Determining the embedding parameters governing long-term dynamics of copper prices

Mineral extraction activities and the commercialisation of mineral-based products in later stages are fundamental for economic, technological and social development, so understanding their long-term market behaviour in order to forecast prices is crucial for governments, companies and society. Mineral commodity markets are multidimensional as they are driven by the dynamic evolution of financial, technological, psychological and geopolitical factors. Thus, identifying the key features governing mineral commodity prices and modelling their long-term behaviour is an intricate problem that becomes even more complex when the relationship with time is also considered. Although several techniques are available to represent the behaviour and reduce the dimensionality of data sets, these techniques often neglect the temporal relationship and evolution of variables. In many cases, the selection of key variables is solely based on the correlation with each other, relevance and degree of variance, and the cause and effect behaviour of the time-relation of variables, is neglected. This is a critical issue in assessing dynamic systems, especially those involving human behaviour such as social sciences and biology where learning and cognition capacities evolve through time. In this sense, before any attempt to forecast mineral commodity prices, a proper understanding of the embedding dimension and time delay of the variables governing the system is fundamental to determining the number of key features driving the system and the extension of the delayed effects of changes in the initial conditions, respectively. Determining these parameters may become even more complicated because of data scarcity, since the high cost, technical or temporal limitations often hamper obtaining large data sets related to complex dynamic systems that occur in nature and the social sciences. Daily and monthly quotations have been commonly used to represent the dynamics of mineral commodity markets and predict prices, yet they are not sufficient to capture and understand the dynamic evolution of prices in the long-term. These quotations are too short as the delayed effects that reflect on the entire economy and arises from changes in macroeconomic variables can normally be perceived about a year ahead, and may last for up to six years. However, chaos theory has provided valuable tools to assess the dynamics of complex systems allowing the proper determination of the embedding parameters (dimension and time delay), even when using small data sets. This paper examines the long-term dynamics of mineral commodity prices via chaos theory based on a short data set of annual copper prices observed between 1900 and 2015. Copper was used as a study case because it is one of the most competitive markets and representative of mineral commodities traded worldwide. It was found that the dynamic-deterministic behaviour of annual copper prices is embedded in a high-dimensional space that fluctuates between five and seven, and a relatively short time delay between two and three periods. Our finding argues for the use of chaos theory as an important technique to assess the long-term behaviour of mineral commodity prices using short data sets because it improves the understanding of those complex dynamic systems and provides important guidelines that remarkably simplifies the forecasting task.