Expected Prediction Error

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• model selection (2)
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1.Fine-tuning Model Parameters for Improved Accuracy[Original Blog]

One of the most important steps in building a forecasting model is to estimate the parameters that govern its behavior. Parameters are the numerical values that determine how the model responds to the input data and the uncertainty in the system. For example, in a linear regression model, the parameters are the slope and the intercept of the line that best fits the data. In a neural network model, the parameters are the weights and biases of the neurons that connect the layers. Estimating the parameters correctly can improve the accuracy and reliability of the forecasts, as well as the interpretability and generalizability of the model. However, parameter estimation is not a trivial task, and it requires careful consideration of several factors, such as:

1. The choice of the estimation method. There are different methods to estimate the parameters of a forecasting model, such as maximum likelihood, least squares, Bayesian inference, gradient descent, etc. Each method has its own advantages and disadvantages, depending on the type of model, the amount and quality of data, the computational resources, and the desired properties of the estimates. For example, maximum likelihood is a popular method that finds the parameters that make the data most probable, but it may not work well if the data is sparse or noisy, or if the model is complex or nonlinear. Bayesian inference is a method that incorporates prior knowledge and uncertainty into the estimation, but it may require more computation and assumptions about the prior distributions. Gradient descent is a method that iteratively updates the parameters by following the direction of the steepest descent of the error function, but it may get stuck in local minima or require fine-tuning of the learning rate and other hyperparameters.

2. The evaluation of the estimation quality. Once the parameters are estimated, it is important to evaluate how well they fit the data and the model. There are different metrics and criteria to assess the quality of the estimation, such as the goodness-of-fit, the confidence intervals, the bias-variance trade-off, the information criteria, the cross-validation, etc. Each metric and criterion measures a different aspect of the estimation quality, such as the accuracy, the precision, the robustness, the complexity, the generalizability, etc. For example, the goodness-of-fit measures how closely the model predictions match the observed data, but it may not reflect how well the model performs on new or unseen data. The confidence intervals measure the range of values that the parameters are likely to take, but they may not account for the model uncertainty or the data variability. The bias-variance trade-off measures the trade-off between the underfitting and overfitting of the model, but it may not capture the optimal balance between the two. The information criteria measure the trade-off between the fit and the complexity of the model, but they may not be comparable across different types of models. The cross-validation measures the generalization error of the model on different subsets of the data, but it may be computationally expensive or sensitive to the choice of the subsets.

3. The optimization of the estimation performance. After evaluating the quality of the estimation, it is possible to optimize the performance of the estimation by adjusting the parameters or the estimation method. There are different techniques and strategies to optimize the estimation performance, such as the regularization, the initialization, the transformation, the feature selection, the model selection, the hyperparameter tuning, etc. Each technique and strategy aims to improve a specific aspect of the estimation performance, such as the stability, the convergence, the scalability, the interpretability, the flexibility, etc. For example, the regularization is a technique that adds a penalty term to the error function to reduce the overfitting of the model, but it may also reduce the sensitivity of the model to the data. The initialization is a technique that sets the initial values of the parameters to speed up the convergence of the estimation method, but it may also affect the final values of the parameters. The transformation is a technique that applies a function to the data or the model to make them more suitable for the estimation method, but it may also change the meaning or the distribution of the data or the model. The feature selection is a technique that selects the most relevant or informative variables for the model, but it may also discard some useful or hidden information. The model selection is a technique that chooses the best model among a set of candidate models, but it may also introduce some bias or uncertainty. The hyperparameter tuning is a technique that optimizes the parameters that control the behavior of the estimation method, but it may also require a lot of trial and error or a systematic search.

To illustrate some of these factors, let us consider an example of a forecasting model that uses a simple exponential smoothing (SES) method to predict the monthly sales of a product. The SES method is a time series forecasting method that uses a weighted average of the past observations, where the weights decay exponentially as the observations get older. The SES method has one parameter, $\alpha$, which is the smoothing factor that determines how much weight is given to the most recent observation. The value of $\alpha$ can range from 0 to 1, where a higher value means more weight to the recent observation and a lower value means more weight to the past observations. The parameter $\alpha$ can be estimated by different methods, such as:

- The method of moments, which equates the sample variance of the data to the theoretical variance of the model and solves for $\alpha$.

- The method of least squares, which minimizes the sum of squared errors between the model predictions and the observed data and solves for $\alpha$.

- The method of maximum likelihood, which maximizes the likelihood function of the data given the model and solves for $\alpha$.

- The method of Bayesian inference, which updates the prior distribution of $\alpha$ with the likelihood function of the data given the model and obtains the posterior distribution of $\alpha$.

Each method may give a different estimate of $\alpha$, depending on the data and the assumptions. For example, if the data is noisy or has outliers, the method of least squares may give a biased estimate of $\alpha$, while the method of Bayesian inference may give a more robust estimate of $\alpha$. If the data is sparse or has missing values, the method of maximum likelihood may give an unreliable estimate of $\alpha$, while the method of moments may give a more consistent estimate of $\alpha$. If the data is non-stationary or has trends or seasonality, the method of moments may give an inaccurate estimate of $\alpha$, while the method of least squares may give a more adaptive estimate of $\alpha$.

Once the estimate of $\alpha$ is obtained, it is important to evaluate its quality by using different metrics and criteria, such as:

- The goodness-of-fit, which can be measured by the coefficient of determination ($R^2$), the mean absolute error (MAE), the root mean squared error (RMSE), the mean absolute percentage error (MAPE), etc. These metrics measure how closely the model predictions match the observed data, where a higher value of $R^2$ or a lower value of MAE, RMSE, or MAPE indicates a better fit. For example, if the estimate of $\alpha$ is 0.8, the $R^2$ is 0.95, the MAE is 10, the RMSE is 15, and the MAPE is 5%, it means that the model explains 95% of the variation in the data, the average absolute error is 10 units, the average squared error is 15 units, and the average percentage error is 5%.

- The confidence intervals, which can be calculated by using the standard error of the estimate, the t-distribution, the bootstrap method, the Bayesian method, etc. These methods provide a range of values that the parameter is likely to take, given a certain level of confidence, such as 95% or 99%. For example, if the estimate of $\alpha$ is 0.8, the standard error of the estimate is 0.05, and the confidence level is 95%, the confidence interval of $\alpha$ is [0.7, 0.9], which means that there is a 95% chance that the true value of $\alpha$ lies between 0.7 and 0.9.

- The bias-variance trade-off, which can be measured by the mean squared error (MSE), the expected prediction error (EPE), the Akaike information criterion (AIC), the Bayesian information criterion (BIC), etc. These measures capture the trade-off between the underfitting and overfitting of the model, where a lower value indicates a better balance. For example, if the estimate of $\alpha$ is 0.8, the MSE is 225, the EPE is 250, the AIC is 300, and the BIC is 320, it means that the model has a moderate bias and a moderate variance, and that it is neither too simple nor too complex.