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Parametric learning algorithms are a class of machine learning algorithms that make specific assumptions about the functional form of the underlying data distribution. These assumptions simplify the learning process by reducing the problem to finding a set of parameters that fit this predetermined model form. These algorithms assume that the data can be adequately represented by a fixed number of parameters, which are estimated from the training data. The primary characteristic of parametric learning algorithms is that the model complexity is not determined by the size of the training dataset but rather by the complexity of the chosen model architecture.

Some key characteristics and examples of parametric learning algorithms are:

1. Fixed Model Structure: Parametric algorithms assume a fixed form or structure for the model that describes the relationship between the input features and the target variable. This structure is defined by a set of parameters, and learning involves finding the optimal values for these parameters. Parametric algorithms fit fixed set of parameters (θ_i) to data.

2. Limited Capacity: Parametric models have a limited capacity to represent complex relationships in the data. If the chosen model is too simple to capture the underlying patterns, it may result in underfitting. On the other hand, if the model is too complex, it may overfit the training data.

3. Efficient Training: Since parametric models have a fixed number of parameters, they are often computationally efficient to train. Training typically involves optimizing the model's parameters using techniques like gradient descent or closed-form solutions.

4. Examples: Some common examples of parametric learning algorithms include linear regression (where the relationship between input features and the target is assumed to be linear), logistic regression (used for binary classification), and various types of neural networks with fixed architectures (e.g., feedforward neural networks with a fixed number of layers and units).

5. Assumptions: Parametric models often make strong assumptions about the distribution of the data, such as Gaussian distributions in the case of linear regression or logistic regression. These assumptions can limit the model's applicability to certain types of data.

6. Scalability: Parametric models can be easier to scale to large datasets compared to non-parametric models, as their complexity is determined by the model structure and not the size of the training data.

In contrast to parametric learning, non-parametric learning methods, such as k-nearest neighbors (KNN) and decision trees, do not assume a fixed functional form and can adapt their complexity based on the data itself. Non-parametric models can be more flexible but may require more data and can be computationally expensive for large datasets.

The choice between parametric and non-parametric learning depends on the nature of the data and the problem at hand. Parametric models are suitable when the underlying data distribution can be reasonably well approximated by the chosen model architecture and assumptions.

A common application of parametric learning algorithms in the semiconductor industry is in the area of yield prediction and optimization. Specifically, linear regression models are often used to predict the yield of semiconductor manufacturing processes. In semiconductor manufacturing, yield refers to the proportion of devices on a wafer that function properly after the manufacturing process. Predicting yield is crucial because it influences production costs and overall profitability. Manufacturers collect extensive data on various process parameters and environmental conditions that could affect the yield, such as temperature, humidity, equipment settings, and material purity. In this example, linear regression model for yield prediction is used:

• Objective: Predict the yield based on various input features that describe the manufacturing conditions and process settings.
• Data: Historical data on yields and corresponding process conditions.
• Model: A linear regression model where the yield is modeled as a function of these inputs. The form of the model might be something like:

  ------ [3889a]

  where β_0​ , β₁​ , … , β_n​ are parameters to be estimated based on the historical data, and ϵ represents the error term.

• The model is trained using historical data where the true yields and corresponding process parameters are known.
• Once the model parameters (β's) are estimated, it can be used to predict the yield for new production batches under given conditions.
• This prediction allows engineers to adjust process parameters proactively to optimize the yield, reducing waste and increasing profitability.

This approach leverages the assumption that the relationship between the process parameters and the yield is linear, which simplifies the modeling and computation. However, if this assumption does not hold true (e.g., if there are nonlinear dependencies), the prediction accuracy might suffer, indicating a potential need for more complex, possibly non-parametric models.

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