The Temporal Markov Transition Field

TL;DR

The formal definition, key structural properties of the representation, and the geometric interpretation of the local transition matrices in terms of process properties such as persistence, mean reversion, and trending behaviour are developed.

Abstract

The Markov Transition Field (MTF), introduced by Wang and Oates (2015), encodes a time series as a two-dimensional image by mapping each pair of time steps to the transition probability between their quantile states, estimated from a single global transition matrix. This construction is efficient when the transition dynamics are stationary, but produces a misleading representation when the process changes regime over time: the global matrix averages across regimes and the resulting image loses all information about \emph{when} each dynamical regime was active. In this paper we introduce the \emph{Temporal Markov Transition Field} (TMTF), an extension that partitions the series into $K$ contiguous temporal chunks, estimates a separate local transition matrix for each chunk, and assembles the image so that each row reflects the dynamics local to its chunk rather than the global average. The resulting $T \times T$ image has $K$ horizontal bands of distinct texture, each encoding the transition dynamics of one temporal segment. We develop the formal definition, establish the key structural properties of the representation, work through a complete numerical example that makes the distinction from the global MTF concrete, analyse the bias--variance trade-off introduced by temporal chunking, and discuss the geometric interpretation of the local transition matrices in terms of process properties such as persistence, mean reversion, and trending behaviour. The TMTF is amplitude-agnostic and order-preserving, making it suitable as an input channel for convolutional neural networks applied to time series characterisation tasks.

Figures (3)

• Figure 1: The 12-point example series. Quantile bands are shaded. The dashed vertical line marks the chunk boundary for $K=2$.
• Figure 2: The assembled $12\times12$ global MTF image for the worked example ($Q=3$). Row-index colours and left-side brackets group the three sets of identical rows: red for state 1 ($i \in \{1,4,7,8\}$), green for state 2 ($i \in \{3,6,9,10\}$), and blue for state 3 ($i \in \{2,5,11,12\}$). Within each group the rows are pixel-for-pixel identical regardless of whether they fall in the mean-reverting first half or the persistent second half of the series. No band structure is visible; the regime transition at $t=6$ leaves no trace.
• Figure 3: The assembled $12\times12$ TMTF image for the worked example ($K=2$, $Q=3$). Rows $i=1$--$6$ (top band, above the dashed line) are governed by $W^{(1)}$ and contain only binary values $\{0,1\}$, producing a stark alternating texture reflecting the deterministic mean-reverting cycle of chunk 1. Rows $i=7$--$12$ (bottom band) are governed by $W^{(2)}$ and contain values in $\{0, 0.5, 1\}$, producing a softer texture reflecting the persistent upward dynamics of chunk 2. A CNN filter traversing the image detects the texture change at the band boundary and associates it with the regime transition at $t=6$.

Theorems & Definitions (17)

• Definition 2.1: Quantile Bins and State Sequence
• Definition 2.2: Empirical Transition Probability Matrix
• Definition 3.1: Global Markov Transition Field
• Proposition 3.1: Row Degeneracy of the Global MTF
• proof
• Example 3.1: Global MTF
• Definition 4.1: Temporal Segmentation
• Definition 4.2: Local Transition Matrix
• Definition 4.3: Temporal Markov Transition Field
• Proposition 4.1: Band Structure