# What is the difference between the quantile mapping method and the empirical quantile mapping method?

Quantile mapping and empirical quantile mapping are both techniques used for bias correction in climate model outputs. While they share the same fundamental goal of aligning model outputs with observed data, they differ in their approach and the specific methods they use to achieve this alignment.

Quantile Mapping (General)

Quantile mapping involves adjusting the cumulative distribution function (CDF) of model-simulated data to match the CDF of observed data. It can be implemented in various ways, including both parametric and non-parametric approaches.

1. Parametric Quantile Mapping: Involves fitting a theoretical distribution (e.g., normal, gamma, or exponential) to both the observed and modeled data. The adjustment is then based on these fitted distributions.

- Process:

1. Fit a chosen distribution to the observed data and another to the model data.

2. Adjust model data by mapping its quantiles to the quantiles of the observed data distribution.

- Advantage: Can handle a wider range of data types and might be more robust when dealing with limited data.

- Disadvantage: Requires the correct specification of the theoretical distribution, which can be challenging.

#### Bias Correction Tool for CMIP6 Data | Example on CanESM5 Model

Empirical Quantile Mapping (EQM)

Empirical quantile mapping (EQM) is a specific type of quantile mapping that uses the empirical distribution of the data, without fitting it to any theoretical distribution. This is a non-parametric approach.

1. Empirical Quantile Mapping: Relies directly on the observed and modeled data without assuming any specific underlying statistical distribution.

- Process:

1. Construct empirical cumulative distribution functions (ECDFs) for both the observed and modeled datasets.

2. For each value in the modeled dataset, find its quantile in the modeled ECDF.

3. Map this quantile to the corresponding value in the observed ECDF.

- Advantage: Does not assume a specific distribution, making it more flexible and directly reflective of the actual data characteristics.

- Disadvantage: Can be less smooth and might require a larger sample size to construct reliable ECDFs.

Key Differences:

1. Distribution Assumption:

- Quantile Mapping (General): Can be either parametric (assuming a specific theoretical distribution) or non-parametric.

- Empirical Quantile Mapping: Strictly non-parametric, relying directly on the empirical distribution of the data.

2. Flexibility:

- Quantile Mapping (General): More flexibility in terms of the method used, as it can be tailored to fit specific distributions.

- Empirical Quantile Mapping: More flexible in terms of data types and patterns, as it does not rely on predefined distribution forms.

3. Implementation Complexity:

- Quantile Mapping (General): Potentially more complex if using parametric methods, requiring distribution fitting and validation.

- Empirical Quantile Mapping: Simpler to implement as it directly uses observed data without fitting distributions.

4. Data Requirements:

- Quantile Mapping (General): Can work with smaller datasets if a robust distribution fit is achieved.

- Empirical Quantile Mapping: Typically requires a larger sample size to accurately reflect the empirical distributions.

Example Scenarios:

- Quantile Mapping (Parametric): Might be preferred in cases where the data closely follows a known distribution, and there is a need to ensure smoothness and robustness in the bias correction process.

- Empirical Quantile Mapping: Preferred when the data does not fit any known distribution well, or when it is crucial to capture the exact empirical characteristics of the observed data.

Both quantile mapping and empirical quantile mapping are powerful techniques for bias correction in climate data, with their use case depending on the specific nature of the data and the goals of the analysis. Quantile mapping offers flexibility with parametric options, while empirical quantile mapping provides a direct and distribution-free approach.

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