Standard Fitting Models

Linear

• c1 - Gradient

• c0 - (y) Intercept

\[y = c_1x + c_0\]

Polynomial

• cn … c0 - Polynomial coefficients

For a polynomial degree n:

\[y = c_{n}x^n + c_{n-1}x^n-1 + ... + c_1 * x^1 + c_0 \]

Gaussian

• amp - The maximum height of the Gaussian above background

• sigma - A scalar for Gaussian width

• x0 - The centre (x) of the Gaussian

• background - The minimum value (y) of the Gaussian

\[y = \text{amp} * e^{-\frac{(x - x0) ^ 2}{2 * \text{sigma}^2}} + \text{background}\]

Lorentzian

• amp - The maximum height of the Lorentzian above background

• sigma - A scalar for Lorentzian width

• center - The centre (x) of the Lorentzian

• background - The minimum value (y) of the Lorentzian

\[y = \frac{\text{amp}}{1 + \frac{x - \text{center}}{\text{sigma}}^2} + \text{background}\]

Damped Oscillator (DampedOsc)

• center - The centre (x) of the oscillation

• amp - The maximum height of the curve above 0

• freq - The frequency of the oscillation

• width - How far away from the centre will oscillations last for

\[y = \text{amp} * \cos((x - \text{center}) * \text{freq}) * e^{-\frac{x - \text{center}}{\text{width}^ 2}}\]

Slit Scan (SlitScan)

• background \(b\) - The minimum value (y) of the model

• inflection0 \(i_0\) - The x coord of the first inflection point

• gradient \(g\) - The gradient of the sloped-linear section of the model

• inflections_diff \(i_{\Delta}\) - The x displacement between the two inflection points

• height_above_inflection1 \(h_1\) - The y displacement between inflection 1 and the model’s asymptote

\[\text{exp_seg} = h_1 \cdot \text{erf} \left( g \cdot \frac{\sqrt{\pi}}{2h_1} \cdot (x - i_0 - \Delta i) \right) + g \cdot \Delta i + b\]

\[\text{lin_seg} = \max(b + g * (x - i_0), b)\]

\[y = \min(\text{lin_seg}, \text{exp_seg})\]

Error Function (ERF)

• cen - The centre (x) of the model

• stretch - A horizontal stretch factor for the model

• scale - A vertical stretch factor for the model

• background - The minimum value (y) of the model

\[y = background + scale * erf(stretch * (x - cen))\]

Complementary Error Function (ERFC)

• cen - The centre (x) of the model

• stretch - A horizontal stretch factor for the model

• scale - A vertical stretch factor for the model

• background - The minimum value (y) of the model

\[y = background + scale * erfc(stretch * (x - cen))\]

Top Hat (TopHat)

• cen - The centre (x) of the model

• width - How wide the ‘hat’ is

• height - The maximum height of the model above background

• background - The minimum value (y) of the model

\[\begin{split}y = \begin{cases} \text{background} + \text{height}, & \text{if } |x - \text{cen}| < \frac{\text{width}}{2} \\ \text{background}, & \text{otherwise} \end{cases}\end{split}\]

Trapezoid

• cen - The centre (x) of the model

• gradient - How steep the edges of the trapezoid are

• height - The maximum height of the model above background

• background - The minimum value (y) of the model

• y_offset - Acts as a width factor for the trapezoid. If you extrapolate the sides of the trapezoid until they meet above the top, this value represents the y coord of this point minus height and background.

\[f(x) = \text{y_offset} + \text{height} + \text{background} - \text{gradient} * |x - \text{cen}|\]

\[g(x) = \max(f(x), \text{background})\]

\[y = \min(g(x), \text{background} + \text{height})\]