\begin{algorithm}
\caption{Kalman filter algorithm}\label{euclidKF1}
\begin{algorithmic}[1]

\State Initialize with initial state $\Hat{X}_{(0)} = x_{(0)}$ and $\Hat{\boldsymbol{P}}_{(0)} = \boldsymbol{Q}^*$ at $t=0$;
\For {$t = 1,...,\textit{T}$}
\State $X_{(t)}^{t-1} = \boldsymbol{A}^* \Hat{X}_{(t-1)} + \boldsymbol{B}^* u_{(t)}$,  \quad \text{State estimate extrapolation};
\State $\boldsymbol{P}_{(t)}^{t-1} = \boldsymbol{A}^* \Hat{\boldsymbol{P}}_{(t-1)}\boldsymbol{A}^{*'}  + \boldsymbol{Q}^*$,  \quad \text{State covariance extrapolation};
\State $\boldsymbol{K}_{(t)} = \boldsymbol{P}_{(t)}^{t-1} \boldsymbol{C}^{*'}[\boldsymbol{R}^* + \boldsymbol{C}^* \boldsymbol{P}_{(t)}^{t-1} \boldsymbol{C}^{*'}]^{-1}$,  \quad \text{Kalman gain matrix};
\State $\Hat{X}_{(t)} = X_{(t)}^{t-1} + \boldsymbol{K}_{(t)} [Y_{(t)} - \boldsymbol{C}^* X_{(t)}^{t-1}]$, \quad \text{State estimate update};
\State $\Hat{\boldsymbol{P}}_{(t)} = [\boldsymbol{I} - \boldsymbol{K}_{(t)} \boldsymbol{C}^*] \boldsymbol{P}_{(t)}^{t-1} $,        \quad \text{State covariance update};
\State Compute the log-likelihood contribution, $l_{(t)}^{\mathrm{KF}}$, at time $t$ through the density $\mbox{MVN}(Y_{(t)} - \boldsymbol{C}^* X_{(t)}^{t-1}, \boldsymbol{R}^* + \boldsymbol{C}^* \boldsymbol{P}_{(t)}^{t-1} \boldsymbol{C}^{*'})$;
\EndFor
\State The complete log-likelihood can be calculated as $L^* = \sum_t l_{(t)}^{\mathrm{KF}}$
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
\caption{Kalman filter algorithm}\begin{algorithmic}
[1]
\State Initialize with initial state $\Hat{X}_{(0)} = x_{(0)}$ and $\Hat{\boldsymbol{P}}_{(0)} = \boldsymbol{Q}^*$ at $t=0$;
\For {$t = 1,...,\textit{T}$}
\State $X_{(t)}^{t-1} = \boldsymbol{A}^* \Hat{X}_{(t-1)} + \boldsymbol{B}^* u_{(t)}$, \quad \text{State estimate extrapolation};
\State $\boldsymbol{P}_{(t)}^{t-1} = \boldsymbol{A}^* \Hat{\boldsymbol{P}}_{(t-1)}\boldsymbol{A}^{*'} + \boldsymbol{Q}^*$, \quad \text{State covariance extrapolation};
\State $\boldsymbol{K}_{(t)} = \boldsymbol{P}_{(t)}^{t-1} \boldsymbol{C}^{*'}[\boldsymbol{R}^* + \boldsymbol{C}^* \boldsymbol{P}_{(t)}^{t-1} \boldsymbol{C}^{*'}]^{-1}$, \quad \text{Kalman gain matrix};
\State $\Hat{X}_{(t)} = X_{(t)}^{t-1} + \boldsymbol{K}_{(t)} [Y_{(t)} - \boldsymbol{C}^* X_{(t)}^{t-1}]$, \quad \text{State estimate update};
\State $\Hat{\boldsymbol{P}}_{(t)} = [\boldsymbol{I} - \boldsymbol{K}_{(t)} \boldsymbol{C}^*] \boldsymbol{P}_{(t)}^{t-1} $, \quad \text{State covariance update};
\State Compute the log-likelihood contribution, $l_{(t)}^{\mathrm{KF}}$, at time $t$ through the density $\mbox{MVN}(Y_{(t)} - \boldsymbol{C}^* X_{(t)}^{t-1}, \boldsymbol{R}^* + \boldsymbol{C}^* \boldsymbol{P}_{(t)}^{t-1} \boldsymbol{C}^{*'})$;
\EndFor
\State The complete log-likelihood can be calculated as $L^* = \sum_t l_{(t)}^{\mathrm{KF}}$
\end{algorithmic}
\end{algorithm}
{
    "title": "innovative approaches in soil carbon sequestration modelling for better   prediction with limited data",
    "id": "2105.04789",
    "abstract": "soil carbon accounting and prediction play a key role in building decision support systems for land managers selling carbon credits, in the spirit of the paris and kyoto protocol agreements. land managers typically rely on computationally complex models fit using sparse datasets to make these accounts and predictions. the model complexity and sparsity of the data can lead to over-fitting, leading to inaccurate results when making predictions with new data. modellers address over-fitting by simplifying their models and reducing the number of parameters, and in the current context this could involve neglecting some soil organic carbon (soc) components. in this study, we introduce two novel soc models and a new rothc-like model and investigate how the soc components and complexity of the soc models affect the soc prediction in the presence of small and sparse time series data. we develop model selection methods that can identify the soil carbon model with the best predictive performance, in light of the available data. through this analysis we reveal that commonly used complex soil carbon models can over-fit in the presence of sparse time series data, and our simpler models can produce more accurate predictions. the published version of this study is available in scientific reports (https://www.nature.com/articles/s41598-024-53516-z/<10.1038/s41598-024-53516-z>)",
    "categories": "stat.co stat.ap",
    "doi": "10.1038/s41598-024-53516-z",
    "created": "2021-05-11",
    "updated": "2024-02-10",
    "authors": [
        "mohammad javad davoudabadi",
        "daniel pagendam",
        "christopher drovandi",
        "jeff baldock",
        "gentry white"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2105.04789"
}
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