Robbery trends in the city showed significant volatility. There were 3 robberies in 2016, increasing sharply to 11 in 2017, before dropping to zero in 2018. This corresponds to rates of 0.54, 1.95, and 0 per 1,000 people respectively. The city's contribution to state robbery statistics rose dramatically from 0.05% in 2016 to 0.23% in 2017, before falling to 0% in 2018. This extreme fluctuation suggests that robbery was a sporadic issue, with 2017 being a particularly problematic year.

Aggravated assault cases in the city showed an upward trend from 2016 to 2017, followed by a decrease in 2018. There were 7 cases in 2016, 14 in 2017, and 9 in 2018. This translates to rates of 1.27, 2.49, and 1.55 per 1,000 people respectively. The city's contribution to state aggravated assault statistics increased from 0.06% in 2016 to 0.14% in 2017, then decreased slightly to 0.11% in 2018. This suggests that aggravated assault remained a persistent issue, with some year-to-year variation.

When examining correlations, a strong relationship emerges between violent crime trends and population density. As the population density increased from 682 per square mile in 2016 to 718 in 2018, there was a corresponding spike in violent crimes in 2017, followed by a sharp decrease in 2018. This suggests that population density may have some influence on crime rates, although the relationship is not straightforward.

Applying predictive models to forecast violent crime trends, we can estimate that by 2029 (five years from now), the city may see a stabilization in violent crime rates if current trends continue. The sporadic nature of serious crimes like murder and the volatility in robbery rates make precise predictions challenging. However, based on the overall downward trend from 2017 to 2018, we might expect to see violent crime rates settle at or slightly below the 2018 levels, assuming no significant changes in socioeconomic factors or law enforcement strategies.

In summary, Lyons experienced significant fluctuations in violent crime rates between 2016 and 2018, with a notable peak in 2017 followed by a sharp decline in 2018. The city's contribution to state crime statistics was disproportionately high in some categories, particularly murder and robbery, despite its small population. The relationship between population density and crime rates suggests that urban dynamics play a role in crime patterns. Moving forward, maintaining vigilance and continuing crime prevention efforts will be crucial for Lyons to sustain the downward trend in violent crime rates observed in the most recent data.