An Intro to Proxima Development Modeling

Additional Context: Intro

Claims evolve significantly from initial reporting to final settlement, with costs changing at each stage:

1. Initial Reporting: Early estimates are based on limited information, often excluding future costs like medical expenses or legal fees.

2. Investigation and Adjustment: As the claim develops, new factors can raise or lower costs. For instance, discovering additional injuries or liability can adjust the claim value.

3. Development Period: Claims can escalate due to complications like prolonged medical treatment or litigation. Economic factors like inflation may also drive up costs.

4. Final Resolution: The claim reaches its ultimate cost once all payments are made, including unexpected settlements or additional expenses.

The Proxima model aggregates this data to capture trends in claim payments and reserves, ultimately achieving more accurate predictions of ultimate costs.

Additional Context: Going to Zero

Using ratios, such as losses to premiums, is an essential technique in insurance claim modeling to normalize data across various accident years, segments, or lines of business. Different segments may have vastly different levels of loss and exposure, with one segment potentially involving millions of dollars and another just thousands. By converting raw data into ratios with similar expected ranges, we allow for direct comparisons between segments of vastly different sizes. This normalization ensures that the model is not skewed by the size of the claims but instead focuses on the trends and relationships within the data.

We also use incremental ratios to enable comparisons between accident years that may have very different cumulative loss ratios. For instance, one year could have a significantly higher cumulative loss ratio due to larger initial claims, while another may start with a smaller loss ratio. However, when examining the incremental claim development over time, both years may show similar patterns or trends after the first year. This allows for a more consistent modeling structure, ensuring the model reflects real-world behavior rather than inflated early-year variations.

Additional Context: Creating a Surface

Spatial dependence is invaluable when working with multidimensional time series, as it allows models to leverage relationships across all time periods. This interconnectedness helps to smooth predictions and ensures that the model accounts for natural correlations within the data. For example, claim periods that are temporally or sequentially close often share similar economic conditions, regulatory environments, or operational policies, leading to shared patterns in their development.

By incorporating spatial dependence into the model, we can improve its ability to generalize, especially in cases where data is sparse or incomplete. For instance, gaps in claim development can be estimated more accurately by inferring from nearby periods. Similarly, anomalies or outliers can be identified and contextualized more effectively when the relationships among neighboring data points are understood.

This approach also enhances predictive accuracy, as the model can identify broader trends that transcend individual claim periods, aligning with the underlying structure of the dataset.

Additional Context: Fitting with Kernels

Kernels serve as the covariance function in Gaussian Process (GP) regression. GPs are a powerful non-parametric method used to model time series data as a multivariate normal distribution. The kernel plays a key role in GPs, as it defines the relationship between data points by specifying how similar they are to each other. This similarity measure is fundamental to making predictions and understanding the underlying patterns in the data.

Kernels provide several key benefits, including:

• Efficiency: Kernels use a small set of hyperparameters to build covariance matrices of any size.
• Flexibility: Kernels, such as the Radial Basis Function (RBF), model non-linear relationships in data.
• Uncertainty Quantification: GPs offer not only predictions but also confidence in these predictions.
• Smooth Extrapolation: GPs can smoothly predict values beyond the observed data.
• Interpretability: Kernel hyperparameters provide insights into the data structure and underlying processes.

Additional Context: Across Claim Periods

The values within an actuarial triangles can exhibit varying trends, such as gradual increases or decreases, which are influenced by a variety of factors including economic conditions, changes in underwriting practices, or shifts in claims handling procedures. Short-term trends often reflect more immediate factors, such as recent changes in claims processes or external conditions, while long-term trends capture broader, more persistent patterns that develop over time.

Short-term averages may be useful for capturing immediate, impactful trends, especially in volatile environments, while longer-term averages can smooth out fluctuations and provide more stable, reliable insights. The goal is to strike a balance between responsiveness and stability in predictions.

Additional Context: Within Claim Periods

Modeling within a claim period can be beneficial for capturing unique patterns that emerge as claims mature. Early development years often contain noise due to reporting delays and initial volatility, but as claims stabilize, clearer trends may emerge. For example, some claim periods may show a higher frequency of claims with regular indemnity payments, while others may indicate that most claims have already been settled. These distinctions help refine the model, allowing it to better reflect the underlying dynamics of each claim period.

By focusing on the specific progression of claims within a given period, we can identify and account for variations that might be overlooked in a broader, cross-period model. This approach allows for a more nuanced understanding of claims development, which can improve the accuracy of future predictions.

Additional Context: The Combined Effect

Combining kernels allows us to model complex relationships in data by capturing different patterns and trends. There are several ways to combine kernels:

• Adding Kernels: This models multiple independent sources of variation. For example, one kernel may capture long-standing patterns, while another accounts for recent pattern changes.
• Multiplying Kernels: This creates an interaction between patterns. A data point's predicted value is high only if both kernels agree, which is useful when multiple dimensions need to align, such as claim period and development period trends.
• Different Kernel Types: Each kernel captures different forms of similarity. For instance, the RBF kernel models smooth trends, while a periodic kernel captures repeating patterns. Combining these kernels enables us to model complex relationships effectively.

Strategic combination of kernels enhances our ability to model intricate patterns in claims data.

Additional Context: Across All Time Periods

The kernel models the relationship between all data points by calculating pairwise similarities, forming a covariance matrix that captures how each data point relates to all others in the dataset.

Signal variance and noise variance are critical parameters. Signal variance controls the strength of relationships modeled by the kernel, influencing how much influence nearby data points have. Noise variance accounts for random fluctuations or errors in the data, ensuring the model isn't overfitting to noise.

Real-world data validation usually includes review of the signal-to-noise ratio which describes the quality of the model fit and projections. A high signal-to-noise ratio suggests the underlying patterns were well-interpreted by the model.

Additional Context: Adding More Datasets

Multioutput models in Gaussian Processes (GPs) capture dependencies between related datasets, improving predictions and reducing uncertainty. Instead of modeling groups of claims independently, these models leverage shared patterns, like market trends or claims handling practices, to provide better insights.

By learning cross-dataset correlations, multioutput GPs use information from credible groups to refine predictions for smaller or noisier datasets. This reduces uncertainty and ensures that both shared trends and unique behaviors are captured effectively.

For example, if multiple claim groups exhibit similar reactions to inflation, a multioutput model can exploit these shared patterns to enhance predictions, especially in areas with sparse data. While an isolated increase in costs for a single triangle during a development period might not significantly alter future expectations, observing similar increases across 20 triangles suggests a broader trend that the model should account for.

Additional Context: Weighing the Relationships

This section describes the Linear Model of Coregionalization (LMC). LMC is a powerful framework for modeling dependencies across multiple datasets in Gaussian Processes (GPs). It achieves this by decomposing the covariance structure into a sum of separable components. Each component consists of a kernel applied to the input space, scaled by a coregionalization matrix that defines relationships between datasets.

In practice, LMC enables the model to capture both shared and dataset-specific patterns. For instance, a single kernel might model a common trend across datasets, while another kernel accounts for unique variations within each dataset. The coregionalization matrix specifies the strength and nature (positive or negative) of correlations, allowing fine-tuned control over how datasets influence one another.

This approach can be extended by summing multiple kernels, enabling the model to capture complex dependencies. The flexibility of LMC makes it an effective tool for multioutput GPs and other cross-dataset analyses.

Additional Context: Predict the Future

Gaussian Processes (GPs) predict values for new data points by leveraging their learned relationships from observed data. These relationships are encoded in the covariance matrix, which captures the similarity between points based on the kernel. The result is a probabilistic prediction that provides both a mean and variance for each new point.

In GP regression, predictions at a new point, \( x_∗ \), rely on the covariance vector (\( k_∗ \)) between \( x_∗ \) and the observed points, and the covariance matrix (\( K \)) of the observed data. The mean prediction is a weighted average of the observed values, where the weights depend on the covariance structure. The variance reflects the uncertainty based on the distance and relationships defined by the kernel.

This flexibility allows GPs to model nonlinear patterns and quantify uncertainty, making them invaluable for tasks requiring both accurate predictions and reliable uncertainty estimates.