<｜begin▁of▁sentence｜>### Instruction:
Imagine you're a data scientist at a rehabilitation facility where you have collected a wealth of data, 100 records each containing 50 variables, about your clients and their recovery journey. These variables might include information ranging from psychological assessments to physiological metrics. In your effort to distil this information into more tangible insights, you've decided that dimensionality reduction could provide a clearer picture of the recovery path and the factors most influential on it.

You're looking to employ a method akin to how a counselor might pinpoint underlying issues from a broad array of behavioral signs—a method called profile likelihood, recommended by researchers Zhu and Godsill for just such a purpose. In this method, finding the 'elbows' or points of inflection in your dataset can be likened to identifying key breakthroughs or turning points in a patient's treatment plan. 

To apply this method digitally, you turn to the `select_dimension` function from the graspologic package; it's designed to assist in isolating these critical elbows that signify the most informative dimensions of your dataset. Your question, as you translate these complex statistical processes into day-to-day improvements in your clients' treatment plans, is how to effectively use graspologic's `select_dimension` to discover these optimal embedding dimensions, and what specific dimension count emerges as the most fitting for your dataset.

This hands-on application in your rehabilitation facility's data-driven approach can potentially illuminate new pathways to recovery by revealing underlying patterns in the data that might otherwise remain obscured by the sheer volume of variables at play.
### Response:
import numpy as np
from graspologic.embed import select_dimension

# Given a 2D array (e.g., an adjacency matrix from a graph)
np.random.seed(42)
X = np.random.rand(100, 50)

# Select the optimal embedding dimension
elbows, sing_vals = select_dimension(X, n_components=None, n_elbows=2, threshold=None, return_likelihoods=False)

# Print the optimal dimensions and their corresponding singular values
for elbow, sing_val in zip(elbows, sing_vals):
    print(f'Optimal dimension: {elbow}, Singular value: {sing_val}')
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