Markov Transition Fields & TMTF Insights

• Markov Transition Field is a method that encodes a univariate time series as a 2D image by mapping transitions between discretized quantile states.
• The Temporal Markov Transition Field segments the series to fit local transition matrices, enabling precise detection of regime change dynamics.
• TMTF maintains amplitude-invariance and order, offering robust features for deep learning models through clear texture patterns and structural insights.

A Markov Transition Field (MTF) encodes the sequential dynamics of a univariate time series as a two-dimensional image, making temporal dependencies amenable to convolutional neural network (CNN) analysis. The MTF, as introduced by Wang and Oates (2015), represents each pair of time points by the transition probability between their discretized quantile states, estimated from a single global transition matrix. While efficient for stationary series, the global MTF construction averages over temporal regimes, thereby obscuring information about when distinct dynamical regimes occur. The Temporal Markov Transition Field (TMTF) addresses this shortcoming by partitioning the time series into $K$ contiguous temporal segments, fitting a local $Q \times Q$ Markov transition matrix in each, and assembling the image such that each row encodes the regime-local dynamics corresponding to its temporal segment. This methodology recovers regime timing and enhances interpretability without sacrificing the amplitude-agnostic, order-preserving properties that make MTF suitable for deep learning. The TMTF enables up to $KQ$ distinct row patterns and admits analysis of geometric and statistical properties central to time-series characterization (Leznik, 9 Mar 2026).

1. Markov Transition Field (MTF): Foundations and Global Construction

Let $\mathbf{x} = (x_1, \dots, x_T)$ be a univariate time series of length $T$. Choose a quantile bin count $Q$. Empirical quantile boundaries $q_0 < q_1 < \cdots < q_Q$ divide the range so that roughly $\lfloor T/Q \rfloor$ points fall into each bin. Each observation $x_t$ is assigned a quantile-state

$$b_t = k \quad \Longleftrightarrow \quad x_t \in [q_{k-1}, q_k)$$

yielding the state sequence $Q \times Q$0. This procedure is amplitude-invariant and preserves order: strictly increasing transformations of $Q \times Q$1 leave $Q \times Q$2 unchanged.

The global empirical transition matrix $Q \times Q$3 has entries

$Q \times Q$4

The global MTF image $Q \times Q$5 is defined by

$Q \times Q$6

Rows with identical $Q \times Q$7 are identical, yielding at most $Q \times Q$8 distinct row patterns; the image is incapable of localizing when regime changes occur, as the matrix is constructed from statistics averaged over the full series.

2. Temporal Markov Transition Field (TMTF): Segmentation and Construction

To recover temporal localization of regimes, TMTF partitions time indices into $Q \times Q$9 contiguous chunks: $KQ$0 Let $KQ$1 iff $KQ$2. For each chunk, compute the local $KQ$3 empirical transition matrix: $KQ$4 Transitions crossing chunk boundaries are omitted.

The TMTF image $KQ$5 is defined as: $KQ$6 Viewed as a block-row matrix, $KQ$7 contains $KQ$8 horizontal bands of height $KQ$9, each encoding the local regime dynamics for its corresponding temporal segment.

3. Structural and Geometric Properties

The TMTF displays several key structural distinctions relative to the global MTF:

• Band Structure: Within chunk $\mathbf{x} = (x_1, \dots, x_T)$0, rows $\mathbf{x} = (x_1, \dots, x_T)$1 are identical if and only if $\mathbf{x} = (x_1, \dots, x_T)$2. Across chunks, this is not guaranteed: even for $\mathbf{x} = (x_1, \dots, x_T)$3, the rows may differ if $\mathbf{x} = (x_1, \dots, x_T)$4.
• Row Diversity: The construction enables up to $\mathbf{x} = (x_1, \dots, x_T)$5 distinct row patterns, compared to the global MTF’s maximum of $\mathbf{x} = (x_1, \dots, x_T)$6.
• Column Asymmetry: The local index applies to rows; columns reference global state labels $\mathbf{x} = (x_1, \dots, x_T)$7. This asymmetry permits consistent comparison across destination states, regardless of originating chunk.
• Graceful Degradation: If $\mathbf{x} = (x_1, \dots, x_T)$8, the TMTF reduces to the global MTF, indicating robustness under stationary dynamics.

Geometrically, the local transition matrices $\mathbf{x} = (x_1, \dots, x_T)$9 encode canonical temporal signatures:

• Persistence: Large diagonal entries ($T$0) generate dark main diagonal texture.
• Mean-Reversion: Small diagonals, with off-diagonal concentration, produce a diffuse texture.
• Trending: Upper-triangular structure, often with $T$1, manifests as brightness above the diagonal.
• Random Walk: Uniform entries ($T$2) yield a homogeneously gray band.

The spectral gap $T$3 of $T$4 signals mixing speed, with a small gap (second eigenvalue near 1) indicating strong persistence.

4. Statistical Trade-Offs in Temporal Chunking

Increasing $T$5 raises the temporal resolution of regime localization but introduces a bias–variance trade-off:

• Variance Inflation: Each $T$6 estimates transitions from approximately $T$7 samples, increasing standard error as $T$8 grows: $T$9 where $Q$0 is the empirical occupancy of state $Q$1 in chunk $Q$2.
• Bias Reduction: When regime dynamics vary between chunks ($Q$3), the global MTF estimator is biased on each chunk; TMTF’s local estimators are unbiased within their segments.
• Practical Guideline: To ensure statistical reliability, at least $Q$4 transitions per state per chunk are recommended, requiring

$Q$5

For $Q$6 and $Q$7, $Q$8 offers robust performance.

5. Worked Example: Regime Change Detection

Given $Q$9, with $q_0 < q_1 < \cdots < q_Q$0 and $q_0 < q_1 < \cdots < q_Q$1, quantile-states partition indices such that $q_0 < q_1 < \cdots < q_Q$2. The global transition matrix is: $q_0 < q_1 < \cdots < q_Q$3 The global MTF lacks any visible mid-series regime shift. In contrast, TMTF segmentation ($q_0 < q_1 < \cdots < q_Q$4, $q_0 < q_1 < \cdots < q_Q$5) yields: $q_0 < q_1 < \cdots < q_Q$6 The assembled $q_0 < q_1 < \cdots < q_Q$7 TMTF presents a stark contrast between the two bands: binary off-diagonal texture in the first band and triangular/diagonal texture in the second, making the regime shift immediately visible to CNN filters.

6. Amplitude-Invariance and Suitability for Deep Learning

TMTF construction uses only the rank-order of series, ensuring invariance under any strictly increasing transformation (amplitude-agnostic and order-preserving). This property obviates normalization for across-series comparability. The horizontal banding of texture localizes regime timing and supports interpretable CNN features; canonical transition signatures (persistence, mean-reversion, trending, random) further enhance interpretability. CNNs can detect band boundaries, making TMTF a practical and effective input channel for automated time series characterization.

7. Algorithmic Implementation

The TMTF construction process proceeds as follows:

1. Compute empirical quantile boundaries for $q_0 < q_1 < \cdots < q_Q$8 bins.
2. Assign quantile-state $q_0 < q_1 < \cdots < q_Q$9 to each $\lfloor T/Q \rfloor$0 via these boundaries.
3. Partition time into $\lfloor T/Q \rfloor$1 contiguous chunks of size $\lfloor T/Q \rfloor$2.
4. For each chunk, count within-chunk transitions to form the local $\lfloor T/Q \rfloor$3, ensuring transitions are not counted across chunk boundaries.
5. Assemble $\lfloor T/Q \rfloor$4.
6. The output is the $\lfloor T/Q \rfloor$5 TMTF image.

The TMTF thus generalizes MTF for nonstationary dynamics, increases the diversity of encoded image patterns, enables precise regime change localization, and preserves all key invariances necessary for feature extraction by deep learning models (Leznik, 9 Mar 2026).