A single anthropometric index such as stunting, wasting, or underweight does not show the holistic picture of under-five children's undernutrition status. To alleviate this problem, we adopted a multifaceted single index known as the composite index for anthropometric failure (CIAF). Using this undernutrition index, we investigated the disparities of Ethiopian under-five children's undernutrition status in space and time. Space refers to the administrative zone which is the second level of the Ethiopian administrative level where the social service delivery decision-making process is made. The time refers to the years 2000, 2005, 2011, and 2016. Besides the spatial-temporal undernutrition investigation, we strive to identify the environmental, socio-economic, household characteristics, and child characteristics risk factors. Machine learning-based exploratory classification methods were used to gain insight into the risk factors and the response variable. Then a generalized mixed model, a generalized additive mixed model with a spatial effect, and spatial-temporal models were used. The duration of breastfeeding, dietary diversity score, mother’s education level, mother’s media exposure, sanitation, source of drinking water, household wealth index and environmental variables precipitation, urbanization, and vegetable indices were found to be significantly associated with under-five children undernutrition status. The results of this study can help policymakers to target appropriate sets of interventions or prevent the use of incorrect interventions for undernutrition control and prevention. This can help in the targeted allocation of limited zone-level health system resources.