basis_fit

Methods based on fitting basis functions to data

pynumdiff.basis_fit.rbfdiff(x, dt_or_t, sigma=1, lmbd=0.01, axis=0)

Find smoothed function and derivative estimates by fitting noisy data with radial-basis-functions. Naively, fill a matrix with basis function samples and solve a linear inverse problem against the data, but truncate tiny values to make columns sparse. Each basis function “hill” is topped with a “tower” of height lmbd to reach noisy data samples, and the final smoothed reconstruction is found by razing these and only keeping the hills.

Parameters:

  • x (np.array[float]) – data to differentiate. May be multidimensional; see axis.

  • dt_or_t (float or array[float]) – This function supports variable step size. This parameter is either the constant \(\Delta t\) if given as a single float, or data locations if given as an array of same length as x.

  • sigma (float) – controls width of radial basis functions

  • lmbd (float) – controls smoothness

  • axis (int) – data dimension along which differentiation is performed

Returns:

  • x_hat (np.array) – estimated (smoothed) x

  • dxdt_hat (np.array) – estimated derivative of x

pynumdiff.basis_fit.spectraldiff(x, dt, params=None, options=None, high_freq_cutoff=None, even_extension=True, pad_to_zero_dxdt=True, axis=0)

Take a derivative in the Fourier domain, with high frequency attentuation.

Parameters:

  • x (np.array[float]) – data to differentiate. May be multidimensional; see axis.

  • dt (float) – step size

  • params (list[float] or float) – (deprecated, prefer high_freq_cutoff)

  • options (dict) – (deprecated, prefer even_extension and pad_to_zero_dxdt) a dictionary consisting of {‘even_extension’: (bool), ‘pad_to_zero_dxdt’: (bool)}

  • high_freq_cutoff (float) – The high frequency cutoff as a multiple of the Nyquist frequency: Should be between 0 and 1. Frequencies below this threshold will be kept, and above will be zeroed.

  • even_extension (bool) – if True, extend the data with an even extension so signal starts and ends at the same value.

  • pad_to_zero_dxdt (bool) – if True, extend the data with extra regions that smoothly force the derivative to zero before taking FFT.

Returns:

  • x_hat (np.array) – estimated (smoothed) x

  • dxdt_hat (np.array) – estimated derivative of x