A multivariate index for estimating seasonal shifts from temperature curves
Seasonal timing is often studied using fixed calendar seasons, temperature thresholds, or a fixed smoothing parameter applied across all years. These may be sensitive to changes in variability over time. To address this, we propose the Seasonal Adaptive LocalisabLe Index (SALLI), a multivariate index, for characterising seasonality from daily temperature curves observed over complete seasons. SALLI uses Significant Zero Crossings of Derivatives (SiZer) to identify significant increases and decreases across a range of smoothing levels. It uses this information to select a smoothing level for each year. The method assumes one main warm peak and one cool trough per annual cycle. For each year, SALLI estimates the annual temperature curve and the midpoints of the four seasons and temperatures. These estimates describe changes in both seasonal timing and temperature. We illustrate SALLI using Australian maximum temperature data from 1965 to 2024. Comparisons of SALLI estimates over the first and last decades show not only a warming, but a shift to earlier seasons, with the midpoints of spring, summer, and winter occurring roughly a week earlier. All four seasonal midpoints become warmer, with the largest increases occurring in summer (1.37 $^\circ$C) and spring (1.22 $^\circ$C). Warming is evident across time, whereas the clearest shift in seasonal timing emerges after the 2000s. By estimating both timing and temperature at each seasonal midpoint, SALLI locates where changes occur on the annual temperature curve, providing information that is more robust and data-driven than fixed thresholds, fixed calendar seasons, or a single smoothing choice.