Peak fitting

Rampy does not (yet) offer a dedicated function for peak fitting. Instead, we invite users to use lmfit or scipy.optimize to perform peak fitting, which is basically the action to fit a model (sum of peaks) to your data.

We provide an example of peak-fitting with the lmfit for instance here: Example notebooks .

Peak shapes

Rampy offers functions for various peak shapes, including:

  • gaussian peaks > rampy.gaussian

  • lorentzian peaks > rampy.lorentzian

  • pseudo-voigt peaks > rampy.pseudovoigt

  • pearson7 peaks > rampy.pearson7

rampy.peak_shapes.gaussian(x: ndarray, amp, freq, HWHM) → ndarray

Computes a Gaussian peak.

Parameters:

  • x (np.ndarray) – Positions at which the signal should be sampled.

  • amp (float or np.ndarray) – Amplitude of the Gaussian peak.

  • freq (float or np.ndarray) – Frequency/position of the Gaussian component.

  • HWHM (float or np.ndarray) – Half-width at half-maximum.

Returns:

The computed Gaussian signal.

Return type:

np.ndarray

Notes

Formula used: ( ext{amp} cdot exp(-log(2) cdot ((x - ext{freq}) / ext{HWHM})^2) ).

Examples

You can create a single peak like:

>>> x = np.linspace(0, 10, 1000)
>>> y = rp.gaussian(x, 10., 3., 1.0)

You can also create an array with several peaks in columns using arrays for the peak parameters:

>>> x = np.linspace(0, 10, 1000)
>>> y = rp.gaussian(x.reshape(-1, 1), np.array([[10, 10.]]), np.array([[3, 7.]]), np.array([[1., 1.]]))

In this case, y will be an array of shape (len(x), 2) with one peak per columns. Your peaks can even share parameters, here the HWHM:

>>> y = rp.gaussian(x.reshape(-1, 1), np.array([[10, 10.]]), np.array([[3, 7.]]), 2.0)

The composite signal is simply given by

>>> y_sum = np.sum(y, axis=1)
rampy.peak_shapes.lorentzian(x: ndarray, amp, freq, HWHM) → ndarray

Computes a Lorentzian peak.

Parameters:

  • x (np.ndarray) – Positions at which the signal should be sampled.

  • amp (float or np.ndarray) – Amplitude of the Lorentzian peak.

  • freq (float or np.ndarray) – Frequency/position of the Lorentzian component.

  • HWHM (float or np.ndarray) – Half-width at half-maximum.

Returns:

The computed Lorentzian signal.

Return type:

np.ndarray

Notes

Formula used: ( ext{amp} / (1 + ((x - ext{freq}) / ext{HWHM})^2) ).

Examples

You can create a single peak like:

>>> x = np.linspace(0, 10, 1000)
>>> y = rp.lorentzian(x, 10., 3., 1.0)

You can also create an array with several peaks in columns using arrays for the peak parameters:

>>> x = np.linspace(0, 10, 1000)
>>> y = rp.lorentzian(x.reshape(-1, 1), np.array([[10, 10.]]), np.array([[3, 7.]]), np.array([[1., 1.]]))

In this case, y will be an array of shape (len(x), 2) with one peak per columns. Your peaks can even share parameters, here the HWHM:

>>> y = rp.lorentzian(x.reshape(-1, 1), np.array([[10, 10.]]), np.array([[3, 7.]]), 2.0)

The composite signal is simply given by

>>> y_sum = np.sum(y, axis=1)
rampy.peak_shapes.pearson7(x, a0, a1, a2, a3)

Computes a Pearson VII peak.

Parameters:

  • x (np.ndarray) – Positions at which the signal should be sampled.

  • a0 (float or np.ndarray) – Amplitude parameter.

  • a1 (float or np.ndarray) – Frequency/position parameter.

  • a2 (float or np.ndarray) – Width parameter.

  • a3 (float or np.ndarray) – Shape parameter.

Returns:

The computed Pearson VII signal.

Return type:

np.ndarray

Notes

Formula used: ( a_0 / ((1 + ((x - a_1) / a_2)^2 cdot (2^{(1/a_3)} - 1))^{a_3}) ).

Examples

You can create a single peak like:

>>> x = np.linspace(0, 10, 1000)
>>> y = rp.pearson7(x, 10., 3., 1.0, 0.5)

You can also create an array with several peaks in columns using arrays for the peak parameters:

>>> x = np.linspace(0, 10, 1000)
>>> y = rp.pearson7(x.reshape(-1, 1), np.array([[10, 10]]), np.array([[3, 7]]), np.array([[1., 1.]]), np.array([[0.5, 0.5]]))

In this case, y will be an array of shape (len(x), 2) with one peak per columns. Your peaks can even share parameters, here the a3:

>>> y = rp.pearson7(x.reshape(-1, 1), np.array([[10, 10]]), np.array([[3, 7]]), np.array([[1., 1.]]), 0.5)

The composite signal is simply given by

>>> y_sum = np.sum(y, axis=1)
rampy.peak_shapes.pseudovoigt(x: ndarray, amp, freq, HWHM, L_ratio) → ndarray

Computes a pseudo-Voigt peak.

Parameters:

  • x (np.ndarray) – Positions at which the signal should be sampled. Can be a vector or array.

  • amp (float) – Amplitude of the pseudo-Voigt peak.

  • freq (float or np.ndarray) – Frequency/position of the Gaussian component.

  • HWHM (float or np.ndarray) – Half-width at half-maximum.

  • L_ratio (float) – Ratio of the Lorentzian component. Must be between 0 and 1 inclusive.

Returns:

The computed pseudo-Voigt signal.

Return type:

np.ndarray

Raises:

ValueError – If L_ratio is not between 0 and 1.

Notes

Formula used: ( (1 - L_{ ext{ratio}}) cdot ext{gaussian}(x, ext{amp}, ext{freq}, ext{HWHM}) +

L_{ ext{ratio}} cdot ext{lorentzian}(x, ext{amp}, ext{freq}, ext{HWHM}) ).

Examples

You can create a single peak like:

>>> x = np.linspace(0, 10, 1000)
>>> y = rp.pseudovoigt(x, 10., 3., 1.0, 0.5)

You can also create an array with several peaks in columns using arrays for the peak parameters:

>>> x = np.linspace(0, 10, 1000)
>>> y = rp.pseudovoigt(x.reshape(-1, 1), np.array([[10, 10]]), np.array([[3, 7]]), np.array([[1., 1.]]), np.array([[0.5, 0.5]]))

In this case, y will be an array of shape (len(x), 2) with one peak per columns. Your peaks can even share parameters, here the L_ratio:

>>> y = rp.pseudovoigt(x.reshape(-1, 1), np.array([[10, 10]]), np.array([[3, 7]]), np.array([[1., 1.]]), 0.5)

The composite signal is simply given by

>>> y_sum = np.sum(y, axis=1)

Peak areas

Peak area can be calculated using the rampy.peakarea function.

rampy.peak_area.gaussianarea(amp, HWHM, **options)

returns the area of a Gaussian peak

Parameters:

  • amp (float or ndarray) – amplitude of the peak

  • HWHM (float or ndarray) – half-width at half-maximum

  • eseAmplitude (float or ndarray, optional) – standard deviation on amp; Default = None

  • eseHWHM (float or ndarray, optional) – standard deviation on HWHM; Default = None

Returns:

peak area esearea (float or ndarray): error on peak area; will be None if no errors on amp and HWHM were provided.

Return type:

area (float or ndarray)

rampy.peak_area.peakarea(shape: str, amp: float, HWHM: float, pos: float = None, a3: float = None, L_ratio: float = None, ese_amp: float = None, ese_HWHM: float = None) → tuple[float, float | None]

Computes the area of a peak given its shape and parameters.

For Gaussian peaks, the area is calculated analytically. For other shapes (Lorentzian, pseudo-Voigt, Pearson VII), the area is calculated using numerical integration (trapezoidal rule).

Parameters:

  • shape (str) – The shape of the peak. Must be one of ‘gaussian’, ‘lorentzian’, ‘pseudovoigt’, or ‘pearson7’.

  • amp (float, list of floats, or np.ndarray) – Amplitude of the peak.

  • HWHM (float, list of floats, or np.ndarray) – Half-width at half-maximum of the peak.

  • pos (float, list of floats, np.ndarray, optional) – Peak position (required for non-Gaussian shapes). Default is None.

  • a3 (float, list of floats, np.ndarray, optional) – Shape parameter for Pearson VII peaks. Required if shape=’pearson7’.

  • L_ratio (float, list of floats, np.ndarray, optional) – Lorentzian component ratio for pseudo-Voigt peaks. Must be between 0 and 1. Required if shape=’pseudovoigt’. Default is None.

  • ese_amp (float, list of floats, np.ndarray, optional) – Standard deviation of the amplitude. Used to calculate error on area. Default is None.

  • ese_HWHM (float, list of floats, np.ndarray, optional) – Standard deviation of HWHM. Used to calculate error on area. Default is None.

Returns:

A tuple containing:

  • area (float): The computed area of the peak.

  • ese_area (float or None): The error on the computed area if ese_amp and ese_HWHM are provided; otherwise, None.

Return type:

tuple[float, float | None]

Raises:

  • ValueError – If required parameters are missing or invalid, or if the list of floats or the np.ndarray for the peak parameters have different sizes.

  • NotImplementedError – If an unsupported shape is specified.

Notes

  • For Gaussian peaks, the formula used is: ( ext{area} = sqrt{pi / ln(2)} cdot ext{amp} cdot ext{HWHM} ).

  • For other shapes, numerical integration is performed over a range of ( pm 10 cdot ext{HWHM} ).

Example

Compute the area of a Gaussian peak:

>>> amp = 10
>>> HWHM = 2
>>> area, ese_area = peakarea("gaussian", amp=amp, HWHM=HWHM)
>>> print(area)

Compute the area of a Lorentzian peak with numerical integration:

>>> amp = 10
>>> HWHM = 2
>>> pos = 5
>>> area, _ = peakarea("lorentzian", amp=amp, HWHM=HWHM, pos=pos)
>>> print(area)

Compute the area of several Lorentzian peaks with numerical integration (beware that in this case, the lists should have equal sizes):

>>> amp = [10., 5., 3.]
>>> HWHM = 2 # they share a common width
>>> pos = [2., 5., 5.5]
>>> area, _ = peakarea("lorentzian", amp=amp, HWHM=HWHM, pos=pos)
>>> print(area)

Propagate uncertainties

The best way to propagate the uncertainties of your model is to directly use the uncertainties package, see the docs here.