# Cram ML

In the article “Introduction & Cram Policy Part 1”, we introduced the Cram method, which enables simultaneous learning and evaluation of a binary policy. In this section, we extend the framework to machine learning tasks through the `cram_ml()` function.

## Output of Cram ML

Cram ML outputs the Expected Loss Estimate, which refers to the following statistical quantity:
\[ R(\hat{\pi}) = \mathbb{E}_{\tilde{X} \sim D} \left[ L(\tilde{X}, \hat{\pi}) \right], \] The Expected Loss Estimate represents the average loss that would be incurred if a model, trained on a given data sample, were deployed across the entire population. In the Cram framework, this corresponds to estimating how the learned model generalizes to unseen data—i.e., how it performs on new observations \(\tilde{X}\) drawn from the true data-generating distribution \(D\), independently of the training data.

This expected loss serves as the population-level performance metric (analogous to a policy value in policy learning), and Cram provides a consistent, low-bias estimate of this quantity by combining models trained on sequential batches and evaluating them on held-out observations.

## Built-in Model

To illustrate the use of `cram_ml()`, we begin by generating a synthetic dataset for a regression task. The data consists of three independent covariates and a continuous outcome.

    set.seed(42)
    X_data <- data.frame(x1 = rnorm(100), x2 = rnorm(100), x3 = rnorm(100))
    Y_data <- rnorm(100)
    data_df <- data.frame(X_data, Y = Y_data)

This section illustrates how to use `cram_ml()` with built-in modeling options available through the `cramR` package. The function integrates with the `caret` framework, allowing users to specify a learning algorithm, a loss function, and a batching strategy to evaluate model performance.

Beyond `caret`, `cram_ml()` also supports fully custom model training, prediction, and loss functions, making it suitable for virtually any machine learning task — including regression and classification.

The `cram_ml()` function offers extensive flexibility through its `loss_name` and `caret_params` arguments.

### loss_name argument

The `loss_name` argument specifies the performance metric used to evaluate the model at each batch. Note that Cram needs to calculate individual losses (i.e. map a data point and prediction to a loss value) that are then internally averaged across batches and observations to form the Expected Loss Estimate. Depending on the task, losses are interpreted as follows:

We denote \(x_i\) a data point and \(\hat{\pi}_k\) a model trained on the first k batches of data to illustrate how the individual losses are computed using the built-in loss names of the package.

#### Regression Losses

- Squared Error (`"se"`):
  \[ L(x_i, \hat{\pi}_k) = (\hat{y}_i - y_i)^2 \] Measures the squared difference between predicted and actual outcomes.

- Absolute Error (`"ae"`):
  \[ L(x_i, \hat{\pi}_k) = |\hat{y}_i - y_i| \] Captures the magnitude of prediction error, regardless of direction.

#### Classification Losses

- Accuracy (`"accuracy"`):
  \[ L(x_i, \hat{\pi}_k) = 1\{\hat{y}_i = y_i\} \] It is more a performance metric than a loss here - Cram allows you to define any performance metric that you want to estimate and accuracy is a built-in example. The metric is 1 for correct predictions, 0 for incorrect ones.

- Logarithmic Loss (`"logloss"`):

  The `"logloss"` loss function measures how well predicted class probabilities align with the true class labels. It applies to both binary and multiclass classification tasks.

  For a given observation \(i\), let:

  - \(y_i \in \{c_1, c_2, \dots, c_K\}\) be the true class label,

  - \(\hat{p}_k(i, c)\) be the predicted probability assigned to class \(c\) by the model.

  The individual log loss is computed as: \[ L(x_i, \hat{\pi}_k) = -\log\left( \hat{p}_k(i, y_i) \right) \] That is, we take the negative log of the probability assigned to the true class.

#### Custom Loss Functions

Users can also define their own custom loss function by providing a `custom_loss(predictions, data)` function that returns a vector of individual losses. This allows evaluation of complex models with domain-specific metrics (more details in the Custom Model part below)

### caret_params argument

The `caret_params` list defines how the model should be trained using the `caret` package. It can include any argument supported by `caret::train()`, allowing full control over model specification and tuning. Common components include:

- `method`: the machine learning algorithm (e.g., `"lm"` for linear regression, `"rf"` for random forest, `"xgbTree"` for XGBoost, `"svmLinear"` for support vector machines)
- `trControl`: the resampling strategy (e.g., `trainControl(method = "cv", number = 5)` for 5-fold cross-validation, or `"none"` for training without resampling)
- `tuneGrid`: a grid of hyperparameters for tuning (e.g., `expand.grid(mtry = c(2, 3, 4))`)
- `metric`: the model selection metric used during tuning (e.g., `"RMSE"` or `"Accuracy"`)
- `preProcess`: optional preprocessing steps (e.g., centering, scaling)
- `importance`: logical flag to compute variable importance (useful for tree-based models)

Refer to the full documentation at caret model training and tuning for the complete list of supported arguments and options.

    caret_params_lm <- list(
      method = "lm",
      trControl = trainControl(method = "none")
    )

    result <- cram_ml(
      data = data_df,
      formula = Y ~ .,
      batch = 5,
      loss_name = "se",
      caret_params = caret_params_lm
    )
    print(result)
    #> $raw_results
    #>                         Metric    Value
    #> 1       Expected Loss Estimate  0.86429
    #> 2 Expected Loss Standard Error  0.73665
    #> 3       Expected Loss CI Lower -0.57952
    #> 4       Expected Loss CI Upper  2.30809
    #>
    #> $interactive_table
    #>
    #> $final_ml_model
    #> Linear Regression
    #>
    #> 100 samples
    #>   3 predictor
    #>
    #> No pre-processing
    #> Resampling: None

### Case of categorical target variable

The `cram_ml()` function can also be used for classification tasks, whether predicting hard labels or class probabilities. This is controlled via the `classify` argument and `loss_name`. Below, we demonstrate two typical use cases.

Also note that all data inputs needs to be of numeric types, hence for `Y` categorical, it should contain numeric values representing the class of each observation. No need to use the type `factor` for `cram_ml()`.

#### Case 1: Predicting Class Labels

In this case, the model outputs hard predictions (labels, e.g. 0, 1, 2 etc.), and the metric used is classification accuracy—the proportion of correctly predicted labels.

- Use `loss_name = "accuracy"`
- Set `classProbs = FALSE` in `trainControl`
- Set `classify = TRUE` in `cram_ml()`

    set.seed(42)

    # Generate binary classification dataset
    X_data <- data.frame(x1 = rnorm(100), x2 = rnorm(100), x3 = rnorm(100))
    Y_data <- rbinom(nrow(X_data), 1, 0.5)
    data_df <- data.frame(X_data, Y = Y_data)

    # Define caret parameters: predict labels (default behavior)
    caret_params_rf <- list(
      method = "rf",
      trControl = trainControl(method = "none")
    )

    # Run CRAM ML with accuracy as loss
    result <- cram_ml(
      data = data_df,
      formula = Y ~ .,
      batch = 5,
      loss_name = "accuracy",
      caret_params = caret_params_rf,
      classify = TRUE
    )

    print(result)
    #> $raw_results
    #>                         Metric    Value
    #> 1       Expected Loss Estimate  0.48750
    #> 2 Expected Loss Standard Error  0.43071
    #> 3       Expected Loss CI Lower -0.35668
    #> 4       Expected Loss CI Upper  1.33168
    #>
    #> $interactive_table
    #>
    #> $final_ml_model
    #> Random Forest
    #>
    #> 100 samples
    #>   3 predictor
    #>   2 classes: 'class0', 'class1'
    #>
    #> No pre-processing
    #> Resampling: None

#### Case 2: Predicting Class Probabilities

In this setup, the model outputs class probabilities, and the loss is evaluated using logarithmic loss (`logloss`)—a standard metric for probabilistic classification.

- Use `loss_name = "logloss"`
- Set `classProbs = TRUE` in `trainControl`
- Set `classify = TRUE` in `cram_ml()`

    set.seed(42)

    # Generate binary classification dataset
    X_data <- data.frame(x1 = rnorm(100), x2 = rnorm(100), x3 = rnorm(100))
    Y_data <- rbinom(nrow(X_data), 1, 0.5)
    data_df <- data.frame(X_data, Y = Y_data)

    # Define caret parameters for probability output
    caret_params_rf_probs <- list(
      method = "rf",
      trControl = trainControl(method = "none", classProbs = TRUE)
    )

    # Run CRAM ML with logloss as the evaluation loss
    result <- cram_ml(
      data = data_df,
      formula = Y ~ .,
      batch = 5,
      loss_name = "logloss",
      caret_params = caret_params_rf_probs,
      classify = TRUE
    )

    print(result)
    #> $raw_results
    #>                         Metric    Value
    #> 1       Expected Loss Estimate  0.93225
    #> 2 Expected Loss Standard Error  0.48118
    #> 3       Expected Loss CI Lower -0.01085
    #> 4       Expected Loss CI Upper  1.87534
    #>
    #> $interactive_table
    #>
    #> $final_ml_model
    #> Random Forest
    #>
    #> 100 samples
    #>   3 predictor
    #>   2 classes: 'class0', 'class1'
    #>
    #> No pre-processing
    #> Resampling: None

Together, these arguments allow users to apply `cram_ml()` using a wide variety of built-in machine learning models and losses. If users need to go beyond these built-in choices, we also provide in the next section a friendly workflow on how to specify custom models and losses with `cram_ml()`.

## Custom Model

In addition to using built-in learners via `caret`, `cram_ml()` also supports fully custom model workflows. You can specify your own:

- Model fitting function (`custom_fit`)
- Prediction function (`custom_predict`)
- Loss function (`custom_loss`)

This offers maximum flexibility, allowing CRAM to evaluate any learning model with any performance criterion, including regression, classification, or even unsupervised losses such as clustering distance.

### 1. `custom_fit(data, ...)`

This function takes a data frame and returns a fitted model. You may define additional arguments such as hyperparameters or training settings.

- `data`: A data frame that includes both predictors and the outcome variable `Y`.

Example: A basic linear model fit on three predictors:

    custom_fit <- function(data) {
      lm(Y ~ x1 + x2 + x3, data = data)
    }

### 2. `custom_predict(model, data)`

This function generates predictions from the fitted model on new data. It returns a numeric vector of predicted outcomes.

- `model`: The fitted model returned by `custom_fit()`
- `data`: A data frame of new observations (typically including all original predictors)

Example: Extract predictors and apply a standard `predict()` call:

    custom_predict <- function(model, data) {
      predictors_only <- data[, setdiff(names(data), "Y"), drop = FALSE]
      predict(model, newdata = predictors_only)
    }

### 3. `custom_loss(predictions, data)`

This function defines the loss metric used to evaluate model predictions. It should return a numeric vector of individual losses, one per observation. These are internally aggregated by `cram_ml()` to compute the overall performance.

- `predictions`: A numeric vector of predicted values from the model
- `data`: The data frame containing the true outcome values (`Y`)

Example: Define a custom loss function using Squared Error (SE)

    custom_loss <- function(predictions, data) {
      actuals <- data$Y
      se_loss <- (predictions - actuals)^2
      return(se_loss)
    }

### 4. Use `cram_ml()` with Custom Functions

Once you have defined your custom training, prediction, and loss functions, you can pass them directly to `cram_ml()` as shown below, note that `caret_params` and `loss_name` that were used for built-in functionalities are now `NULL`:

    set.seed(42)
    X_data <- data.frame(x1 = rnorm(100), x2 = rnorm(100), x3 = rnorm(100))
    Y_data <- rnorm(100)
    data_df <- data.frame(X_data, Y = Y_data)

    result <- cram_ml(
      data = data_df,
      formula = Y ~ .,
      batch = 5,
      custom_fit = custom_fit,
      custom_predict = custom_predict,
      custom_loss = custom_loss
    )
    print(result)
    #> $raw_results
    #>                         Metric    Value
    #> 1       Expected Loss Estimate  0.86429
    #> 2 Expected Loss Standard Error  0.73665
    #> 3       Expected Loss CI Lower -0.57952
    #> 4       Expected Loss CI Upper  2.30809
    #>
    #> $interactive_table
    #>
    #> $final_ml_model
    #>
    #> Call:
    #> lm(formula = Y ~ x1 + x2 + x3, data = data)
    #>
    #> Coefficients:
    #> (Intercept)           x1           x2           x3
    #>    0.031503     0.057754     0.008829    -0.031611