Structural and practical parameter non-identifiability issues are common when mathematical models are used to interpret data. Such issues motivate model reparameterisation and reduction methods. Here, we consider Invariant Image Reparameterisation (IIR), which asks when symbolic reparameterisation conditions can be replaced by numerical derivative calculations at a single reference point. The central object is the invariant image: a reduced, basis-independent representation of the parameter combinations controlling observable model behaviour. We show that when a one-to-one componentwise transformation makes observable behaviour depend only on fixed linear combinations of the transformed parameters, a single numerical Jacobian determines the associated lower-dimensional reparameterisation space. This includes models depending on monomial combinations of the original parameters. We also give a first-order invariance condition that distinguishes minimal from non-minimal but exact reductions via the invariant part of the local null space. In structurally identifiable but practically weakly informed settings, the same calculations separate strongly and weakly informed parameter combinations. The invariant image admits multiple coordinate representations: the SVD gives a default orthonormal basis ordered by local identifiability, while sparse monomial bases are often more interpretable. Treating these coordinates as interest parameters in Profile-Wise Analysis gives likelihood-based uncertainty quantification and prediction. We demonstrate the method on parameterised normal models with Poisson-limit, extended Poisson-limit, and non-limit cases, and on the repressilator, a nonlinear differential equation model of gene regulation. A Julia implementation of IIR, with these and further examples, is available at https://github.com/omaclaren/reparam.

翻译：结构与实践中参数非可辨识性问题在使用数学模型解释数据时普遍存在。此类问题促使了模型重参数化与简化方法的发展。本文考虑不变图像重参数化（IIR），探究符号重参数化条件何时可被单个参考点处的数值导数计算替代。核心概念是不变图像：一种约简的、与基无关的参数组合表示，用以控制可观测的模型行为。我们证明，当一对一的分量变换使可观测行为仅依赖于变换后参数的固定线性组合时，单个数值雅可比矩阵即可确定相关的低维重参数化空间。这依赖于原始参数单项组合的模型。我们还提出一阶不变性条件，通过局部零空间的不变部分区分最小化与非最小化但精确的约简。在结构可辨识但实际弱信息场景中，相同的计算可分离强信息与弱信息参数组合。不变图像支持多种坐标表示：奇异值分解（SVD）提供按局部可辨识性排序的默认正交基，而稀疏单项基通常更具可解释性。将这些坐标作为剖面分析（Profile-Wise Analysis）中的兴趣参数，可得到基于似然的不确定性量化与预测。我们通过泊松极限、广义泊松极限及非极限情形下的参数化正态模型，以及基因调控非线性微分方程模型repressilator演示该方法。IIR的Julia实现（包含上述及更多示例）可在https://github.com/omaclaren/reparam获取。