabla z_t = \sum_{i=1}^{q}b_i\nabla z_{t-i}+\xi_{t},\quad \xi_{t}\sim WN (0,\sigma^{2}_{\xi}), $$

  where b _i is an unknown constant and ∇ z _t ≡ z _t  − z _t−1 and ξ _t is a random variable following a zero-mean white noise process.

• (b) Time varying coefficients autoregressive model (TVAR) (Kitagawa and Gersch 1985)

  $$ \begin{aligned} z_t=&\, \sum_{i=1}^q \beta_{i,t} z_{t-i} +\xi_{t}, \quad \xi_{t}\sim WN(0,\Upsigma), \\ &\beta_{i,t}-\beta_{i,t-1}=u_{i,t}, \quad i=1,\ldots,q, \\ &(u_{1,t},\ldots,u_{q,t})\sim N_{q}(0,\Upsigma_q), \\ \end{aligned} $$

  where ξ _t is a random variable following a zero-mean white noise process, and {u _i,t } is Gaussian.

• (c) A nonstationary multivariate time series model (VAR-chla)

  $$ \begin{aligned} \nabla{\mathbf{z}}_{t} =& \,G_{1}\nabla{\mathbf{z}}_{t-1} +\cdots+ G_{q}\nabla{\mathbf{z}}_{t-q}+ {\varvec{\xi}}_{t}, \quad {\varvec{\xi}}_{t}\sim WN(0, \Upsigma), \\ {\mathbf{z}_{t}} =&\, (V_t, \,C_t)', \end{aligned} $$

  where $G_{i}\,(i=1,\ldots,p$) is an unknown coefficient matrix, and ${\varvec{\xi}}_{t}$ denote random white-noise variables with mean zero and covariance matrix $\Upsigma$.