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In machine learning, "Predicted Values" (often represented as ŷ, pronounced "yhat") refer to the values or outcomes that a machine learning model predicts for a given set of input data points. These predicted values are the model's estimates or approximations of the target variable(s) it is designed to predict.
Here are a few key points about predicted values (ŷ) in machine learning:

Target Variable: Predicted values are associated with the target variable(s) in a supervised learning problem. In supervised learning, you have a dataset consisting of input features (X) and corresponding target values (y). The goal of the machine learning model is to learn a mapping from X to y so that it can make predictions (ŷ) for new, unseen input data.

Regression: In regression tasks, where the target variable is continuous (e.g., predicting house prices, temperature), the predicted values ŷ are continuous as well. These represent numerical estimates of the target variable.

Classification: In classification tasks, where the target variable is categorical (e.g., classifying emails as spam or not spam, identifying types of objects in images), the predicted values ŷ typically represent the predicted class labels or class probabilities.

Evaluation: Predicted values are used to evaluate the performance of a machine learning model. You can compare ŷ to the actual target values (y) to calculate various performance metrics, such as mean squared error (MSE) for regression problems or accuracy, precision, recall, and F1score for classification problems.

Visualization: Predicted values can be visualized to understand how well the model's predictions align with the actual data. Scatter plots, line plots, or confusion matrices are common visualization tools.

Model Selection: Predicted values are essential for model selection and hyperparameter tuning. Different models or configurations may produce different ŷ values, and you can choose the bestperforming model based on how well its predictions match the actual data.
For instance, the predicted age (y^) of a dog in an example would be the output of the network without any activation function applied to it:
 [3977a]
where,
w_{1}^{(i)} is the weight for the single neuron in the output layer.
b_{1}^{(i)} is the bias for the single neuron in the output layer.
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