Random projection architectures such as Echo state networks (ESNs) and Extreme Learning Machines (ELMs) use a network containing a randomly connected hidden layer and train only the output weights, overcoming the problems associated with the complex and computationally demanding training algorithms traditionally used to train neural networks, particularly recurrent neural networks. In this study an ESN is shown to contain an antagonistic trade-off between the amount of non-linear mapping and short-term memory it can exhibit when applied to time-series data which are highly non-linear. To overcome this trade-off a new architecture, Reservoir with Random Static Projections (R2SP) is investigated, that is shown to offer a significant improvement in performance. A similar approach using an ELM whose input is presented through a time delay (TD-ELM) is shown to further enhance performance where it significantly outperformed the ESN and R2SP as well other architectures when applied to a novel task which allows the short-term memory and non-linearity to be varied. The hard-limiting memory of the TD-ELM appears to be best suited for the data investigated in this study, although ESN-based approaches may offer improved performance when processing data which require a longer fading memory.