# Are your Items in Order?

## TL;DR

This paper proposes a statistically grounded method to evaluate the quality of an ordering of dataset items by fitting an order-sensitive model and comparing its score to random orders, aiding data exploration.

## Contribution

It introduces a new order quality score based on a Markov chain-like model and a dynamic programming algorithm to efficiently compute it, along with a greedy method for finding good orders.

## Key findings

- The score effectively distinguishes meaningful orders from random ones.
- The method performs well on both synthetic and real datasets.
- A greedy algorithm can find high-quality orders efficiently.

## Abstract

Items in many datasets can be arranged to a natural order. Such orders are useful since they can provide new knowledge about the data and may ease further data exploration and visualization. Our goal in this paper is to define a statistically well-founded and an objective score measuring the quality of an order. Such a measure can be used for determining whether the current order has any valuable information or can it be discarded. Intuitively, we say that the order is good if dependent attributes are close to each other. To define the order score we fit an order-sensitive model to the dataset. Our model resembles a Markov chain model, that is, the attributes depend only on the immediate neighbors. The score of the order is the BIC score of the best model. For computing the measure we introduce a fast dynamic program. The score is then compared against random orders: if it is better than the scores of the random orders, we say that the order is good. We also show the asymptotic connection between the score function and the number of free parameters of the model. In addition, we introduce a simple greedy approach for finding an order with a good score. We evaluate the score for synthetic and real datasets using different spectral orders and the orders obtained with the greedy method.