loglikelihood(DM::DataModel, θ::AbstractVector) -> Real

Calculates the logarithm of the likelihood $L$, i.e. $\ell(\mathrm{data} \, | \, \theta) \coloneqq \mathrm{ln} \big( L(\mathrm{data} \, | \, \theta) \big)$ given a DataModel and a parameter configuration $\theta$.

MLE(DM::DataModel) -> Vector

Returns the parameter configuration $\theta_\text{MLE} \in \mathcal{M}$ which is estimated to have the highest likelihood of producing the observed data (under the assumption that the specified model captures the true relationship present in the data). For performance reasons, the maximum likelihood estimate is stored as a part of the DataModel type.

MLEuncert(DM::AbstractDataModel, mle::AbstractVector=MLE(DM), F::AbstractMatrix=FisherMetric(DM, mle); Safe::Bool=false, threshold::Real=1e-10, verbose::Bool=true)

Returns vector of type Measurements.Measurement where the parameter uncertainties are approximated via the diagonal of the inverse Fisher metric. That is, the stated uncertainties are a linearized symmetric approximation of the true parameter uncertainties around the MLE.

If Safe=false, infinite values are only imputed for parameters which are structurally non-identifiable themselves, i.e. about which no information contained in the data. However, as a secondary effect, many more parameters than these can also exhibit flat profile likelihoods, if their effects are related to (and can be compensated for by) the degenerate parameters. Nevertheless, upon fixing or removing the purely degenerate parameters, the additional parameters whose profiles were only flat as a secondary effect become structurally identifiable. For Safe=true, infinite values are also imputed for any parameters related to the degenerate parameters, whose profiles may be secondarily affected.

LogLikeMLE(DM::DataModel) -> Real

Returns the value of the log-likelihood $\ell$ when evaluated at the maximum likelihood estimate, i.e. $\ell(\mathrm{data} \, | \, \theta_\text{MLE})$. For performance reasons, this value is stored as a part of the DataModel type.

StructurallyIdentifiable(DM::AbstractDataModel, mle::AbstractVector{<:Number}=MLE(DM); showall::Bool=false, noise::Real=1e-5, threshold::Real=1e-10, N::Int=3)

Checks if jacobian of model wrt parameters has singular values below threshold and provides associated singular directions.

PracticallyIdentifiable(DM::AbstractDataModel, Confnum::Real=3; plot::Bool=isloaded(:Plots), IsCost::Bool=false, kwargs...) -> Real

Determines the maximum confidence level (in units of standard deviations σ) at which the given DataModel is still practically identifiable.

PracticallyIdentifiable(P::ParameterProfiles) -> Real

Determines the maximum level at which ALL the given profiles in ParameterProfiles are still practically identifiable. If IsCost=true was chosen for the profiles, the output is the maximal deviation in cost function value W = 2(L_MLE - PL_i(θ)). If instead IsCost=false was chosen, so that cost function deviations have already been rescaled to confidence levels, the output of PracticallyIdentifiable is the maximal confidence level in units of standard deviations σ where the model is still practically identifiability.

IsLinearParameter(DM::DataModel, MLE::AbstractVector) -> BitVector

Checks with respect to which parameters the model function model(x,θ) is linear and returns vector of booleans where true indicates linearity. This test is performed by comparing the Jacobians of the model for two random configurations $\theta_1, \theta_2 \in \mathcal{M}$ column by column.

Score(DM::DataModel, θ::AbstractVector{<:Number}; ADmode::Val=Val(:ForwardDiff))

Calculates the gradient of the log-likelihood $\ell$ with respect to a set of parameters $\theta$. ADmode=Val(false) computes the Score using the Jacobian dmodel provided in DM, i.e. by having to separately evaluate both the model as well as dmodel. Other choices of ADmode directly compute the Score by differentiating the formula the log-likelihood, i.e. only one evaluation on a dual variable is performed.

FisherMetric(DM::DataModel, θ::AbstractVector{<:Number})

Computes the Fisher metric $g$ given a DataModel and a parameter configuration $\theta$ under the assumption that the likelihood $L(\mathrm{data} \, | \, \theta)$ is a multivariate normal distribution.

\[g_{ab}(\theta) \coloneqq -\int_{\mathcal{D}} \mathrm{d}^m y_{\mathrm{data}} \, L(y_{\mathrm{data}} \,|\, \theta) \, \frac{\partial^2 \, \mathrm{ln}(L)}{\partial \theta^a \, \partial \theta^b} = -\mathbb{E} \bigg( \frac{\partial^2 \, \mathrm{ln}(L)}{\partial \theta^a \, \partial \theta^b} \bigg)\]

GeometricDensity(DM::AbstractDataModel, θ::AbstractVector) -> Real

Computes the square root of the determinant of the Fisher metric $\sqrt{\mathrm{det}\big(g(\theta)\big)}$ at the point $\theta$.

ChristoffelSymbol(DM::DataModel, θ::AbstractVector; BigCalc::Bool=false)
ChristoffelSymbol(Metric::Function, θ::AbstractVector; BigCalc::Bool=false)

Calculates the components of the $(1,2)$ Christoffel symbol $\Gamma$ at a point $\theta$ (i.e. the Christoffel symbol "of the second kind") through finite differencing of the Metric. Accurate to ≈ 3e-11. BigCalc=true increases accuracy through BigFloat calculation.

Riemann(DM::DataModel, θ::AbstractVector; BigCalc::Bool=false)
Riemann(Metric::Function, θ::AbstractVector; BigCalc::Bool=false)

Calculates the components of the $(1,3)$ Riemann tensor by finite differencing of the Metric. BigCalc=true increases accuracy through BigFloat calculation.

Ricci(DM::DataModel, θ::AbstractVector; BigCalc::Bool=false)
Ricci(Metric::Function, θ::AbstractVector; BigCalc::Bool=false)

Calculates the components of the $(0,2)$ Ricci tensor by finite differencing of the Metric. BigCalc=true increases accuracy through BigFloat calculation.

RicciScalar(DM::DataModel, θ::AbstractVector; BigCalc::Bool=false) -> Real
RicciScalar(Metric::Function, θ::AbstractVector; BigCalc::Bool=false) -> Real

Calculates the Ricci scalar by finite differencing of the Metric. BigCalc=true increases accuracy through BigFloat calculation.

GeodesicDistance(DM::DataModel, P::AbstractVector{<:Number}, Q::AbstractVector{<:Number}; tol::Real=1e-10)
GeodesicDistance(Metric::Function, P::AbstractVector{<:Number}, Q::AbstractVector{<:Number}; tol::Real=1e-10)

Computes the length of a geodesic connecting the points P and Q.

AIC(DM::DataModel, θ::AbstractVector) -> Real

Calculates the Akaike Information Criterion given a parameter configuration $\theta$ defined by $\mathrm{AIC} = 2 \, \mathrm{length}(\theta) -2 \, \ell(\mathrm{data} \, | \, \theta)$. Lower values for the AIC indicate that the associated model function is more likely to be correct. For linearly parametrized models and small sample sizes, the AICc should be used instead. The pure AIC tends to select overfitting models. An asymptotically consistent generalization is given by the CAIC and CAICF.

AICc(DM::DataModel, θ::AbstractVector) -> Real

Computes Akaike Information Criterion with an added correction term that prevents the AIC from selecting models with too many parameters (i.e. overfitting) in the case of small sample sizes. $\mathrm{AICc} = \mathrm{AIC} + \frac{2\mathrm{length}(\theta)^2 + 2 \mathrm{length}(\theta)}{N - \mathrm{length}(\theta) - 1}$ where $N$ is the number of data points. Whereas AIC constitutes a first order estimate of the information loss, the AICc constitutes a second order estimate. This correction term is only exact for linearly parametrized models, and an approximation otherwise.

BIC(DM::DataModel, θ::AbstractVector) -> Real

Calculates the Bayesian Information Criterion given a parameter configuration $\theta$ defined by $\mathrm{BIC} = \mathrm{ln}(N) \cdot \mathrm{length}(\theta) -2 \, \ell(\mathrm{data} \, | \, \theta)$ where $N$ is the number of data points.

CAIC(DM::DataModel, θ::AbstractVector) -> Real

Calculates the consistent Akaike Information Criterion developed by Bozdogan in https://doi.org/10.1007/BF02294361 Given a parameter configuration $\theta$, it is defined by $\mathrm{CAIC} = (1 + \mathrm{ln}(N)) \cdot \mathrm{length}(\theta) -2 \, \ell(\mathrm{data} \, | \, \theta) = \mathrm{BIC} + \mathrm{length}(\theta)$ where $N$ is the number of data points. Therefore, it gives a slightly stronger penalization of the parameter degrees of freedom than BIC. While this approximates exactly the same quantity as the AIC, namely minus twice the expected entropy, it in contrast provides a more accurate and asymptotically consistent approximation. That is, in the limit of $N \longrightarrow \infty$, the both the probabilities of under- and overfitting the true model dimensionality go to zero for the CAIC and CAICF.

CAICF(DM::DataModel, θ::AbstractVector) -> Real

Calculates the consistent Akaike Information Criterion with Fisher information developed by Bozdogan in https://doi.org/10.1007/BF02294361 Given a parameter configuration $\theta$, it is defined by $\mathrm{CAIC} = (2 + \mathrm{ln}(N)) \cdot \mathrm{length}(\theta) + 2\mathrm{logdet}(F) -2 \, \ell(\mathrm{data} \, | \, \theta) = \mathrm{BIC} + \mathrm{length}(\theta)$ where $F$ is the Fisher information at $\theta$ and $N$ is the number of data points. Therefore, $\mathrm{CAICF} = \mathrm{AIC} + \mathrm{length}(\theta) \cdot \mathrm{ln}(N) + \mathrm{logdet}(F)$. While this approximates exactly the same quantity as the AIC, namely minus twice the expected entropy, it in contrast provides a more accurate and asymptotically consistent approximation. That is, in the limit of $N \longrightarrow \infty$, the both the probabilities of under- and overfitting the true model dimensionality go to zero for the CAIC and CAICF. Intuitively, the idea is that a model with good fit but lower Fisher information generalizes better, since many parameter values fit the data well.

ModelComparison(DM1::AbstractDataModel, DM2::AbstractDataModel) -> Tuple{Int,Real}

Compares the AICc values of both models at best fit and estimates probability that one model is more likely than the other. First entry of tuple returns which model is more likely to be correct (1 or 2) whereas the second entry returns the ratio of probabilities p_better/p_worse.