Various mathematical functions

Submodule containing functions used in fitting

Fitting the continuum

specmodels.specmodels_functions.polyf_o1(x, a1, a0)[source]

Affine function

(1)\[P(x) = a_{1} \cdot x + a_{0}\]

Parameters:

x (array[float]) – The independant variable
a1 (float) – term for order 1
a0 (float) – constant term

Returns:

The value of the polyf_o1 function at each value of x

Return type:

array[float]

specmodels.specmodels_functions.polyf_o2(x, a2, a1, a0)[source]

Polynomial function of order 2

(2)\[P(x) = a_{2} \cdot x^{2} + a_{1} \cdot x + a_{0}\]

Parameters:

x (array[float]) – The independant variable
a2 (float) – term for order 2
a1 (float) – term for order 1
a0 (float) – constant term

Returns:

The value of the polyf_o2 function at each value of x

Return type:

array[float]

specmodels.specmodels_functions.polyf_o3(x, a3, a2, a1, a0)[source]

Polynomial function of order 3

(3)\[P(x) = a_{3} \cdot x^{3} + a_{2} \cdot x^{2} + a_{1} \cdot x + a_{0}\]

Parameters:

x (array[float]) – The independant variable
a3 (float) – term for order 3
a2 (float) – term for order 2
a1 (float) – term for order 1
a0 (float) – constant term

Returns:

The value of the polyf_o3 function at each value of x

Return type:

array[float]

specmodels.specmodels_functions.polyf_o4(x, a4, a3, a2, a1, a0)[source]

Polynomial function of order 4

(4)\[P(x) = \sum_{i=0}^{4} a_{i} \cdot x^{i}\]

Parameters:

x (array[float]) – The independant variable
a4 (float) – term for order 4
a3 (float) – term for order 3
a2 (float) – term for order 2
a1 (float) – term for order 1
a0 (float) – constant term

Returns:

The value of the polyf_o4 function at each value of x

Return type:

array[float]

specmodels.specmodels_functions.polyf_o5(x, a5, a4, a3, a2, a1, a0)[source]

Polynomial function of order 6

(5)\[P(x) = \sum_{i=0}^{5} a_{i} \cdot x^{i}\]

Parameters:

x (array[float]) – The independant variable
a5 (float) – term for order 5
a4 (float) – term for order 4
a3 (float) – term for order 3
a2 (float) – term for order 2
a1 (float) – term for order 1
a0 (float) – constant term

Returns:

The value of the polyf_o5 function at each value of x

Return type:

array[float]

specmodels.specmodels_functions.polyf_o6(x, a6, a5, a4, a3, a2, a1, a0)[source]

Polynomial function of order 6

(6)\[P(x) = \sum_{i=0}^{6} a_{i} \cdot x^{i}\]

Parameters:

x (array[float]) – The independant variable
a6 (float) – term for order 6
a5 (float) – term for order 5
a4 (float) – term for order 4
a3 (float) – term for order 3
a2 (float) – term for order 2
a1 (float) – term for order 1
a0 (float) – constant term

Returns:

The value of the polyf_o6 function at each value of x

Return type:

array[float]

specmodels.specmodels_functions.polyf_o7(x, a7, a6, a5, a4, a3, a2, a1, a0)[source]

Polynomial function of order 7

(7)\[P(x) = \sum_{i=0}^{7} a_{i} \cdot x^{i}\]

Parameters:

x (array[float]) – The independant variable
a7 (float) – term for order 7
a6 (float) – term for order 6
a5 (float) – term for order 5
a4 (float) – term for order 4
a3 (float) – term for order 3
a2 (float) – term for order 2
a1 (float) – term for order 1
a0 (float) – constant term

Returns:

The value of the polyf_o7 function at each value of x

Return type:

array[float]

specmodels.specmodels_functions.polyf_o8(x, a8, a7, a6, a5, a4, a3, a2, a1, a0)[source]

Polynomial function of order 8

(8)\[P(x) = \sum_{i=0}^{8} a_{i} \cdot x^{i}\]

Parameters:

x (array[float]) – The independant variable
a8 (float) – term for order 8
a7 (float) – term for order 7
a6 (float) – term for order 6
a5 (float) – term for order 5
a4 (float) – term for order 4
a3 (float) – term for order 3
a2 (float) – term for order 2
a1 (float) – term for order 1
a0 (float) – constant term

Returns:

The value of the polyf_o8 function at each value of x

Return type:

array[float]

specmodels.specmodels_functions.polyf_o9(x, a9, a8, a7, a6, a5, a4, a3, a2, a1, a0)[source]

Polynomial function of order 9

(9)\[P(x) = \sum_{i=0}^{9} a_{i} \cdot x^{i}\]

Parameters:

x (array[float]) – The independant variable
a9 (float) – term for order 9
a8 (float) – term for order 8
a7 (float) – term for order 7
a6 (float) – term for order 6
a5 (float) – term for order 5
a4 (float) – term for order 4
a3 (float) – term for order 3
a2 (float) – term for order 2
a1 (float) – term for order 1
a0 (float) – constant term

Returns:

The value of the polyf_o9 function at each value of x

Return type:

array[float]

specmodels.specmodels_functions.polyf_o10(x, a10, a9, a8, a7, a6, a5, a4, a3, a2, a1, a0)[source]

Polynomial function of order 10

(10)\[P(x) = \sum_{i=0}^{10} a_{i} \cdot x^{i}\]

Parameters:

x (array[float]) – The independant variable
a10 (float) – term for order 10
a9 (float) – term for order 9
a8 (float) – term for order 8
a7 (float) – term for order 7
a6 (float) – term for order 6
a5 (float) – term for order 5
a4 (float) – term for order 4
a3 (float) – term for order 3
a2 (float) – term for order 2
a1 (float) – term for order 1
a0 (float) – constant term

Returns:

The value of the polyf_o10 function at each value of x

Return type:

array[float]

specmodels.specmodels_functions.polyf_gen(order=1)[source]

Setup switch for polynomial function

Parameters:: order (int) – Order of polynomial function to return.
Returns:: The polynomial function of the required order
Return type:: function

Fitting bands

specmodels.specmodels_functions.gaussian(x, amplitude, mean, stddev)[source]

Canonical gaussian function

Parameters:

x (array[float]) – The independant variable
amplitude (float) – The amplitude of the band
mean (float) – The central position of the band
stddev (float) – The root-square of the width of the band

Returns:

The value of the gaussian function at each value of x

Return type:

array[float]

specmodels.specmodels_functions.emg(x, amplitude, mean, stddev, rfactor)[source]

Exponentially Modified Gaussian (Chromatography expression)

Check Pottin+2020 and Grushka+1972 for details on the function

Parameters:

x (array[float]) – The independant variable (e.g. wavelength)
amplitude (float) – The amplitude of the band
mean (float) – The central position of the band
stddev (float) – The root-square of the width of the band
rfactor (float) – The asymmetry factor of the band

Returns:

The value of the EMG function at each value of x

Return type:

array[float]

specmodels.specmodels_functions.emg_alt(x, amplitude, mean, stddev)[source]

Exponentially Modified Gaussian (Canonical expression)

Check Pottin+2020 and Grushka+1972 for details on the function

Parameters:

x (array[float]) – The independant variable (e.g. wavelength)
amplitude (float) – The amplitude of the band
mean (float) – The central position of the band
stddev (float) – The root-square of the width of the band

Returns:

The value of the EMG function at each value of x

Return type:

array[float]