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In machine learning, "Predicted Values" (often represented as ŷ, pronounced "y-hat") 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:

1. 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.

2. 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.

3. 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.

4. 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 F1-score for classification problems.

5. 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.

6. 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 best-performing 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,

          w₁⁽ⁱ⁾​ is the weight for the single neuron in the output layer.

          b₁⁽ⁱ⁾​ is the bias for the single neuron in the output layer.

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