A lightweight MATLAB framework for recursive LSTM-based correction of numerical weather prediction wind speed forecasts.
This Long Short-Term Memory (LSTM) model is designed to predict $x_t$, the forecast error, given the history of the process available up to time $t-1$, i.e, $\mathcal{F}_{t-1} := { x_{t-1}, x_{t-2}, \dots, x_{t-p} }$, where $p$ is the length of the history considered for the prediction.
$$
x_t = w_t - g_{t-h}^{(h)} \Leftrightarrow w_t = g_{t-h}^{(h)} + x_t
$$
where $w_t$ denotes the observed wind speed at time $t$, and $g_{t-h}^{(h)}$ is the GFS forecast issued at time $t-h$ for a lead time $h \in \mathbb{N}_{0}$. The latter is fully determined at time $t$, since both $w_t$ and $g_{t-h}^{(h)}$ are available and refer to the same time instant. Thus, the corrected GFS forecast is defined as
$$
\begin{equation}
\hat{w}_t^{(h)} = g_t^{(h)} + \tilde{x}_t^{(h)}
\end{equation}
$$
where $\tilde{x}_t^{(h)}$ denotes an estimate of the unknown future forecast error. In this work a correction composed of two components is considered
$$
\begin{equation}
\tilde{x}_t^{(h)} = \bar{m}_{t} + \hat{x}_{t}^{(h)}
\end{equation}
$$
where $\bar{m}_{t}$ is a mean bias correction term and $\hat{x}_{t}^{(h)}$ is the forecast error predicted by the LSTM model. The bias correction term is defined as
$$
\begin{equation}
\bar{m}_t=\frac{1}{t}\sum_{i=1}^{t}x_i,
\end{equation}
$$
corresponding to the average of past forecast errors available up to time $t$.
Therefore, the LSTM model predicts $x_t$ given the information in $\mathcal{F}_{t-1}$, or mathematically,
$$
\hat{x}_t = f(\mathcal{F}_{t-1})
$$
where $f(.)$ is a general function that expresses the operations of the LSTM model.
To predict for $h = 1, 2, \dots$ steps ahead, the model is adapted to predict $x_{t+h}$ by recursively feeding its own predictions back into the input for each subsequent time step, i.e.,
$$
\hat{x}_{t}^{(h)} = f(\mathcal{F}_{t-1+h}),
$$
where $\mathcal{F}_{t-1+h} := {\hat{x}_{t-1+h}, \dots, \hat{x}_t, x_{t-1}, x_{t-2}, \dots, x_{t-p-h}}$ is the input history for future steps, including previous predictions $\hat{x}$ and original observed data $x$. Since the model is sequence-to-one, it receives as input the vector $\mathcal{F}_{t-1}$ of length $p$ and predicts a single continuous value, denoted as $x_t$. The fundamental component in this architecture is the LSTM cell, which consists of three gates, namely forget $f$, input $i$ and output $o$, summarized as follows
$$
\begin{align}
f_t &= \sigma \big(W_f [h_{t-1}, x_t] + b_f\big), \nonumber \\
i_t &= \sigma \big(W_i [h_{t-1}, x_t] + b_i\big), \nonumber \\
\tilde{C}_t &= \tanh \big(W_C [h_{t-1}, x_t] + b_C\big), \nonumber \\
C_t &= f_t \odot C_{t-1} + i_t \odot \tilde{C}_t, \nonumber \\
o_t &= \sigma \big(W_o [h_{t-1}, x_t] + b_o\big), \nonumber \\
h_t &= o_t \odot \tanh(C_t), \nonumber
\end{align}
$$
where $\sigma$ denotes the sigmoid function, $W_{\cdot}$ and $b_{\cdot}$ are weight matrices and bias vectors, respectively, and $\odot$ denotes element-wise multiplication.
The model outputs the predicted value for $x_t$ and is computed from the final hidden state $h_t$ as
$$
\hat{x}_t = W_y h_t + b_y,
$$
where $W_y$ and $b_y$ are weights and biases of the fully connected layer. The referred $W_{\cdot}$ matrices and $b_{\cdot}$ vectors are trainable parameters of the LSTM model.
This framework has been implemented in MATLAB, and synthetic data is provided along to illustrate how the model is employed.
To properly cite this work use: Gomes V, Martins A, Carvalho D, Gouveia S. (2026) Recursive Multi-step-ahead LSTM Correction of GFS Wind Speed Forecasts in Portugal. Pattern Analysis and Applications.
