CARMA¶

CARMA GP kernels¶

Note

term is a celerite nomenclature, a term is just a GP kernel. For example, a DRW_term object defines a DRW kernel.

A collection of GP kernels that express the autovariance structure of CARMA models using celerite.

class eztao.carma.CARMATerm.CARMA_term(log_arpars, log_mapars, *args, **kwargs)¶

A general-purpose CARMA term (kernel)

Parameters:

log_arpars (array(float)) – Natural log of the AR coefficients.
log_mapars (array(float)) – Natural log of the MA coefficients.

static carma2fcoeffs(log_arpars, log_mapars)¶

Get the representation of a CARMA kernel in the factored polynomial space

A wrapper of CARMA_term.carma2fcoeffs_log for backward compatibility. This function will be deprecated in future releases.

static carma2fcoeffs_log(log_arpars, log_mapars)¶

Get the representation of a CARMA kernel in the factored polynomial space

Parameters:

log_arpars (array(float)) – Natural log of the AR coefficients.
log_mapars (array(float)) – Natural log of the MA coefficients.

Returns:

The coefficients (in natural log) of the factored polymoical: for the CARMA kernel specified by the input parameters. The last coeff is a multiplying factor of the returned polynomial.

Return type:

array(float)

static fcoeffs2carma(log_fcoeffs, p)¶

Get the representation of a CARMA kernel in the nominal CARMA parameter space

A wrapper of CARMA_term.fcoeffs2carma_log for backward compatibility. This function will be deprecated in future releases.

static fcoeffs2carma_log(log_fcoeffs, p)¶

Get the representation of a CARMA kernel in the nominal CARMA parameter space

Parameters:

log_coeffs (array(float)) – The array of coefficients for the factored polynomial with the last coeff being a multiplying factor of the polynomial.
p (int) – The p order of the CARMA kernel.

Returns:

Natural log of the AR and MA parameters in two separate arrays.

get_complex_coefficients(params)¶

Get arrays of alpha_complex_real, alpha_complex_imag, beta_complex_real and beta_complex_imag (coefficients of celerite’s complex kernel)

Parameters:: params (array(float)) – Parameters of this kernel.
Returns:: Arrays of alpha_complex_real, alpha_complex_imag, beta_complex_real and beta_complex_imag, one for each.

get_real_coefficients(params)¶

Get arrays of alpha_real and beta_real (coefficients of celerite’s real kernel)

Parameters:: params (array(float)) – Parameters of this kernel.
Returns:: Arrays of alpha_real and beta_real, one for each.

get_rms_amp()¶

Get the RMS amplitude of this CARMA kernel

Returns:: The RMS amplitude of this CARMA kernel.

static rms_amp(log_arpars, log_mapars)¶

Compute the RMS amplitude of a CARMA kernel

Parameters:

log_arpars (array(float)) – Natural log of the AR coefficients.
log_mapars (array(float)) – Natural log of the MA coefficients.

Returns:

The RMS amplitude of the CARMA kernel specified by the input parameters.

set_log_fcoeffs(log_fcoeffs)¶

Set kernel parameters

Use coeffs of the factored polynomial to set CARMA paramters, note that the last input coeff is always the coeff for the highest-order MA differential. While performing the conversion, 1 is added to the AR coeffs to maintain the same formatting (AR polynomials always have the highest order coeff to be 1).

Parameters:: log_fcoeffs (array(float)) – Natural log of the coefficients for the factored characteristic polynomial, with the last coeff being an additional multiplying factor on this polynomial.

class eztao.carma.CARMATerm.DHO_term(log_a1, log_a2, log_b0, log_b1, *args, **kwargs)¶

Damped Harmonic Oscillator (DHO) term (kernel)

Parameters:

log_a1 (float) – Natural log of the DHO parameter a1.
log_a2 (float) – Natural log of the DHO parameter a2.
log_b0 (float) – Natural log of the DHO parameter b0.
log_b1 (float) – Natural log of the DHO parameter b1.

class eztao.carma.CARMATerm.DRW_term(*args, **kwargs)¶

Damped Random Walk (DRW) term (kernel)

\[k_{DRW}(\Delta t) = \sigma^2\,e^{-\Delta t/\tau}\]

with the parameters log_amp and log_tau.

Parameters:

log_amp (float) – The natural log of the RMS amplitude of the DRW process.
log_tau (float) – The natural log of the characteristic timescale of the DRW process.

Note

Conversions between EzTao DRW parameters and some other DRW representations seen in the literature:

\[\mathrm{SF_{\infty}} = 2*\sigma^2\]

\[\sigma^2 = \tau*\sigma_{KBS}^2/2\]

\[\tau = 1/\alpha_1; \,\sigma_{KBS} = \beta_0\]

see MacLeod et al. (2010) for SF_{infty}} and Kelly et al. (2009) for sigma_{KBS}. alpha_1 and beta_0 are the AR and MA parameters for a CARMA(1,0) model, respectively.

get_carma_parameter()¶

Get DRW parameters in CARMA notation (alpha_*/beta_*).

Returns:: [alpha_1, beta_0].

get_perturb_amp()¶

Get the amplitude of the perturbing noise (beta_0) in DRW

Returns:: The amplitude of the perturbing noise (beta_0) in the current DRW.

get_real_coefficients(params)¶

Get alpha_real and beta_real (coeffs of celerite’s real kernel)

Parameters:: params (array(float)) – Parameters of this kernel.
Returns:: (alpha_real, beta_real).

get_rms_amp()¶

Get the RMS amplitude of this DRW process.

Returns:: The RMS amplitude of this DRW process.

static perturb_amp(log_amp, log_tau)¶

Compute the amplitude of the perturbing noise (beta_0) in DRW.

Parameters:

log_amp (float) – The natural log of the RMS amplitude of the DRW process.
log_tau (float) – The natural log of the characteristic timescale of the DRW process.

Returns:

The amplitude of the perturbing noise (beta_0) in the DRW specified by the input parameters.

eztao.carma.CARMATerm.acf(arroots, arparam, maparam)¶

Get ACVF coefficients given CARMA parameters

The CARMA noation (index) folows that in Brockwell et al. (2001).

Parameters:

arroots (array(complex)) – AR roots in a numpy array
arparam (array(float)) – AR parameters in a numpy array
maparam (array(float)) – MA parameters in a numpy array

Returns:

ACVF coefficients, each element correspond to a root.

Return type:

array(complex)

CARMA utility functions¶

A set of functions to computer 2nd order statistics of CARMA models.

eztao.carma.model_utils.carma_acf(arparams, maparams)¶

Return a function that computes the model autocorrelation function (ACF) of CARMA.

Parameters:

arparams (array(float)) – AR coefficients.
maparams (array(float)) – MA coefficients.

Returns:

A function that takes in time lags and returns ACF at the given lags.

eztao.carma.model_utils.carma_psd(arparams, maparams)¶

Return a function that computes CARMA Power Spectral Density (PSD).

Parameters:

arparams (array(float)) – AR coefficients.
maparams (array(float)) – MA coefficients

Returns:

A function that takes in frequencies and returns PSD at the given frequencies.

eztao.carma.model_utils.carma_sf(arparams, maparams)¶

Return a function that computes the CARMA structure function (SF).

Parameters:

arparams (array(float)) – AR coefficients.
maparams (array(float)) – MA coefficients.

Returns:

A function that takes in time lags and returns CARMA SF at the given lags.

eztao.carma.model_utils.drw_acf(tau)¶

Return a function that computes the DRW autocorrelation function (ACF).

Parameters:: tau (float) – DRW decorrelation/characteristic timescale.
Returns:: A function that takes in time lags and returns ACF at the given lags.

eztao.carma.model_utils.drw_psd(amp, tau)¶

Return a function that computes DRW Power Spectral Density (PSD).

Parameters:

amp (float) – DRW RMS amplitude
tau (float) – DRW decorrelation/characteristic timescale

Returns:

A function that takes in frequencies and returns PSD at the given frequencies.

eztao.carma.model_utils.drw_sf(amp, tau)¶

Return a function that computes the structure function (SF) of DRW.

Parameters:

amp (float) – DRW RMS amplitude
tau (float) – DRW decorrelation/characteristic timescale.

Returns:

A function that takes in time lags and returns DRW SF at the given lags.

eztao.carma.model_utils.gp_psd(carmaTerm)¶

Return a function that computes native GP Power Spectral Density (PSD).

Parameters:: carmaTerm (object) – A celerite CARMA term.
Returns:: A function that takes in frequencies and returns PSD at the given frequencies.