Material parameter identification is a technique that is used to calibrate material models, often a precursor to perform an industrial analysis. Conventional material parameter identification methods estimate the material parameters for a material model by solving an optimisation problem. An alternative but lesser-known approach, called a direct inverse map, directly maps the measured response to the parameters of a material model. In this study we investigate the potential pitfalls of the well-known stochastic noise and lesser-known model errors when constructing direct inverse maps. We show how to address these problems, explaining in particular the importance of projecting the measured response onto the domain of the simulated responses before mapping it to the material parameters. This paper concludes by proposing partial least squares regression as an elegant and computationally efficient approach to address stochastic and systematic (model) errors. This paper also gives insight into the nature of the inverse problem under consideration.