peakandtrough

Peak and trough is a term used in data analysis, mathematics, and signal processing to describe local extrema in a sequence, function, or time series. A peak, or local maximum, is a point whose value is greater than those of its neighboring points; a trough, or local minimum, is a point whose value is lower than its neighbors. These extrema can be local, affecting only a portion of the data, or global, representing the absolute highest or lowest value in the dataset.

In continuous functions, peaks occur where the first derivative is zero and the second derivative is negative; troughs occur where the first derivative is zero and the second derivative is positive. In discrete data, peaks and troughs are identified by comparing a sample with adjacent points. Noise and irregular sampling can create spurious extrema, so smoothing, filtering, or thresholding are common preprocessing steps.

Methods to detect peaks and troughs include derivative-based approaches, zero-crossing analysis of the signal’s gradient, and algorithms that use prominence, width, or distance constraints to distinguish significant extrema from minor fluctuations. Prominence measures how much an extremum stands out relative to surrounding features, while width describes the extent of the peak or trough at a chosen height.

Applications span finance (identifying turning points in prices), meteorology (extreme weather indicators), engineering and signal processing (envelope and peak detection), and physiology (notable events in ECG or other biosignals). Challenges include noise, nonstationarity, and uneven sampling, all of which can affect the reliability of peak and trough detection.