Assignment 2: Search of Associative Memory (SAM) Model

Assignment 2: Search of Associative Memory (SAM) Model#

Overview#

In this assignment, you will implement the Search of Associative Memory (SAM) model as described in Atkinson & Shiffrin (1968). The SAM model is a probabilistic model of free recall that assumes items are stored in short-term memory (STS) and long-term memory (LTS), with retrieval governed by associative processes. You will fit your implementation to Murdock (1962) free recall data and evaluate how well the model explains the observed recall patterns.

Data Format and Preprocessing#

The dataset consists of sequences of recalled items from multiple trials of a free recall experiment. Each row represents a participant’s recall sequence from a single list presentation.

Numbers (1, 2, …, N): Serial position of recalled items.
88: Denotes errors (recalls of items that were never presented)
Example:
```
6 1 4 7 10 2 8  
10 9 6 8 2 1 88  
10 7 9 8 1  
```
This represents three recall trials. Each row contains the order in which items were recalled for one list.

Model Description#

You will implement and fit a Search of Associative Memory (SAM) model that consists of:

1. Encoding Stage: Memory Storage#

When an item is presented, it is stored in both STS and LTS:

STS (Short-Term Store): Holds a limited number of items (size \(S\)), with older items displaced as new ones enter.
LTS (Long-Term Store): Each item is encoded with a strength \(S_i\) that depends on:
- The number of times it was rehearsed in STS.
- A decay function over time.

Encoding Rules#

Rehearsal Buffer Size (\(S\)): At any time, only \(S\) most recent items are actively rehearsed.
Probability of Rehearsal (\(P_r\)): Given by:

\[P_r(i) = \frac{R_i}{\sum_j R_j}\]

where \(R_i\) is the number of times item \(i\) was rehearsed.
Strength in LTS (\(S_i\)): Memory strength decays with time and interference:

\[S_i = \alpha \cdot R_i \cdot e^{-\beta (T - t_i)}\]

where:
- \(\alpha\) is the encoding rate parameter (controls how well rehearsed items are stored).
- \(\beta\) is the decay rate (how quickly memory strength fades over time).
- \(T\) is the time since the start of the list, and \(t_i\) is when the item was last rehearsed.

2. Retrieval Stage: Memory Search#

The recall process is modeled as a probabilistic search through LTS.

Retrieval Probability#

Each item’s retrievability depends on its memory strength:

\[P_{\text{recall}}(i) = \frac{S_i}{\sum_j S_j}\]

Items with higher \(S_i\) are more likely to be retrieved.
Retrieval stochasticity is modeled via a retrieval threshold—if an item’s strength is too low, recall terminates.

Search Dynamics#

Recall starts with a high-probability item (often first or last presented).
Associative transitions determine what item comes next:
- If item \(i\) was just recalled, the next recall follows a probability based on the similarity (lag) between memories.
\[P(l) = \frac{S_{i+l}}{\sum_j S_j}\]

where:
- \(l\) is the lag (distance in serial position from the previously recalled item).
- \(S_{i+l}\) represents memory strength at that position.
Recall continues until a failure occurs (e.g., no item surpasses a retrieval threshold).

3. Fitting the Model#

You will fit three key parameters to optimize the match to human recall data:

Rehearsal Buffer Size \(S\): Number of items actively rehearsed in STS.
Encoding Rate \(\alpha\): Governs how well rehearsed items are stored in LTS.
Decay Rate \(\beta\): Controls how quickly memory traces degrade over time.

Implementation Tasks#

Step 1: Implement Memory Encoding#

Create an STS buffer of size \(S\).
Assign memory strengths \(S_i\) to each item based on rehearsal frequency.
Implement the decay function for LTS.

Step 2: Implement Retrieval Process#

Use probabilistic sampling to simulate recall.
Implement stopping rules for retrieval failures.
Simulate associative transitions in recall.

Step 3: Load and Process Data#

Read in the recall dataset and parse trials correctly.
Compute observed recall probabilities.

Step 4: Fit Model Parameters#

Adjust \(S\), \(\alpha\), and \(\beta\) to optimize match to observed data.

Step 5: Generate Key Plots#

For each combination of list length and presentation rate, generate the following plots:

Plot 1: Probability of First Recall as a Function of Serial Position#

X-axis: Serial position in the presented list.
Y-axis: Probability that the item is the first item recalled.
Overlay model predictions on observed data.

Plot 2: Conditional Recall Probability as a Function of Lag#

X-axis: Lag (\(l\)) (distance between successive recalls).
Y-axis: Probability of recalling an item at position \(i+l\) after recalling \(i\).
Overlay model predictions on observed data.

Plot 3: Overall Probability of Recall as a Function of Serial Position#

X-axis: Serial position in the presented list.
Y-axis: Probability of recall at any output position.
Overlay model predictions on observed data.

Each plot should include:

Observed human recall data (solid line).
Model predictions (dashed line).
Error bars (95% confidence intervals).

Step 6: Evaluate the Model#

Write a brief discussion (3-5 sentences) addressing:
- Does the model explain the data well?
- Which patterns are well captured?
- Where does the model fail, and why?
- Potential improvements or limitations of SAM.

Grading Criteria#

Model Implementation (50%):
- Correct encoding, retrieval, and stopping rule logic.
- Accurate probability calculations.
- Clear and well-documented code.
Plot Accuracy (45%):
- 15% for each of the three required plots.
- Must correctly overlay observed data and model predictions.
Interpretation and Discussion (5%):
- Thoughtful analysis of model fit.
- Identifies strengths and weaknesses of SAM.

Submission Instructions#

Submit a Google Colaboratory notebook (or similar) that includes:
- Your full implementation of the SAM model.
- Markdown cells explaining your code, methodology, and results.
- All required plots comparing model predictions to observed data.
- A short written interpretation of model performance.

Good luck, and happy modeling!