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Joint Models

Models that fit donut and spot images simultaneously for a shared set of wavefront coefficients.

Base class

danish.joint_model.JointModel

Joint donut + spot model for simultaneous fitting.

Composes a donut sub-model and a spot sub-model that share atmospheric and wavefront parameters. Per-star parameters (fluxes, dxs, dys, bkgs) are independent between the two modalities.

Parameters:

Name Type Description Default
donut_model BaseMultiDonutModel subclass

Pre-constructed donut fitter.

 required
spot_model BaseMultiSpotModel subclass

Pre-constructed spot fitter.

 required
spot_weight float

Relative weight of spot chi residuals. The spot chi values are multiplied by sqrt(spot_weight) so that the cost contribution is scaled by spot_weight. Default 1.0.

 1.0
Source code in danish/joint_model.py 
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class JointModel:
    """Joint donut + spot model for simultaneous fitting.

    Composes a donut sub-model and a spot sub-model that share atmospheric and
    wavefront parameters.  Per-star parameters (fluxes, dxs, dys, bkgs) are
    independent between the two modalities.

    Parameters
    ----------
    donut_model : BaseMultiDonutModel subclass
        Pre-constructed donut fitter.
    spot_model : BaseMultiSpotModel subclass
        Pre-constructed spot fitter.
    spot_weight : float, optional
        Relative weight of spot chi residuals.  The spot chi values are
        multiplied by sqrt(spot_weight) so that the cost contribution is
        scaled by spot_weight.  Default 1.0.
    """
    def __init__(self, donut_model, spot_model, spot_weight=1.0):
        if donut_model.atm_mode != spot_model.atm_mode:
            raise ValueError(
                f"atm_mode mismatch: donut={donut_model.atm_mode}, "
                f"spot={spot_model.atm_mode}"
            )
        if donut_model.wavefront_step != spot_model.wavefront_step:
            raise ValueError(
                f"wavefront_step mismatch: donut={donut_model.wavefront_step}, "
                f"spot={spot_model.wavefront_step}"
            )

        self.donut_model = donut_model
        self.spot_model = spot_model
        self.spot_weight = spot_weight
        self.sqrt_spot_weight = np.sqrt(spot_weight)

        self.nd = donut_model.nstar
        self.ns = spot_model.nstar
        self.atm_mode = donut_model.atm_mode
        self.natm = donut_model.natm
        self.nbkg_d = donut_model.nbkg
        self.nbkg_s = spot_model.nbkg
        self.wavefront_step = donut_model.wavefront_step

        self.npix_d = donut_model.npix
        self.npix_s = spot_model.npix
        self.nchi_d = self.nd * self.npix_d ** 2
        self.nchi_s = self.ns * self.npix_s ** 2

    def pack_params(
        self, *,
        d_fluxes, d_dxs, d_dys,
        s_fluxes, s_dxs, s_dys,
        fwhm=None, Ixx=None, Ixy=None, Iyy=None,
        wavefront_params,
        d_bkgs=None, s_bkgs=None
    ):
        """Pack joint parameters into a single tuple.

        Layout: [d_fluxes | d_dxs | d_dys | s_fluxes | s_dxs | s_dys |
                 atm(natm) | wavefront | d_bkgs | s_bkgs]
        """
        if d_bkgs is None:
            d_bkgs = [()] * self.nd
        if s_bkgs is None:
            s_bkgs = [()] * self.ns
        params = []
        params.extend(d_fluxes)
        params.extend(d_dxs)
        params.extend(d_dys)
        params.extend(s_fluxes)
        params.extend(s_dxs)
        params.extend(s_dys)
        if self.atm_mode == 'fwhm':
            params.append(fwhm)
        else:
            params.extend([Ixx, Ixy, Iyy])
        params.extend(wavefront_params)
        for bkg in d_bkgs:
            params.extend(bkg)
        for bkg in s_bkgs:
            params.extend(bkg)
        return tuple(params)

    def unpack_params(self, params):
        """Unpack joint parameters from optimization tuple.

        Returns
        -------
        dict with keys:
            d_fluxes, d_dxs, d_dys, s_fluxes, s_dxs, s_dys,
            fwhm or (Ixx, Ixy, Iyy), wavefront_params, d_bkgs, s_bkgs
        """
        nd = self.nd
        ns = self.ns
        natm = self.natm

        idx = 0
        d_fluxes = params[idx:idx+nd]; idx += nd
        d_dxs = params[idx:idx+nd]; idx += nd
        d_dys = params[idx:idx+nd]; idx += nd
        s_fluxes = params[idx:idx+ns]; idx += ns
        s_dxs = params[idx:idx+ns]; idx += ns
        s_dys = params[idx:idx+ns]; idx += ns

        out = dict(
            d_fluxes=d_fluxes, d_dxs=d_dxs, d_dys=d_dys,
            s_fluxes=s_fluxes, s_dxs=s_dxs, s_dys=s_dys,
        )

        if self.atm_mode == 'fwhm':
            out['fwhm'] = params[idx]; idx += 1
        else:
            out['Ixx'] = params[idx]; idx += 1
            out['Ixy'] = params[idx]; idx += 1
            out['Iyy'] = params[idx]; idx += 1

        # Wavefront params: everything between atm and bkgs
        nbkg_total = self.nbkg_d * nd + self.nbkg_s * ns
        nwf = len(params) - idx - nbkg_total
        out['wavefront_params'] = params[idx:idx+nwf]; idx += nwf

        d_bkgs = []
        for i in range(nd):
            d_bkgs.append(tuple(params[idx:idx+self.nbkg_d]))
            idx += self.nbkg_d
        out['d_bkgs'] = d_bkgs

        s_bkgs = []
        for i in range(ns):
            s_bkgs.append(tuple(params[idx:idx+self.nbkg_s]))
            idx += self.nbkg_s
        out['s_bkgs'] = s_bkgs

        return out

    def _donut_packed(self, joint_dict):
        """Build packed params for the donut sub-model from joint dict."""
        kw = dict(
            fluxes=joint_dict['d_fluxes'],
            dxs=joint_dict['d_dxs'],
            dys=joint_dict['d_dys'],
            wavefront_params=joint_dict['wavefront_params'],
            bkgs=joint_dict['d_bkgs'],
        )
        if self.atm_mode == 'fwhm':
            kw['fwhm'] = joint_dict['fwhm']
        else:
            kw['Ixx'] = joint_dict['Ixx']
            kw['Ixy'] = joint_dict['Ixy']
            kw['Iyy'] = joint_dict['Iyy']
        return self.donut_model.pack_params(**kw)

    def _spot_packed(self, joint_dict):
        """Build packed params for the spot sub-model from joint dict."""
        kw = dict(
            fluxes=joint_dict['s_fluxes'],
            dxs=joint_dict['s_dxs'],
            dys=joint_dict['s_dys'],
            wavefront_params=joint_dict['wavefront_params'],
            bkgs=joint_dict['s_bkgs'],
        )
        if self.atm_mode == 'fwhm':
            kw['fwhm'] = joint_dict['fwhm']
        else:
            kw['Ixx'] = joint_dict['Ixx']
            kw['Ixy'] = joint_dict['Ixy']
            kw['Iyy'] = joint_dict['Iyy']
        return self.spot_model.pack_params(**kw)

    def model(self, **kwargs):
        """Compute model images for both donuts and spots.

        Parameters
        ----------
        **kwargs : as returned by unpack_params

        Returns
        -------
        donut_imgs : array, shape (nd, npix_d, npix_d)
        spot_imgs : array, shape (ns, npix_s, npix_s)
        """
        d_packed = self._donut_packed(kwargs)
        s_packed = self._spot_packed(kwargs)
        d_imgs = self.donut_model.model(
            **self.donut_model.unpack_params(d_packed)
        )
        s_imgs = self.spot_model.model(
            **self.spot_model.unpack_params(s_packed)
        )
        return d_imgs, s_imgs

    def chi(self, params, donut_data, donut_vars, spot_data, spot_vars):
        """Compute joint chi residuals.

        Parameters
        ----------
        params : sequence of float
            Joint packed parameters.
        donut_data : array, shape (nd, npix_d, npix_d)
        donut_vars : sequence of float or array
        spot_data : array, shape (ns, npix_s, npix_s)
        spot_vars : sequence of float or array

        Returns
        -------
        chi : array of float
            Concatenated [donut_chi, sqrt_spot_weight * spot_chi].
        """
        joint_dict = self.unpack_params(params)
        d_packed = self._donut_packed(joint_dict)
        s_packed = self._spot_packed(joint_dict)
        d_chi = self.donut_model.chi(d_packed, donut_data, donut_vars)
        s_chi = self.spot_model.chi(s_packed, spot_data, spot_vars)
        return np.concatenate([d_chi, self.sqrt_spot_weight * s_chi])

    def jac(self, params, donut_data, donut_vars, spot_data, spot_vars):
        """Compute joint jacobian d(chi)/d(param).

        Uses block-sparse structure: donut per-star params only affect donut
        rows, spot per-star params only affect spot rows, shared atm+wavefront
        params affect all rows.

        Parameters
        ----------
        params : sequence of float
        donut_data, donut_vars, spot_data, spot_vars : arrays

        Returns
        -------
        jac : array of float, shape (nchi_d + nchi_s, nparams)
        """
        nd = self.nd
        ns = self.ns
        natm = self.natm
        nbkg_d = self.nbkg_d
        nbkg_s = self.nbkg_s
        nchi_d = self.nchi_d
        nchi_s = self.nchi_s
        nparams = len(params)

        # Compute the number of wavefront params
        nperstar = 3*nd + 3*ns
        nbkg_total = nbkg_d*nd + nbkg_s*ns
        nwf = nparams - nperstar - natm - nbkg_total

        out = np.zeros((nchi_d + nchi_s, nparams))
        chi0 = self.chi(params, donut_data, donut_vars, spot_data, spot_vars)
        chi0_d = chi0[:nchi_d]
        chi0_s = chi0[nchi_d:]

        joint_dict = self.unpack_params(params)
        d_packed = np.array(self._donut_packed(joint_dict), dtype=float)
        s_packed = np.array(self._spot_packed(joint_dict), dtype=float)

        # --- Donut per-star sparse params (d_fluxes, d_dxs, d_dys) ---
        # These only affect donut chi rows.
        # d_fluxes columns: [0, nd)
        npix_d = self.npix_d
        dflux = 0.01
        d_dict = self.donut_model.unpack_params(d_packed)
        d_dict["fluxes"] = np.array(d_dict["fluxes"]) + dflux
        chi_d = self.donut_model.chi(
            self.donut_model.pack_params(**d_dict), donut_data, donut_vars
        )
        for i in range(nd):
            s = slice(i*npix_d**2, (i+1)*npix_d**2)
            out[s, i] = (chi_d[s] - chi0_d[s]) / dflux

        # d_dxs columns: [nd, 2*nd)
        dx = 0.01
        d_dict = self.donut_model.unpack_params(d_packed)
        d_dict["dxs"] = np.array(d_dict["dxs"]) + dx
        chi_d = self.donut_model.chi(
            self.donut_model.pack_params(**d_dict), donut_data, donut_vars
        )
        for i in range(nd):
            s = slice(i*npix_d**2, (i+1)*npix_d**2)
            out[s, nd+i] = (chi_d[s] - chi0_d[s]) / dx

        # d_dys columns: [2*nd, 3*nd)
        dy = 0.01
        d_dict = self.donut_model.unpack_params(d_packed)
        d_dict["dys"] = np.array(d_dict["dys"]) + dy
        chi_d = self.donut_model.chi(
            self.donut_model.pack_params(**d_dict), donut_data, donut_vars
        )
        for i in range(nd):
            s = slice(i*npix_d**2, (i+1)*npix_d**2)
            out[s, 2*nd+i] = (chi_d[s] - chi0_d[s]) / dy

        # --- Spot per-star sparse params (s_fluxes, s_dxs, s_dys) ---
        # These only affect spot chi rows.
        # s_fluxes columns: [3*nd, 3*nd+ns)
        npix_s = self.npix_s
        s_dict = self.spot_model.unpack_params(s_packed)
        s_dict["fluxes"] = np.array(s_dict["fluxes"]) + dflux
        chi_s = self.spot_model.chi(
            self.spot_model.pack_params(**s_dict), spot_data, spot_vars
        )
        for i in range(ns):
            s = slice(i*npix_s**2, (i+1)*npix_s**2)
            out[nchi_d+s.start:nchi_d+s.stop, 3*nd+i] = (
                self.sqrt_spot_weight * (chi_s[s] - chi0_s[s] / self.sqrt_spot_weight)
            ) / dflux

        # s_dxs columns: [3*nd+ns, 3*nd+2*ns)
        s_dict = self.spot_model.unpack_params(s_packed)
        s_dict["dxs"] = np.array(s_dict["dxs"]) + dx
        chi_s = self.spot_model.chi(
            self.spot_model.pack_params(**s_dict), spot_data, spot_vars
        )
        for i in range(ns):
            s = slice(i*npix_s**2, (i+1)*npix_s**2)
            out[nchi_d+s.start:nchi_d+s.stop, 3*nd+ns+i] = (
                self.sqrt_spot_weight * (chi_s[s] - chi0_s[s] / self.sqrt_spot_weight)
            ) / dx

        # s_dys columns: [3*nd+2*ns, 3*nd+3*ns)
        s_dict = self.spot_model.unpack_params(s_packed)
        s_dict["dys"] = np.array(s_dict["dys"]) + dy
        chi_s = self.spot_model.chi(
            self.spot_model.pack_params(**s_dict), spot_data, spot_vars
        )
        for i in range(ns):
            s = slice(i*npix_s**2, (i+1)*npix_s**2)
            out[nchi_d+s.start:nchi_d+s.stop, 3*nd+2*ns+i] = (
                self.sqrt_spot_weight * (chi_s[s] - chi0_s[s] / self.sqrt_spot_weight)
            ) / dy

        # --- Shared atm + wavefront params (dense: affect all rows) ---
        shared_start = 3*nd + 3*ns
        for k in range(natm):
            params1 = np.array(params, dtype=float)
            params1[shared_start + k] += 0.01
            chi1 = self.chi(
                params1, donut_data, donut_vars, spot_data, spot_vars
            )
            out[:, shared_start + k] = (chi1 - chi0) / 0.01

        for k in range(nwf):
            col = shared_start + natm + k
            params1 = np.array(params, dtype=float)
            params1[col] += self.wavefront_step
            chi1 = self.chi(
                params1, donut_data, donut_vars, spot_data, spot_vars
            )
            out[:, col] = (chi1 - chi0) / self.wavefront_step

        # --- Donut bkg params (sparse: only donut rows) ---
        dbkg = 0.01
        bkg_d_start = shared_start + natm + nwf
        for k in range(nbkg_d):
            d_dict = self.donut_model.unpack_params(d_packed)
            for j in range(nd):
                bkgj = list(d_dict["bkgs"][j])
                bkgj[k] += dbkg
                d_dict["bkgs"][j] = tuple(bkgj)
            chi_d = self.donut_model.chi(
                self.donut_model.pack_params(**d_dict), donut_data, donut_vars
            )
            for j in range(nd):
                s = slice(j*npix_d**2, (j+1)*npix_d**2)
                out[s, bkg_d_start + nbkg_d*j + k] = (
                    chi_d[s] - chi0_d[s]
                ) / dbkg

        # --- Spot bkg params (sparse: only spot rows) ---
        bkg_s_start = bkg_d_start + nbkg_d * nd
        for k in range(nbkg_s):
            s_dict = self.spot_model.unpack_params(s_packed)
            for j in range(ns):
                bkgj = list(s_dict["bkgs"][j])
                bkgj[k] += dbkg
                s_dict["bkgs"][j] = tuple(bkgj)
            chi_s = self.spot_model.chi(
                self.spot_model.pack_params(**s_dict), spot_data, spot_vars
            )
            for j in range(ns):
                s = slice(j*npix_s**2, (j+1)*npix_s**2)
                chi0_sj = chi0_s[s] / self.sqrt_spot_weight
                out[nchi_d+s.start:nchi_d+s.stop, bkg_s_start + nbkg_s*j + k] = (
                    self.sqrt_spot_weight * (chi_s[s] - chi0_sj)
                ) / dbkg

        return out

    def _jac2(self, params, donut_data, donut_vars, spot_data, spot_vars):
        """Naive column-by-column jacobian for validation."""
        nd = self.nd
        ns = self.ns
        natm = self.natm
        nbkg_d = self.nbkg_d
        nbkg_s = self.nbkg_s
        nparams = len(params)

        nperstar = 3*nd + 3*ns
        nbkg_total = nbkg_d*nd + nbkg_s*ns
        nwf = nparams - nperstar - natm - nbkg_total

        nchi = self.nchi_d + self.nchi_s
        out = np.empty((nchi, nparams))

        step = [0.01]*(3*nd)  # d_fluxes, d_dxs, d_dys
        step += [0.01]*(3*ns)  # s_fluxes, s_dxs, s_dys
        step += [0.01]*natm
        step += [self.wavefront_step]*nwf
        step += [0.01]*(nbkg_d*nd)
        step += [0.01]*(nbkg_s*ns)

        chi0 = self.chi(params, donut_data, donut_vars, spot_data, spot_vars)
        for i, h in enumerate(step):
            params1 = np.array(params, dtype=float)
            params1[i] += h
            chi1 = self.chi(
                params1, donut_data, donut_vars, spot_data, spot_vars
            )
            out[:, i] = (chi1 - chi0) / h

        return out

pack_params 

pack_params(*, d_fluxes, d_dxs, d_dys, s_fluxes, s_dxs, s_dys, fwhm=None, Ixx=None, Ixy=None, Iyy=None, wavefront_params, d_bkgs=None, s_bkgs=None)

Pack joint parameters into a single tuple.

Layout: [d_fluxes | d_dxs | d_dys | s_fluxes | s_dxs | s_dys | atm(natm) | wavefront | d_bkgs | s_bkgs]

Source code in danish/joint_model.py 
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def pack_params(
    self, *,
    d_fluxes, d_dxs, d_dys,
    s_fluxes, s_dxs, s_dys,
    fwhm=None, Ixx=None, Ixy=None, Iyy=None,
    wavefront_params,
    d_bkgs=None, s_bkgs=None
):
    """Pack joint parameters into a single tuple.

    Layout: [d_fluxes | d_dxs | d_dys | s_fluxes | s_dxs | s_dys |
             atm(natm) | wavefront | d_bkgs | s_bkgs]
    """
    if d_bkgs is None:
        d_bkgs = [()] * self.nd
    if s_bkgs is None:
        s_bkgs = [()] * self.ns
    params = []
    params.extend(d_fluxes)
    params.extend(d_dxs)
    params.extend(d_dys)
    params.extend(s_fluxes)
    params.extend(s_dxs)
    params.extend(s_dys)
    if self.atm_mode == 'fwhm':
        params.append(fwhm)
    else:
        params.extend([Ixx, Ixy, Iyy])
    params.extend(wavefront_params)
    for bkg in d_bkgs:
        params.extend(bkg)
    for bkg in s_bkgs:
        params.extend(bkg)
    return tuple(params)

unpack_params 

unpack_params(params)

Unpack joint parameters from optimization tuple.

Returns:

Type Description
dict with keys:

d_fluxes, d_dxs, d_dys, s_fluxes, s_dxs, s_dys, fwhm or (Ixx, Ixy, Iyy), wavefront_params, d_bkgs, s_bkgs
Source code in danish/joint_model.py 
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def unpack_params(self, params):
    """Unpack joint parameters from optimization tuple.

    Returns
    -------
    dict with keys:
        d_fluxes, d_dxs, d_dys, s_fluxes, s_dxs, s_dys,
        fwhm or (Ixx, Ixy, Iyy), wavefront_params, d_bkgs, s_bkgs
    """
    nd = self.nd
    ns = self.ns
    natm = self.natm

    idx = 0
    d_fluxes = params[idx:idx+nd]; idx += nd
    d_dxs = params[idx:idx+nd]; idx += nd
    d_dys = params[idx:idx+nd]; idx += nd
    s_fluxes = params[idx:idx+ns]; idx += ns
    s_dxs = params[idx:idx+ns]; idx += ns
    s_dys = params[idx:idx+ns]; idx += ns

    out = dict(
        d_fluxes=d_fluxes, d_dxs=d_dxs, d_dys=d_dys,
        s_fluxes=s_fluxes, s_dxs=s_dxs, s_dys=s_dys,
    )

    if self.atm_mode == 'fwhm':
        out['fwhm'] = params[idx]; idx += 1
    else:
        out['Ixx'] = params[idx]; idx += 1
        out['Ixy'] = params[idx]; idx += 1
        out['Iyy'] = params[idx]; idx += 1

    # Wavefront params: everything between atm and bkgs
    nbkg_total = self.nbkg_d * nd + self.nbkg_s * ns
    nwf = len(params) - idx - nbkg_total
    out['wavefront_params'] = params[idx:idx+nwf]; idx += nwf

    d_bkgs = []
    for i in range(nd):
        d_bkgs.append(tuple(params[idx:idx+self.nbkg_d]))
        idx += self.nbkg_d
    out['d_bkgs'] = d_bkgs

    s_bkgs = []
    for i in range(ns):
        s_bkgs.append(tuple(params[idx:idx+self.nbkg_s]))
        idx += self.nbkg_s
    out['s_bkgs'] = s_bkgs

    return out

model 

model(**kwargs)

Compute model images for both donuts and spots.

Parameters:

Name Type Description Default
**kwargs as returned by unpack_params
{}

Returns:

Name Type Description
donut_imgs (array, shape(nd, npix_d, npix_d))
spot_imgs (array, shape(ns, npix_s, npix_s))
Source code in danish/joint_model.py 
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def model(self, **kwargs):
    """Compute model images for both donuts and spots.

    Parameters
    ----------
    **kwargs : as returned by unpack_params

    Returns
    -------
    donut_imgs : array, shape (nd, npix_d, npix_d)
    spot_imgs : array, shape (ns, npix_s, npix_s)
    """
    d_packed = self._donut_packed(kwargs)
    s_packed = self._spot_packed(kwargs)
    d_imgs = self.donut_model.model(
        **self.donut_model.unpack_params(d_packed)
    )
    s_imgs = self.spot_model.model(
        **self.spot_model.unpack_params(s_packed)
    )
    return d_imgs, s_imgs

chi 

chi(params, donut_data, donut_vars, spot_data, spot_vars)

Compute joint chi residuals.

Parameters:

Name Type Description Default
params sequence of float

Joint packed parameters.

 required
donut_data (array, shape(nd, npix_d, npix_d))
required
donut_vars sequence of float or array
required
spot_data (array, shape(ns, npix_s, npix_s))
required
spot_vars sequence of float or array
required

Returns:

Name Type Description
chi array of float

Concatenated [donut_chi, sqrt_spot_weight * spot_chi].
Source code in danish/joint_model.py 
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def chi(self, params, donut_data, donut_vars, spot_data, spot_vars):
    """Compute joint chi residuals.

    Parameters
    ----------
    params : sequence of float
        Joint packed parameters.
    donut_data : array, shape (nd, npix_d, npix_d)
    donut_vars : sequence of float or array
    spot_data : array, shape (ns, npix_s, npix_s)
    spot_vars : sequence of float or array

    Returns
    -------
    chi : array of float
        Concatenated [donut_chi, sqrt_spot_weight * spot_chi].
    """
    joint_dict = self.unpack_params(params)
    d_packed = self._donut_packed(joint_dict)
    s_packed = self._spot_packed(joint_dict)
    d_chi = self.donut_model.chi(d_packed, donut_data, donut_vars)
    s_chi = self.spot_model.chi(s_packed, spot_data, spot_vars)
    return np.concatenate([d_chi, self.sqrt_spot_weight * s_chi])

jac 

jac(params, donut_data, donut_vars, spot_data, spot_vars)

Compute joint jacobian d(chi)/d(param).

Uses block-sparse structure: donut per-star params only affect donut rows, spot per-star params only affect spot rows, shared atm+wavefront params affect all rows.

Parameters:

Name Type Description Default
params sequence of float
required
donut_data arrays
required
donut_vars arrays
required
spot_data arrays
required
spot_vars arrays
required

Returns:

Name Type Description
jac array of float, shape (nchi_d + nchi_s, nparams)
Source code in danish/joint_model.py 
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def jac(self, params, donut_data, donut_vars, spot_data, spot_vars):
    """Compute joint jacobian d(chi)/d(param).

    Uses block-sparse structure: donut per-star params only affect donut
    rows, spot per-star params only affect spot rows, shared atm+wavefront
    params affect all rows.

    Parameters
    ----------
    params : sequence of float
    donut_data, donut_vars, spot_data, spot_vars : arrays

    Returns
    -------
    jac : array of float, shape (nchi_d + nchi_s, nparams)
    """
    nd = self.nd
    ns = self.ns
    natm = self.natm
    nbkg_d = self.nbkg_d
    nbkg_s = self.nbkg_s
    nchi_d = self.nchi_d
    nchi_s = self.nchi_s
    nparams = len(params)

    # Compute the number of wavefront params
    nperstar = 3*nd + 3*ns
    nbkg_total = nbkg_d*nd + nbkg_s*ns
    nwf = nparams - nperstar - natm - nbkg_total

    out = np.zeros((nchi_d + nchi_s, nparams))
    chi0 = self.chi(params, donut_data, donut_vars, spot_data, spot_vars)
    chi0_d = chi0[:nchi_d]
    chi0_s = chi0[nchi_d:]

    joint_dict = self.unpack_params(params)
    d_packed = np.array(self._donut_packed(joint_dict), dtype=float)
    s_packed = np.array(self._spot_packed(joint_dict), dtype=float)

    # --- Donut per-star sparse params (d_fluxes, d_dxs, d_dys) ---
    # These only affect donut chi rows.
    # d_fluxes columns: [0, nd)
    npix_d = self.npix_d
    dflux = 0.01
    d_dict = self.donut_model.unpack_params(d_packed)
    d_dict["fluxes"] = np.array(d_dict["fluxes"]) + dflux
    chi_d = self.donut_model.chi(
        self.donut_model.pack_params(**d_dict), donut_data, donut_vars
    )
    for i in range(nd):
        s = slice(i*npix_d**2, (i+1)*npix_d**2)
        out[s, i] = (chi_d[s] - chi0_d[s]) / dflux

    # d_dxs columns: [nd, 2*nd)
    dx = 0.01
    d_dict = self.donut_model.unpack_params(d_packed)
    d_dict["dxs"] = np.array(d_dict["dxs"]) + dx
    chi_d = self.donut_model.chi(
        self.donut_model.pack_params(**d_dict), donut_data, donut_vars
    )
    for i in range(nd):
        s = slice(i*npix_d**2, (i+1)*npix_d**2)
        out[s, nd+i] = (chi_d[s] - chi0_d[s]) / dx

    # d_dys columns: [2*nd, 3*nd)
    dy = 0.01
    d_dict = self.donut_model.unpack_params(d_packed)
    d_dict["dys"] = np.array(d_dict["dys"]) + dy
    chi_d = self.donut_model.chi(
        self.donut_model.pack_params(**d_dict), donut_data, donut_vars
    )
    for i in range(nd):
        s = slice(i*npix_d**2, (i+1)*npix_d**2)
        out[s, 2*nd+i] = (chi_d[s] - chi0_d[s]) / dy

    # --- Spot per-star sparse params (s_fluxes, s_dxs, s_dys) ---
    # These only affect spot chi rows.
    # s_fluxes columns: [3*nd, 3*nd+ns)
    npix_s = self.npix_s
    s_dict = self.spot_model.unpack_params(s_packed)
    s_dict["fluxes"] = np.array(s_dict["fluxes"]) + dflux
    chi_s = self.spot_model.chi(
        self.spot_model.pack_params(**s_dict), spot_data, spot_vars
    )
    for i in range(ns):
        s = slice(i*npix_s**2, (i+1)*npix_s**2)
        out[nchi_d+s.start:nchi_d+s.stop, 3*nd+i] = (
            self.sqrt_spot_weight * (chi_s[s] - chi0_s[s] / self.sqrt_spot_weight)
        ) / dflux

    # s_dxs columns: [3*nd+ns, 3*nd+2*ns)
    s_dict = self.spot_model.unpack_params(s_packed)
    s_dict["dxs"] = np.array(s_dict["dxs"]) + dx
    chi_s = self.spot_model.chi(
        self.spot_model.pack_params(**s_dict), spot_data, spot_vars
    )
    for i in range(ns):
        s = slice(i*npix_s**2, (i+1)*npix_s**2)
        out[nchi_d+s.start:nchi_d+s.stop, 3*nd+ns+i] = (
            self.sqrt_spot_weight * (chi_s[s] - chi0_s[s] / self.sqrt_spot_weight)
        ) / dx

    # s_dys columns: [3*nd+2*ns, 3*nd+3*ns)
    s_dict = self.spot_model.unpack_params(s_packed)
    s_dict["dys"] = np.array(s_dict["dys"]) + dy
    chi_s = self.spot_model.chi(
        self.spot_model.pack_params(**s_dict), spot_data, spot_vars
    )
    for i in range(ns):
        s = slice(i*npix_s**2, (i+1)*npix_s**2)
        out[nchi_d+s.start:nchi_d+s.stop, 3*nd+2*ns+i] = (
            self.sqrt_spot_weight * (chi_s[s] - chi0_s[s] / self.sqrt_spot_weight)
        ) / dy

    # --- Shared atm + wavefront params (dense: affect all rows) ---
    shared_start = 3*nd + 3*ns
    for k in range(natm):
        params1 = np.array(params, dtype=float)
        params1[shared_start + k] += 0.01
        chi1 = self.chi(
            params1, donut_data, donut_vars, spot_data, spot_vars
        )
        out[:, shared_start + k] = (chi1 - chi0) / 0.01

    for k in range(nwf):
        col = shared_start + natm + k
        params1 = np.array(params, dtype=float)
        params1[col] += self.wavefront_step
        chi1 = self.chi(
            params1, donut_data, donut_vars, spot_data, spot_vars
        )
        out[:, col] = (chi1 - chi0) / self.wavefront_step

    # --- Donut bkg params (sparse: only donut rows) ---
    dbkg = 0.01
    bkg_d_start = shared_start + natm + nwf
    for k in range(nbkg_d):
        d_dict = self.donut_model.unpack_params(d_packed)
        for j in range(nd):
            bkgj = list(d_dict["bkgs"][j])
            bkgj[k] += dbkg
            d_dict["bkgs"][j] = tuple(bkgj)
        chi_d = self.donut_model.chi(
            self.donut_model.pack_params(**d_dict), donut_data, donut_vars
        )
        for j in range(nd):
            s = slice(j*npix_d**2, (j+1)*npix_d**2)
            out[s, bkg_d_start + nbkg_d*j + k] = (
                chi_d[s] - chi0_d[s]
            ) / dbkg

    # --- Spot bkg params (sparse: only spot rows) ---
    bkg_s_start = bkg_d_start + nbkg_d * nd
    for k in range(nbkg_s):
        s_dict = self.spot_model.unpack_params(s_packed)
        for j in range(ns):
            bkgj = list(s_dict["bkgs"][j])
            bkgj[k] += dbkg
            s_dict["bkgs"][j] = tuple(bkgj)
        chi_s = self.spot_model.chi(
            self.spot_model.pack_params(**s_dict), spot_data, spot_vars
        )
        for j in range(ns):
            s = slice(j*npix_s**2, (j+1)*npix_s**2)
            chi0_sj = chi0_s[s] / self.sqrt_spot_weight
            out[nchi_d+s.start:nchi_d+s.stop, bkg_s_start + nbkg_s*j + k] = (
                self.sqrt_spot_weight * (chi_s[s] - chi0_sj)
            ) / dbkg

    return out

Concrete models

danish.joint_model.DZJointModel

Bases: JointModel

Joint donut + spot model using double Zernike parameterization.

Both sub-models must be DZMultiDonutModel and DZMultiSpotModel with identical dz_terms.

Parameters:

Name Type Description Default
donut_model DZMultiDonutModel

Pre-constructed donut fitter.

 required
spot_model DZMultiSpotModel

Pre-constructed spot fitter. Must have the same dz_terms as donut_model.

 required
spot_weight float

Relative weight of spot chi residuals. Default 1.0.

 1.0
Source code in danish/joint_model.py 
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class DZJointModel(JointModel):
    """Joint donut + spot model using double Zernike parameterization.

    Both sub-models must be DZMultiDonutModel and DZMultiSpotModel with
    identical dz_terms.

    Parameters
    ----------
    donut_model : DZMultiDonutModel
        Pre-constructed donut fitter.
    spot_model : DZMultiSpotModel
        Pre-constructed spot fitter.  Must have the same ``dz_terms`` as
        ``donut_model``.
    spot_weight : float, optional
        Relative weight of spot chi residuals.  Default 1.0.
    """
    def __init__(self, donut_model, spot_model, spot_weight=1.0):
        if not isinstance(donut_model, DZMultiDonutModel):
            raise TypeError("donut_model must be a DZMultiDonutModel")
        if not isinstance(spot_model, DZMultiSpotModel):
            raise TypeError("spot_model must be a DZMultiSpotModel")
        if tuple(donut_model.dz_terms) != tuple(spot_model.dz_terms):
            raise ValueError(
                "dz_terms mismatch between donut and spot models"
            )
        super().__init__(donut_model, spot_model, spot_weight=spot_weight)
        self.dz_terms = donut_model.dz_terms
        self.nwavefront = len(self.dz_terms)

danish.joint_model.DZBasisJointModel

Bases: JointModel

Joint donut + spot model using sensitivity matrix parameterization.

Both sub-models must be DZBasisMultiDonutModel and DZBasisMultiSpotModel with identical sensitivity matrices.

Parameters:

Name Type Description Default
donut_model DZBasisMultiDonutModel

Pre-constructed donut fitter.

 required
spot_model DZBasisMultiSpotModel

Pre-constructed spot fitter. Must have the same sensitivity matrix as donut_model.

 required
spot_weight float

Relative weight of spot chi residuals. Default 1.0.

 1.0
Source code in danish/joint_model.py 
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class DZBasisJointModel(JointModel):
    """Joint donut + spot model using sensitivity matrix parameterization.

    Both sub-models must be DZBasisMultiDonutModel and DZBasisMultiSpotModel
    with identical sensitivity matrices.

    Parameters
    ----------
    donut_model : DZBasisMultiDonutModel
        Pre-constructed donut fitter.
    spot_model : DZBasisMultiSpotModel
        Pre-constructed spot fitter.  Must have the same ``sensitivity``
        matrix as ``donut_model``.
    spot_weight : float, optional
        Relative weight of spot chi residuals.  Default 1.0.
    """
    def __init__(self, donut_model, spot_model, spot_weight=1.0):
        if not isinstance(donut_model, DZBasisMultiDonutModel):
            raise TypeError("donut_model must be a DZBasisMultiDonutModel")
        if not isinstance(spot_model, DZBasisMultiSpotModel):
            raise TypeError("spot_model must be a DZBasisMultiSpotModel")
        if not np.array_equal(donut_model.sensitivity, spot_model.sensitivity):
            raise ValueError(
                "sensitivity mismatch between donut and spot models"
            )
        super().__init__(donut_model, spot_model, spot_weight=spot_weight)
        self.sensitivity = donut_model.sensitivity
        self.nwavefront = donut_model.nmode