Noise generators

Pints contains a module pints.noise that contains methods that generate

different kinds of noise.

This can then be added to simulation output to create “realistic” experimental

data.

Overview:

pints.noise.ar1(rho, sigma, n)[source]

Generates first-order autoregressive (AR1) noise that can be added to a vector of simulated data.

The generated noise follows the distribution

\[e(t) = \rho e(t - 1) + v(t),\]

where \(v(t) \stackrel{\text{iid}}{\sim }\mathcal{N}(0, \sigma \sqrt{1 - \rho ^2})\).

Returns an array of length n containing the generated noise.

Parameters:

  • rho – Determines the magnitude of the noise \(\rho\) (see above). Must be less than 1.

  • sigma – The marginal standard deviation \(\sigma\) of e(t) (see above). Must be greater than zero.

  • n – The length of the signal. (Only single time-series are supported.)

Example

values = model.simulate(parameters, times)
noisy_values = values + noise.ar1(0.9, 5, len(values))
pints.noise.ar1_unity(rho, sigma, n)[source]

Generates noise following an autoregressive order 1 process of mean 1, that a vector of simulated data can be multiplied with.

Returns an array of length n containing the generated noise.

Parameters:

  • rho – Determines the magnitude of the noise (see ar1()). Must be less than or equal to 1.

  • sigma – The marginal standard deviation of e(t) (see ar()). Must be greater than 0.

  • n (int) – The length of the signal. (Only single time-series are supported.)

Example

values = model.simulate(parameters, times)
noisy_values = values * noise.ar1_unity(0.5, 0.8, len(values))
pints.noise.arma11(rho, theta, sigma, n)[source]

Generates an ARMA(1,1) error process of the form:

\[e(t) = (1 - \rho) + \rho * e(t - 1) + v(t) + \theta * v(t-1),\]

where \(v(t) \stackrel{\text{iid}}{\sim }\mathcal{N}(0, \sigma ')\), and

\[\sigma ' = \sigma \sqrt{\frac{1 - \rho ^ 2}{1 + 2 \theta \rho + \theta ^ 2}}.\]
pints.noise.arma11_unity(rho, theta, sigma, n)[source]

Generates an ARMA(1,1) error process of the form:

e(t) = (1 - rho) + rho * e(t - 1) + v(t) + theta * v[t-1],

where v(t) ~ iid N(0, sigma'),

and sigma' = sigma * sqrt((1 - rho^2) / (1 + 2 * theta * rho + theta^2)).

Returns an array of length n containing the generated noise.

Parameters:

  • rho – Determines the long-run persistence of the noise (see ar1()). Must be less than 1.

  • theta – Contributes to first order autocorrelation of noise. Must be less than 1.

  • sigma – The marginal standard deviation of e(t) (see ar()). Must be greater than 0.

  • n (int) – The length of the signal. (Only single time-series are supported.)

Example

values = model.simulate(parameters, times)
noisy_values = values * noise.ar1_unity(0.5, 0.8, len(values))
pints.noise.independent(sigma, shape)[source]

Generates independent Gaussian noise iid \(\mathcal{N}(0,\sigma)\).

Returns an array of shape shape containing the generated noise.

Parameters:

  • sigma – The standard deviation of the noise. Must be zero or greater.

  • shape – A tuple (or sequence) defining the shape of the generated noise array.

Example

values = model.simulate(parameters, times)
noisy_values = values + noise.independent(5, values.shape)
pints.noise.multiplicative_gaussian(eta, sigma, f)[source]

Generates multiplicative Gaussian noise for a single output.

With multiplicative noise, the measurement error scales with the magnitude of the output. Given a model taking the form,

\[X(t) = f(t; \theta) + \epsilon(t)\]

multiplicative Gaussian noise models the noise term as:

\[\epsilon(t) = f(t; \theta)^\eta v(t)\]

where v(t) is iid Gaussian:

\[v(t) \stackrel{\text{ iid }}{\sim} \mathcal{N}(0, \sigma)\]

The output magnitudes f are required as an input to this function. The noise terms are returned in an array of the same shape as f.

Parameters:

  • eta – The exponential power controlling the rate at which the noise scales with the output. The argument must be either a float (for single-output or multi-output noise) or an array_like of floats (for multi-output noise only, with one value for each output).

  • sigma – The baseline standard deviation of the noise (must be greater than zero). The argument must be either a float (for single-output or multi-output noise) or an array_like of floats (for multi-output noise only, with one value for each output).

  • f – A NumPy array giving the time-series for the output over time. For multiple outputs, the array should have shape (n_outputs, n_times).