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Uncertainty in Multiclass Classification

Dr. Rishabh Pathak
1. What is Uncertainty in Classification? Uncertainty refers to the model’s confidence or doubt in its predictions. Quantifying uncertainty is important to understand how reliable each prediction is. In multiclass classification , uncertainty estimates provide probabilities over multiple classes, reflecting how sure the model is about each possible class. 2. Methods to Estimate Uncertainty in Multiclass Classification Most multiclass classifiers provide methods such as: predict_proba: Returns a probability distribution across all classes. decision_function: Returns scores or margins for each class (sometimes called raw or uncalibrated confidence scores). The probability distribution from predict_proba captures the uncertainty by assigning a probability to each class. 3. Shape and Interpretation of predict_proba in Multiclass Output shape: (n_samples, n_classes) Each row corresponds to the probabilities of ...
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Analytical Model: Growing Cortex on elastic subcortex

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Dr. Rishabh Pathak

In the analytical model of brain development, specifically focusing on cortical folding, the scenario of a growing cortex on an elastic subcortex is considered. Here are the key aspects of this analytical model:


1.   Model Description: The model involves interpreting the subcortex as an infinite half-space and imposing a sinusoidal deflection on its upper boundary. The deflection is considered as the sum of an elastic subcortical deflection and subcortical growth, reflecting the dynamic nature of the tissue.


2.  Deflection Analysis: The deflection of the growing cortex on the elastic subcortex is analyzed using the Föppl–von Kármán theory and the classical fourth-order plate equation. This analysis helps in understanding the deformation behavior of the cortical tissue as it grows and interacts with the underlying subcortical layer.


3.   Parameter Variation: The model explores the effects of varying parameters such as cortical thickness, stiffness ratios between the cortex and subcortex, and growth rates. By systematically changing these parameters, researchers can investigate how different mechanical properties influence the folding patterns and surface morphologies of the brain.


4. Sensitivity Studies: Sensitivity studies are conducted to analyze how changes in cortical thickness and stiffness ratios impact the wavelength of folding patterns. These studies provide insights into the relationship between mechanical properties and the resulting brain surface morphology.


5.  Computational Validation: The analytical estimates derived from this model are validated computationally using finite element analysis. Computational modeling allows for a more detailed exploration of the complex folding patterns and surface morphologies that arise from the interactions between the growing cortex and elastic subcortex.


6.     Implications: By studying the growth of the cortex on the elastic subcortex, researchers can gain a better understanding of the mechanical mechanisms underlying cortical folding in the brain. This model helps in predicting realistic surface morphologies and provides insights into the development of complex brain structures.


In summary, the analytical model of a growing cortex on an elastic subcortex provides a framework for investigating the mechanical interactions that drive cortical folding during brain development. By combining analytical and computational approaches, researchers can elucidate the role of growth, stiffness, and other factors in shaping the intricate surface morphologies of the mammalian brain.

 

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