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Open toolDynamic Time Warping (DTW) Calculator
Compute the exact **Dynamic Time Warping (DTW)** distance, normalized per-step cost, optimal alignment path `(i, j)`, and accumulated dynamic-programming cost matrix `D(i, j)` between two numeric sequences of equal or unequal length.
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Enter the inputs, then select Calculate DTW Distance & Path.
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How the Dynamic Time Warping Calculator Works
Paste two numeric series of any length (up to 150 observations each) to run the exact O(n×m) dynamic programming solver.
D(i, j) = c(x_i, y_j) + min( D(i−1, j), D(i, j−1), D(i−1, j−1) ) subject to |i − j| ≤ w (when w > 0); Normalized Cost = D(n, m) / K, where K is the number of aligned steps in the optimal warping path.
Computes exact O(n×m) dynamic programming alignment satisfying boundary ((1,1) to (n,m)), monotonicity, and step-continuity constraints, and compares DTW against lock-step Euclidean/Manhattan distance.
Dynamic Time Warping (DTW) Metrics & Recurrence Reference Table
Summary of local cost options, dynamic programming recurrence rules, and output metrics computed on this page.
| Metric / Parameter | Mathematical Definition | Default Sample Value (`X` vs `Y`) | Practical Interpretation |
|---|---|---|---|
| Local Cost `c(x_i, y_j)` | `|x_i − y_j|` or `(x_i − y_j)²` | `|x_i − y_j|` (Absolute) | Point-to-point discrepancy before cumulative DP warping |
| DP Recurrence `D(i, j)` | `c(x_i,y_j) + min(D(i−1,j−1), D(i−1,j), D(i,j−1))` | **`DTW = 1.00`** | Minimum cumulative cost to align prefixes `X[1..i]` and `Y[1..j]` |
| Normalized Path Cost | `DTW / K` (`K` = warping path length) | **`0.1250 / step`** (`K = 8`) | Average alignment error per step along the optimal path |
| Lock-Step Baseline | `∑ |x_i − y_i|` (when `n = m`) | `16.00` (vs. `1.00` DTW) | Shows how much phase-shift distortion DTW removes (`93.75%` lower) |
How to use Dynamic Time Warping (DTW) Calculator step by step
Enter the values for your scenario, verify the units, and run the calculation.
Enter the inputs
Start by entering the required fields in the Dynamic Time Warping (DTW) Calculator form above.
Verify units and assumptions
Match each value to the unit or format shown next to the input box before running the calculation.
Run and export your result
Select Calculate DTW Distance & Path, inspect the primary metric and breakdown cards, then use Copy result or Download result.txt.
Worked examples for Dynamic Time Warping (DTW) Calculator
Compare baseline and alternative runs for Dynamic Time Warping (DTW) Calculator before exporting the result.
Baseline example
Calculate Dynamic Time Warping (DTW) distance and optimal alignment path between two time series in your browser. Supports absolute, squared, and root-squared costs plus Sakoe-Chiba window band.
Comparison scenario
D(i, j) = c(x_i, y_j) + min( D(i−1, j), D(i, j−1), D(i−1, j−1) ) subject to |i − j| ≤ w (when w > 0); Normalized Cost = D(n, m) / K, where K is the number of aligned steps in the optimal warping path.
Boundary check before exporting
Computes exact O(n×m) dynamic programming alignment satisfying boundary ((1,1) to (n,m)), monotonicity, and step-continuity constraints, and compares DTW against lock-step Euclidean/Manhattan distance.
How to interpret the Dynamic Time Warping (DTW) Calculator output
Review the calculated output alongside your entered parameters and unit conventions.
Computes exact O(n×m) dynamic programming alignment satisfying boundary ((1,1) to (n,m)), monotonicity, and step-continuity constraints, and compares DTW against lock-step Euclidean/Manhattan distance.
DTW Analysis Verification Checklist
Check these three items when interpreting DTW distances.
D(i, j) = c(x_i, y_j) + min( D(i−1, j), D(i, j−1), D(i−1, j−1) ) subject to |i − j| ≤ w (when w > 0); Normalized Cost = D(n, m) / K, where K is the number of aligned steps in the optimal warping path. Computes exact O(n×m) dynamic programming alignment satisfying boundary ((1,1) to (n,m)), monotonicity, and step-continuity constraints, and compares DTW against lock-step Euclidean/Manhattan distance.
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Open toolDynamic Time Warping (DTW) Calculator: common questions
Answers about using Dynamic Time Warping (DTW) Calculator and understanding its results.
Why is Dynamic Time Warping (DTW) better than Euclidean distance for shifted time series?
Lock-step Euclidean or Manhattan distance compares index i of series X strictly against the same index i of series Y (and cannot compare series of unequal length). If two identical waveforms are shifted by even a single time step—such as X = [1, 3, 4, 9, 8, 2, 1] and Y = [1, 1, 3, 4, 9, 8, 2]—lock-step distance reports a large error (16.0), whereas DTW nonlinearly stretches the time axis and finds a near-zero alignment cost (1.0).
What are the three mathematical constraints on a valid DTW warping path?
A standard DTW warping path W = [w_1, ..., w_K] must satisfy: (1) Boundary condition: start at w_1 = (1, 1) and end at w_K = (n, m); (2) Continuity: each step moves by at most 1 index in i and j, i.e., (i_k − i_{k−1}, j_k − j_{k−1}) ∈ {(1,0), (0,1), (1,1)}; and (3) Monotonicity: indices never move backward in time (i_k ≥ i_{k−1} and j_k ≥ j_{k−1}).
What does the Sakoe-Chiba window bandwidth w do?
The Sakoe-Chiba band constrains the warping path so that index i in series X can only align with index j in series Y if |i − j| ≤ max(w, |n − m|). Setting a locality window prevents pathological warping (where a single point in X stretches across dozens of points in Y) and speeds up computation.
What is the difference between raw DTW distance and Normalized DTW Cost per step?
Raw DTW distance sums the local costs across all K steps of the warping path, so longer sequences naturally accumulate larger raw sums. Dividing the total DTW cost by the warping path length K yields the Normalized Path Cost per aligned step, allowing fair similarity comparisons across pairs of different lengths.
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