README.md

---
title: matching_series
tags:
- evaluate
- metric
description: "Matching-based time-series generation metric"
sdk: gradio
sdk_version: 3.50
app_file: app.py
pinned: false
---

# Metric Card for matching_series

## Metric Description
Matching Series is a metric for evaluating time-series generation models. It is based on the idea of matching the generated time-series with the original time-series. The metric calculates the Mean Squared Error (distance) between the generated time-series and the original time-series between matched instances. The metric outputs a score greater or equal to 0, where 0 indicates a perfect generation.

## How to Use
At minium, the metric requires the original time-series and the generated time-series as input. The metric can be used to evaluate the performance of time-series generation models.

```python
>>> num_generation = 100
>>> num_reference = 10
>>> seq_len = 100
>>> num_features = 10
>>> references = np.random.rand(num_reference, seq_len, num_features)
>>> predictions = np.random.rand(num_generation, seq_len, num_features)
>>> metric = evaluate.load("bowdbeg/matching_series")
>>> results = metric.compute(references=references, predictions=predictions, batch_size=1000, return_all=True)
>>> print(results)
{'precision_distance': 0.1573285013437271, 'recall_distance': 0.15106813609600067, 'mean_distance': 0.1541983187198639, 'index_distance': 0.16858606040477753, 'matching_precision': 0.06, 'matching_recall': 1.0, 'matching_f1': 0.11320756503381972, 'cuc': 0.12428571428571429, 'macro_precision_distance': 0.13803552389144896, 'macro_recall_distance': 0.12179495096206665, 'macro_mean_distance': 0.1299152374267578, 'macro_index_distance': 0.16858604848384856, 'macro_matching_precision': 0.094, 'macro_matching_recall': 0.97, 'macro_matching_f1': 0.17132608782381706, 'macro_cuc': 0.11419285714285714, 'distance': array([[[0.20763363, 0.16514072, 0.18695284, ..., 0.15037987,
         0.19424284, 0.15943716],
        [0.17150438, 0.18020014, 0.17024504, ..., 0.18492931,
         0.18814348, 0.204207  ],
        [0.1769202 , 0.15609328, 0.17568389, ..., 0.17731658,
         0.2027854 , 0.13216409],
        ...,
        [0.1838122 , 0.19475608, 0.14176111, ..., 0.1635111 ,
         0.1652672 , 0.17145865],
        [0.16084194, 0.14208058, 0.17567575, ..., 0.15595785,
         0.16614595, 0.17834347],
        [0.16388315, 0.14126392, 0.18021484, ..., 0.16791071,
         0.18403953, 0.16666758]],

       [[0.16838932, 0.18878576, 0.17654441, ..., 0.1747057 ,
         0.16590554, 0.16901629],
        [0.16553226, 0.1882645 , 0.17863466, ..., 0.19269662,
         0.20451452, 0.19941731],
        [0.16502398, 0.16619626, 0.18069996, ..., 0.16124909,
         0.18933088, 0.1495165 ],
        ...,
        [0.15946846, 0.19988221, 0.17965002, ..., 0.12951666,
         0.2067793 , 0.13811146],
        [0.16227122, 0.17736743, 0.18641905, ..., 0.15038314,
         0.20186146, 0.17849396],
        [0.16410898, 0.18323919, 0.16945514, ..., 0.15783694,
         0.21556957, 0.17172968]],

       [[0.18094379, 0.1364854 , 0.18436092, ..., 0.187335  ,
         0.16240291, 0.13713893],
        [0.18005298, 0.15323727, 0.15788248, ..., 0.19451861,
         0.12822135, 0.14064161],
        [0.1564556 , 0.17312287, 0.1856657 , ..., 0.17237219,
         0.1596888 , 0.16547912],
        ...,
        [0.15611127, 0.16121496, 0.15533476, ..., 0.16520709,
         0.1427248 , 0.19455005],
        [0.17268528, 0.17360437, 0.15962966, ..., 0.18134868,
         0.15509704, 0.20222983],
        [0.18704675, 0.15934442, 0.14928888, ..., 0.18904984,
         0.16192877, 0.18576236]],

       ...,

       [[0.13717972, 0.15645625, 0.16123378, ..., 0.19453087,
         0.14441733, 0.1487963 ],
        [0.1454296 , 0.13368016, 0.18665504, ..., 0.16096605,
         0.15130125, 0.18332979],
        [0.14654924, 0.19097947, 0.19629759, ..., 0.15887487,
         0.19266474, 0.17430782],
        ...,
        [0.161704  , 0.16357127, 0.18512094, ..., 0.16441964,
         0.13961458, 0.17298506],
        [0.1366249 , 0.15852758, 0.1982772 , ..., 0.18822236,
         0.16153064, 0.19617072],
        [0.14570995, 0.15005183, 0.19667573, ..., 0.1856473 ,
         0.18603194, 0.19179863]],

       [[0.17813908, 0.176182  , 0.16847256, ..., 0.16903524,
         0.17150073, 0.15068175],
        [0.17632519, 0.1404587 , 0.16388708, ..., 0.16873878,
         0.15744762, 0.198475  ],
        [0.14986345, 0.1517829 , 0.17624639, ..., 0.18365957,
         0.17399347, 0.15581599],
        ...,
        [0.16128553, 0.1974935 , 0.13766351, ..., 0.14026196,
         0.15450196, 0.16110381],
        [0.16281141, 0.14699166, 0.16935429, ..., 0.1394466 ,
         0.1717883 , 0.16191883],
        [0.14886455, 0.1603608 , 0.15172943, ..., 0.12851712,
         0.19859877, 0.15576601]],

       [[0.20230632, 0.19680001, 0.17143433, ..., 0.18601838,
         0.15998998, 0.16043548],
        [0.19753966, 0.19073424, 0.15046756, ..., 0.18833323,
         0.16755773, 0.20127842],
        [0.16012056, 0.16638812, 0.16493171, ..., 0.15849902,
         0.20269662, 0.1857642 ],
        ...,
        [0.16341361, 0.19168772, 0.16597596, ..., 0.15715535,
         0.18122095, 0.17266828],
        [0.1570099 , 0.18294124, 0.16713732, ..., 0.17442709,
         0.17020254, 0.18804537],
        [0.16752282, 0.1295177 , 0.18792175, ..., 0.13976808,
         0.21054329, 0.18118018]]], dtype=float32), 'match': array([4, 7, 3, 9, 4, 0, 7, 5, 4, 7, 9, 7, 7, 5, 7, 0, 0, 7, 4, 3, 3, 2,
       8, 9, 4, 4, 5, 1, 4, 9, 0, 2, 7, 3, 6, 5, 6, 3, 2, 2, 2, 6, 9, 4,
       4, 9, 1, 6, 0, 6, 9, 2, 0, 6, 7, 2, 0, 4, 5, 2, 3, 9, 2, 3, 9, 1,
       6, 4, 8, 9, 7, 4, 6, 5, 5, 6, 9, 5, 6, 2, 9, 4, 9, 3, 2, 9, 9, 7,
       9, 5, 9, 1, 7, 6, 4, 4, 5, 4, 7, 5]), 'match_inv': array([15, 91, 79,  4,  4,  4, 49,  4, 49, 45]), 'coverages': [0.10000000000000002, 0.16666666666666666, 0.3666666666666667, 0.6333333333333333, 0.8333333333333334, 0.9, 1.0], 'precision_distance_features': [0.1383965164422989, 0.13804036378860474, 0.1388234943151474, 0.1392393559217453, 0.1357768476009369, 0.1364508718252182, 0.14039862155914307, 0.13417008519172668, 0.1368638128042221, 0.14219526946544647], 'recall_distance_features': [0.11730053275823593, 0.12232911586761475, 0.12200610339641571, 0.12571024894714355, 0.12081331014633179, 0.11693283170461655, 0.12660981714725494, 0.12248671054840088, 0.11726576089859009, 0.12649507820606232], 'mean_distance_features': [0.1278485246002674, 0.13018473982810974, 0.13041479885578156, 0.13247480243444443, 0.12829507887363434, 0.12669185176491737, 0.133504219353199, 0.12832839787006378, 0.1270647868514061, 0.1343451738357544], 'index_distance_features': [0.17064405977725983, 0.17019756138324738, 0.17373089492321014, 0.17575454711914062, 0.15942324697971344, 0.1615942418575287, 0.16519878804683685, 0.1714271903038025, 0.17072594165802002, 0.16716401278972626], 'matching_precision_features': [0.1, 0.09, 0.1, 0.1, 0.09, 0.09, 0.1, 0.08, 0.09, 0.1], 'matching_recall_features': [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 0.9, 0.9, 0.9, 1.0], 'matching_f1_features': [0.18181819851239656, 0.16513763164885095, 0.18181819851239656, 0.18181819851239656, 0.16513763164885095, 0.16513763164885095, 0.18000001639999985, 0.14693879251145342, 0.16363638033057834, 0.18181819851239656], 'cuc_features': [0.11935714285714286, 0.11578571428571431, 0.11814285714285715, 0.12407142857142857, 0.11207142857142856, 0.11821428571428572, 0.10807142857142855, 0.09635714285714285, 0.10700000000000001, 0.12285714285714286], 'coverages_features': [[0.10000000000000002, 0.20000000000000004, 0.26666666666666666, 0.4666666666666666, 0.7666666666666666, 0.8666666666666667, 1.0], [0.10000000000000002, 0.20000000000000004, 0.3666666666666667, 0.5666666666666668, 0.6, 0.8333333333333334, 1.0], [0.10000000000000002, 0.16666666666666666, 0.26666666666666666, 0.4666666666666666, 0.6999999999999998, 0.8666666666666667, 1.0], [0.10000000000000002, 0.20000000000000004, 0.3, 0.6, 0.7333333333333333, 0.9333333333333332, 1.0], [0.10000000000000002, 0.20000000000000004, 0.3, 0.5, 0.6666666666666666, 0.7666666666666666, 1.0], [0.10000000000000002, 0.20000000000000004, 0.3333333333333333, 0.5333333333333333, 0.7666666666666666, 0.8333333333333334, 1.0], [0.10000000000000002, 0.20000000000000004, 0.3, 0.5333333333333333, 0.6999999999999998, 0.7666666666666666, 0.9], [0.10000000000000002, 0.20000000000000004, 0.2333333333333333, 0.4666666666666666, 0.5333333333333333, 0.6333333333333333, 0.9], [0.10000000000000002, 0.16666666666666666, 0.26666666666666666, 0.4666666666666666, 0.5666666666666667, 0.8000000000000002, 0.9], [0.10000000000000002, 0.16666666666666666, 0.30000000000000004, 0.5666666666666667, 0.7999999999999999, 0.9, 1.0]]}
```

### Inputs
- **predictions**: (list of list of list of float or numpy.ndarray): The generated time-series. The shape of the array should be `(num_generation, seq_len, num_features)`.
- **references**: (list of list of list of float or numpy.ndarray): The original time-series. The shape of the array should be `(num_reference, seq_len, num_features)`.
- **batch_size**: (int, optional): The batch size for computing the metric. This affects quadratically. Default is None.
- **cuc_n_calculation**: (int, optional): The number of samples to compute the coverage because sampling exists. Default is 3.
- **cuc_n_samples**: (list of int, optional): The number of samples to compute the coverage. Default is $[2^i \text{for} i \leq \log_2 n] + [n]$.
- **metric**: (str, optional): The metric to measure distance between examples. Default is "mse". Available options are "mse", "mae", "rmse".
- **num_processes**: (int, optional): The number of processes to use for computing the distance. Default is 1.
- **instance_normalization**: (bool, optional): Whether to normalize the instances along the time axis. Default is False.
- **return_distance**: (bool, optional): Whether to return the distance matrix. Default is False.
- **return_matching**: (bool, optional): Whether to return the matching matrix. Default is False.
- **return_each_features**: (bool, optional): Whether to return the results for each feature. Default is False.
- **return_coverages**: (bool, optional): Whether to return the coverages. Default is False.
- **return_all**: (bool, optional): Whether to return all the results. Default is False.
- **dtype**: (str, optional): The data type used for computation. Default is "float32".
- **eps**: (float, optional): The epsilon value to avoid division by zero. Default is 1e-8.

### Output Values

Let prediction instances be $P = \{p_1, p_2, \ldots, p_n\}$ and reference instances be $R = \{r_1, r_2, \ldots, r_m\}$.

- **precision_distance**: (float): Average of the distance between the generated instance and the reference instance with the lowest distance. Intuitively, this is similar to precision in classification. In the equation, $\frac{1}{n} \sum_{i=1}^{n} \min_{j} \mathrm{distance}(p_i, r_j)$.
- **recall_distance**: (float): Average of the distance between the reference instance and the  with the lowest distance. Intuitively, this is similar to recall in classification. In the equation, $\frac{1}{m} \sum_{j=1}^{m} \min_{i} \mathrm{distance}(p_i, r_j)$.
- **mean_disntance**: (float): Average of the precision_distance and recall_distance.
- **index_distance**: (float): Average of the distance between the generated instance and the reference instance with the same index. In the equation, $\frac{1}{n} \sum_{i=1}^{n} \mathrm{distance}(p_i, r_i)$.
- **matching_precision**: (float): Precision of the matching instances, which means how predictions are covered by references, i.e., how accurate the predictions are. In the equation, $\frac{ | \{i | \argmin_{i} \mathrm{distance}(p_i, r_j)\} | }{n}$.
- **matching_recall**: (float): Recall of the matching instances, which means how predictions cover references. In the equation, $\frac{ | \{j | \argmin_{j} \mathrm{distance}(p_i, r_j)\} | }{m}$.
- **matching_f1**: (float): F1-score of the matching instances, harmonic mean of the matching_precision and matching_recall.
- **coverages**: (list of float): Coverage of the matching instances computed on the sampled generated data in cuc_n_samples. In the equation, $[\frac{1}{m} | \{ j \mid \argmin_{j} \mathrm{distance}(p_i, r_j)~\text{where $p_i \in \mathrm{sample}(P, \mathrm{n\_sample})$} \}  | ~\text{for}~\mathrm{n\_sample} \in \mathrm{cuc\_n\_samples} ]$.
- **cuc**: (float): Under the curve of the coverage. In the equation, $\int_{0}^{n} \mathrm{coverage}(x) dx$. As an approximation, the trapezoidal rule is used.
- **.\*_features**: (list of float): The values computed individually for each feature.
- **macro_.\***: (float): Averaged values computed for each feature, average of the \*\_features.
- **distance**: (numpy.ndarray): The distance matrix between the generated instances and the reference instances.
- **match**: (numpy.ndarray): The matching matrix between the generated instances and the reference instances.
- **match_inv**: (numpy.ndarray): The matching matrix between the reference instances and the generated instances.

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## Limitations and Bias
This metric is based on the assumption that the generated time-series should match the original time-series. This may not be the case in some scenarios. The metric may not be suitable for evaluating time-series generation models that are not required to match the original time-series.

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