Project: s41467-024-45514-6 | Function: forecast (python)

system: You are an expert Python programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow Python best practices and maintain consistent code style.
Return **only** the complete Python function code.

user: Complete this Python function:

**DESCRIPTION:** Estimate a forecast using DGA with memory for computing forward committor and mean first passage time

**KEY TERMS:** {
  "basis": "basis for estimating the solution",
  "in_domain": "whether each frame is in the domain",
  "function": "function to integrate",
  "guess": "guess for the solution",
  "lag": "maximum lag time",
  "mem": "number of memory terms"
}

**CONTEXT CODE:**         Basis for estimating the stationary distribution. Each basis
        function must have zero mean with respect to `weights`.
    weights : Sequence[NDArray[float]], shape=(n_trajs, n_frames[traj])
        Weight of each frame. The last `lag` frames of each trajectory
        must be zero.
    lag : int
        Maximum lag time in units of frames.
    mem : int
        Number of memory terms to use. These are evaluated at equally
        spaced times between time 0 and time `lag`, so `mem+1` must
        evenly divide `lag`. For example, with a `lag=32`, `mem=3` and
        `mem=7` are fine since 7+1=8 and 3+1=4 evenly divide 32. Setting
        `mem=0` corresponds to not using memory.
    coef : NDArray[float], shape=(n_basis,)
        Projection coefficients.
    mem_coef : NDArray[float], shape=(mem, n_basis)
        Memory-correction coefficients.

    Returns
    -------
    Sequence[NDArray[float]], shape=(n_trajs, n_frames[traj])
        Estimate of the stationary distribution.

    """
    assert lag % (mem + 1) == 0
    dlag = lag // (mem + 1)
    n_basis = basis[0].shape[1]

    out = []
    for y, w in zip(basis, weights, strict=True):
        n_frames = y.shape[0]
        assert y.shape == (n_frames, n_basis)
        assert w.shape == (n_frames,)

        if n_frames <= lag:
            assert np.all(w == 0.0)
            out.append(np.zeros(n_frames))
            continue
        assert np.all(w[-lag:] == 0.0)

        pad = np.zeros(dlag)
        u = w * (y @ coef + 1.0)
        for v in mem_coef:
            u = np.concatenate([pad, u[:-dlag]])
            u -= w * (y @ v)
        u = np.concatenate([pad, u[:-dlag]])
        out.append(u)
    return out


def forecast(
    basis,
    weights,
    in_domain,
    function,
    guess,
    lag,
    mem,
    test_basis=None,
    *,
    return_projection=False,
    return_solution=True,
    return_coef=False,
    return_mem_coef=False,
):
    """
    Estimate a forecast using DGA with memory.

    To solve for the forward committor, set `function=0`.
    To solve for the mean first passage time, set `function=dt` where
    `dt` is the elapsed time between frames.

    Parameters
    ----------
    basis : Sequence[NDArray[float] | SparseArray[float]], \
            shape=(n_trajs, n_frames[traj], n_basis)
        Basis for estimating the solution. Must be zero outside of the
        domain.
    weights : Sequence[NDArray[float]], shape=(n_trajs, n_frames[traj])
        Weight of each frame. The last `lag` frames of each trajectory
        must be zero.
    in_domain : Sequence[NDArray[bool]], shape=(n_trajs, n_frames[traj])
        Whether each frame is in the domain.
    function : Sequence[NDArray[float]], shape=(n_trajs, n_frames[traj]-1)
        Function to integrate. This is defined over *steps*, not frames.
    guess : Sequence[NDArray[float]], shape=(n_trajs, n_frames[traj])
        Guess for the solution. Must satisfy boundary conditions.
    lag : int
        Maximum lag time in units of frames.
    mem : int
        Number of memory terms to use. These are evaluated at equally
        spaced times between time 0 and time `lag`, so `mem+1` must
        evenly divide `lag`. For example, with a `lag=32`, `mem=3` and
        `mem=7` are fine since 7+1=8 and 3+1=4 evenly divide 32. Setting
        `mem=0` corresponds to not using memory.
    test_basis : Sequence[NDArray[float] | SparseArray[float]], \
            shape=(n_trajs, n_frames[traj], n_basis), optional
        Test basis against which to minimize the error. Must have the
        same shape as `basis`. If `None`, use `basis`.
    return_projection : bool, optional
        If True, return the estimated projection.
    return_solution : bool, optional
        If True (default), return the estimated solution.
    return_coef : bool, optional
        If True, return the projection coefficients.
    return_mem_coef : bool, optional
        If True, return the memory-correction coefficients.

    Returns
    -------
    projection : Sequence[NDArray[float]], shape=(n_trajs, n_frames[traj])
        Estimate of the projected forecast.
    solution : Sequence[NDArray[float]], shape=(n_trajs, n_frames[traj])
        Estimate of the forecast.
    coef : NDArray[float], shape=(n_basis,)
        Projection coefficients.
    mem_coef : NDArray[float], shape=(mem, n_basis)
        Memory-correction coefficients.

    """
    # MASKED: Estimate a forecast using DGA with memory for computing forward committor and mean first passage time implementation
    raise NotImplementedError("This function is masked for testing purposes.")


def forecast_matrices(
    basis, weights, in_domain, function, guess, lag, mem, test_basis=None
):
    """
    Compute matrices for estimating a forecast using DGA with memory.

    Parameters
    ----------
    basis : Sequence[NDArray[float] | SparseArray[float]], \
            shape=(n_trajs, n_frames[traj], n_basis)
        Basis for estimating the solution. Must be zero outside of the
        domain.
    weights : Sequence[NDArray[float]], shape=(n_trajs, n_frames[traj])
        Weight of each frame. The last `lag` frames of each trajectory
        must be zero.
    in_domain : Sequence[NDArray[bool]], shape=(n_trajs, n_frames[traj])
        Whether each frame is in the domain.
    function : Sequence[NDArray[float]], shape=(n_trajs, n_frames[traj]-1)
        Function to integrate. This is defined over *steps*, not frames.
    guess : Sequence[NDArray[float]], shape=(n_trajs, n_frames[traj])
        Guess for the solution. Must satisfy boundary conditions.
    lag : int
        Maximum lag time in units of frames.
    mem : int
        Number of memory terms to use. These are evaluated at equally
        spaced times between time 0 and time `lag`, so `mem+1` must


Complete the function implementation replacing "raise NotImplementedError".