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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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Föppl–von Kármán Theory

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Dr. Rishabh Pathak

The Föppl–von Kármán theory is a fundamental theory in the field of solid mechanics, specifically in the study of the deformation of thin plates and shells. This theory provides a mathematical framework for analyzing the behavior of thin elastic structures subjected to bending and stretching loads. Here is an overview of the key aspects of the Föppl–von Kármán theory:


1.  Plate and Shell Deformation: The theory is commonly applied to analyze the deformation of thin plates and shells under various loading conditions. It considers the nonlinear effects of both bending and stretching in these structures.


2.  Nonlinear Elasticity: The theory accounts for the nonlinear elasticity of thin plates and shells, where the deformations are significant enough to warrant a nonlinear analysis. This is in contrast to linear elasticity theories that assume small deformations.


3.   Equilibrium Equations: The theory provides equilibrium equations that govern the deformation of thin plates and shells. These equations consider the balance of internal stresses, external loads, and geometric properties of the structure.


4.  Von Kármán Equations: The equations derived from the Föppl–von Kármán theory describe the equilibrium and compatibility conditions for thin plates and shells. These equations are essential for understanding the complex deformations that occur in these structures.


5.  Applications: The Föppl–von Kármán theory has applications in various fields, including aerospace engineering, civil engineering, and biomechanics. In the context of brain development, the theory is used to model the deformation of the cortical tissue during folding processes.


6.    Limitations: While the theory is powerful for analyzing the behavior of thin plates and shells, it has limitations, especially when dealing with highly nonlinear and complex deformations. In such cases, numerical methods like finite element analysis are often employed for more accurate predictions.


In the study of brain development, the Föppl–von Kármán theory is utilized to model the deformation of the cortical tissue and analyze the critical conditions at the onset of folding. By incorporating this theory into analytical and computational models, researchers can gain insights into the mechanical aspects of cortical folding and the formation of brain surface morphologies.

 

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