Chapter 7 Noise Settings (Noise Tab)

In the Noise tab, you specify the model for the noise contained in the observed spectrum.

7.1 Noise model

In the Noise model section, you can select Gaussian, Modified Gaussian, or Poisson.

Model	Description
Gaussian	Treats the noise width with a single \(\sigma\)
Modified Gaussian	Treats signal-intensity-dependent noise with three parameters: \(\sigma_0\) / \(\sigma_1\) / \(\sigma_2\)
Poisson	Treats fluctuations in count data with a Poisson likelihood. A fixed \(\lambda\) is specified in the Noise tab

Gaussian Equation

Each observation \(y_i\) is assumed to follow a normal distribution with the model prediction \(f(x_i)\) as its mean and a common variance \(\sigma^2\):

\[ y_i \mid f(x_i), \sigma \sim \mathcal{N}(f(x_i), \sigma^2) \]

The Gaussian here is the likelihood model for the observation noise; it is distinct from the Gaussian peak shape in Chapter 5.

Poisson Equation

\[ \mathrm{Error} = \frac{1}{N}\sum_{i=1}^{N}\left\{ f(x_i) - y_i\log(f(x_i)) \right\} \]

• \(f(x_i)\) is the model prediction (fitted curve) and \(y_i\) is the observed value
• For Poisson, the Estimate / Fixed toggle of Noise parameter σ is disabled, and a fixed Noise Parameter (λ) is entered
• The initial value of \(\lambda\) is 1

Modified Gaussian Equation

Modified Gaussian uses \(\sigma_0\) / \(\sigma_1\) / \(\sigma_2\) to represent intensity-dependent and constant noise components.

• \(\sigma_0\): noise that grows in proportion to the signal intensity \(f(x)\)
• \(\sigma_1\): noise that grows in proportion to \(\{f(x)\}^2\)
• \(\sigma_2\): constant noise independent of the signal intensity
• \(\sigma(x_i)\) is the noise parameter at each data point
• \(f(x_i)\) is the model prediction (fitted curve) and \(y_i\) is the observed value

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7.2 Noise parameter \(\sigma\) (Estimation Mode)

The Noise parameter σ segmented control selects how the noise parameter \(\sigma\) is handled.

When Poisson is selected, this segmented control is disabled and a fixed \(\lambda\) is entered.

Mode	Behavior
Estimate	The noise parameter is included in the Bayesian estimation
Fixed	The noise parameter is given as a fixed value and not estimated

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7.3 Prior Settings

In the Noise Estimation Priors section, set the lower-to-upper range for each \(\sigma\) parameter.

Gaussian

For Gaussian, set a single \(\sigma\).

• Specify the range with the slider or numeric input
• Initial values are set automatically from the current data.
• In Fixed mode, the value is treated as a fixed value

Modified Gaussian

For Modified Gaussian, set the following three parameters.

Parameter	Role
\(\sigma_0\)	Noise that grows in proportion to the signal intensity \(f(x)\)
\(\sigma_1\)	Noise that grows in proportion to \(\{f(x)\}^2\)
\(\sigma_2\)	Constant noise independent of the signal intensity

• Specify the ranges with the sliders or numeric inputs
• The slider ranges are computed automatically from the Y-value scale of the loaded data
• Reloading the data recomputes the slider ranges

Poisson

For Poisson, set a single \(\lambda\).

• \(\lambda\) is treated as a fixed value and cannot be switched to an estimation target in the Noise tab
• The initial value is 1, and clicking the reset icon restores it to 1

When to Use Fixed \(\sigma\) Mode

• When the noise standard deviation is known
• When you want faster convergence of the analysis (fewer parameters)

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7.4 Reset Icon

Clicking the reset icon at the top right of a slider restores each \(\sigma\) prior range to initial values recomputed from the current data.