Drift Positional CMR

Different drift rate for repeated items

Drift Positional CMR extends Positional CMR with a separate context drift rate for repeated presentations. When an item appears for the second (or later) time, context drifts at a different rate than for first presentations.

The Mechanism

In standard encoding, context drifts at rate \(\beta_{enc}\) for all items. Drift Positional CMR uses:

\[\beta_{item} = \begin{cases} \beta_{enc} & \text{if first presentation} \\ \beta_{rep} & \text{if repetition} \end{cases}\]

This allows the model to capture phenomena where repetitions have different contextual effects than first presentations.

Why Different Drift?

Several theoretical motivations exist for repetition-specific drift:

  1. Reduced novelty: Repeated items may update context less because they’re already somewhat familiar
  2. Chunking: Repetitions might integrate more strongly, linking the two presentations
  3. Attention differences: People may attend differently to repeated items

Mathematical Specification

Encoding Decision

When item \(k\) is presented at position \(i\):

Code 
if item_k ∈ studied[0:i]:
    drift_rate = β_rep  # Repetition
else:
    drift_rate = β_enc  # First presentation

Context Update

\[\mathbf{c}_i = \rho_i \mathbf{c}_{i-1} + \beta_{item} \mathbf{c}^{IN}_i\]

where \(\beta_{item}\) is determined by the check above.

Parameters

Parameter Symbol Description
encoding_drift_rate \(\beta_{enc}\) Drift rate for first presentations
repetition_drift_rate \(\beta_{rep}\) Drift rate for repetitions

All other parameters are inherited from Positional CMR.

Predictions

\(\beta_{rep}\) Interpretation
\(< \beta_{enc}\) Repetitions update context less (reduced novelty)
\(= \beta_{enc}\) Equivalent to standard Positional CMR
\(> \beta_{enc}\) Repetitions update context more (enhanced chunking)
\(= 0\) Repetitions don’t update context at all
\(= 1\) Repetitions completely reset context

Usage

Code 
from jaxcmr.models.drift_positional_cmr import CMR

params = {
    "encoding_drift_rate": 0.6,      # First presentations
    "repetition_drift_rate": 0.3,    # Repetitions (less drift)
    "start_drift_rate": 0.5,
    "recall_drift_rate": 0.5,
    "learning_rate": 0.5,
    "primacy_scale": 2.0,
    "primacy_decay": 0.8,
    "shared_support": 0.05,
    "item_support": 0.25,
    "choice_sensitivity": 0.6,
    "mfc_sensitivity": 3.0,
    "stop_probability_scale": 0.05,
    "stop_probability_growth": 0.2,
    "learn_after_context_update": True,
    "allow_repeated_recalls": False,
}

model = CMR(list_length=16, parameters=params)

Experimental Predictions

Spacing Effects

With \(\beta_{rep} < \beta_{enc}\):

  • Massed repetitions (immediate): Second presentation barely shifts context → strong overlap with first
  • Spaced repetitions (delayed): Context has drifted far; repetition at lower rate creates moderate overlap

This can produce the classic spacing effect: spaced repetitions are better remembered than massed repetitions.

Lag-Recency Analysis

The model predicts different lag-CRP curves for: - Transitions from first presentations - Transitions from repetitions (if \(\beta_{rep} \neq \beta_{enc}\))

Lower \(\beta_{rep}\) would produce flatter lag-CRP for repetitions (less context change = less contiguity).

When To Use

Use Drift Positional CMR when:

  • Fitting data with repeated items
  • Spacing effects are important
  • You hypothesize that repetitions affect context differently

Implementation

Key difference from Positional CMR:

Code 
def experience_item(self, item_index):
    # Check if item has been studied before
    is_repetition = jnp.isin(item_index + 1, self.studied)

    # Use appropriate drift rate
    new_context = lax.cond(
        is_repetition,
        lambda: self.context.integrate(context_input, self.repetition_drift_rate),
        lambda: self.context.integrate(context_input, self.encoding_drift_rate),
    )
    ...

Relationship to Other Variants

Variant Key Mechanism
Positional CMR Position-based encoding
Drift Positional CMR Position + different drift for repetitions
Reinforcement CMR Position + first-presentation boosting
Outlist CMR Item-based + out-of-list context