Partial Duration Series

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Partial Duration Series (PDS), also known as Peak Over Threshold (POT) method, is a statistical approach used in various fields such as hydrology, finance, and environmental studies to analyze extreme values or peaks over a specified threshold. This method focuses on the magnitudes and occurrences of events that exceed a predefined threshold, rather than considering all data points. PDS is particularly useful for assessing risks and designing systems to withstand extreme events.

Concept

The core idea behind Partial Duration Series is to identify and analyze only those data points that represent significant deviations from the norm, i.e., values that exceed a certain threshold level. This contrasts with the Annual Maximum Series (AMS) approach, which considers only the single highest value in each year.

Mathematical Representation

Given a time series of data , where is the number of observations, a threshold value is selected. The partial duration series consists of all such that:

The selection of an appropriate threshold is critical in PDS analysis. It should be high enough to exclude non-extreme events but not so high that it leads to too few data points for reliable statistical analysis.

Advantages of PDS

Focus on Extremes: By concentrating on values above a threshold, PDS effectively isolates the most critical events for analysis.
Flexibility: The method allows for adjusting the threshold according to specific needs or objectives.

Applications

Hydrology: In flood risk assessment, PDS can help determine flood frequencies and magnitudes by analyzing peak river flows.
Finance: The method is used to assess risk by analyzing extreme market movements or insurance claims.
Environmental Science: For studying extremes in temperature or precipitation changes due to climate variability.

Challenges

Threshold Selection: Finding an optimal threshold can be subjective and may significantly affect results.
Data Requirements: A sufficiently long record of data is necessary to capture rare events accurately.
Statistical Assumptions: The method assumes independence between exceedances; however, in reality, clustering of extreme events can occur.

Example

Consider a simplified example where we have daily rainfall data for one year ( days). If we set a threshold at 20mm/day as our criterion for an extreme rainfall event:

import numpy as np

# Sample daily rainfall data for one year
rainfall_data = np.random.uniform(0, 50, 365)  # Simulated rainfall amounts between 0mm and 50mm
threshold = 20  # Threshold set at 20mm/day

# Extracting partial duration series
pds = rainfall_data[rainfall_data > threshold]

print(f"Number of extreme rainfall events: {len(pds)}")

In this example code snippet, rainfall_data represents our hypothetical daily rainfall measurements over one year. By applying the Partial Duration Series method with a chosen threshold of 20mm/day, we extract all instances considered as extreme rainfall events based on our criteria.