## compare two time series (simulation results)

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I want to do unit testing of simulation models and for that, I run a simulation once and store the results (a time series) as reference in a csv file (see an example here). Now when I change my model, I run the simulation again, store the new reults as a csv file as well and then I compare the results.

The results are usually not 100% identical, an example plot is shown below:
The reference results are plotted in **black** and the new results are plotted in **green**.
The difference of the two is plotted in the second plot, in **blue**.
As can be seen, at a step the difference can become arbitrarily high, while everywhere else the difference is almost zero.

Therefore, I would prefer to use a different algorithms for comparison than just subtracting the two, but I can only describe my idea graphically:
When plotting the reference line twice, first in a light color with a high line width and then again in a dark color and a small line width, then it will look like it has a **pink tube** around the centerline.

Note that during a step that tube will not only be in the direction of the ordinate axis, but also in the direction of the abscissa. When doing my comparison, I want to know whether the green line stays within the pink tube.

Now comes my question: I do not want to compare the two time series using a graph, but using a python script. There must be something like this already, but I cannot find it because I am missing the right vocabulary, I believe. Any ideas? Is something like that in numpy, scipy, or similar? Or would I have to write the comparison myself?

Additional question: When the script says the two series are not sufficiently similar, I would like to plot it as described above (using matplotlib), but the line width has to be defined somehow in other units than what I usually use to define line width.

I would assume here that your problem can be simplified by assuming that your function has to be close to another function (e.g. the center of the tube) with the very same support points and then a certain number of discontinuities are allowed.
Then, I would implement a different discretization of function compared to the typical one that is used for `L^2`

norm (See for example some reference here).

Basically, in the continuous case, the `L^2`

norm relaxes the constrain of the two function being close everywhere, and allow it to be different on a finite number of points, called singularities
This works because there are an infinite number of points where to calculate the integral, and a finite number of points will not make a difference there.

However, since there are no continuous functions here, but only their discretization, the naive approach will not work, because any singularity will contribute potentially significantly to the final integral value.

Therefore, what you could do is to perform a point by point check whether the two functions are close (within some tolerance) and allow at most `num_exceptions`

points to be off.

import numpy as np def is_close_except(arr1, arr2, num_exceptions=0.01, **kwargs): # if float, calculate as percentage of number of points if isinstance(num_exceptions, float): num_exceptions = int(len(arr1) * num_exceptions) num = len(arr1) - np.sum(np.isclose(arr1, arr2, **kwargs)) return num <= num_exceptions

By contrast the standard `L^2`

norm discretization would lead to something like this integrated (and normalized) metric:

import numpy as np def is_close_l2(arr1, arr2, **kwargs): norm1 = np.sum(arr1 ** 2) norm2 = np.sum(arr2 ** 2) norm = np.sum((arr1 - arr2) ** 2) return np.isclose(2 * norm / (norm1 + norm2), 0.0, **kwargs)

This however will fail for arbitrarily large peaks, unless you set such a large tolerance than basically anything results as "being close".

Note that the `kwargs`

is used if you want to specify a additional tolerance constraints to `np.isclose()`

or other of its options.

As a test, you could run:

import numpy as np import numpy.random np.random.seed(0) num = 1000 snr = 100 n_peaks = 5 x = np.linspace(-10, 10, num) # generate ground truth y = np.sin(x) # distributed noise y2 = y + np.random.random(num) / snr # distributed noise + peaks y3 = y + np.random.random(num) / snr peak_positions = [np.random.randint(num) for _ in range(n_peaks)] for i in peak_positions: y3[i] += np.random.random() * snr # for distributed noise, both work with a 1/snr tolerance is_close_l2(y, y2, atol=1/snr) # output: True is_close_except(y, y2, atol=1/snr) # output: True # for peak noise, since n_peaks < num_exceptions, this works is_close_except(y, y3, atol=1/snr) # output: True # and if you allow 0 exceptions, than it fails, as expected is_close_except(y, y3, num_exceptions=0, atol=1/snr) # output: False # for peak noise, this fails because the contribution from the peaks # in the integral is much larger than the contribution from the rest is_close_l2(y, y3, atol=1/snr) # output: False

There are other approaches to this problem involving higher mathematics (e.g. Fourier or Wavelet transforms), but I would stick to the simplest.

**EDIT** (updated):

However, if the working assumption does not hold or you do not like, for example because the two functions have different sampling or they are described by non-injective relations. In that case, you can follow the center of the tube using (x, y) data and the calculate the Euclidean distance from the target (the tube center), and check that this distance is point-wise smaller than the maximum allowed (the tube size):

import numpy as np # assume it is something with shape (N, 2) meaning (x, y) target = ... # assume it is something with shape (M, 2) meaning again (x, y) trajectory = ... # calculate the distance minimum distance between each point # of the trajectory and the target def is_close_trajectory(trajectory, target, max_dist): dist = np.zeros(trajectory.shape[0]) for i in range(len(dist)): dist[i] = np.min(np.sqrt( (target[:, 0] - trajectory[i, 0]) ** 2 + (target[:, 1] - trajectory[i, 1]) ** 2)) return np.all(dist < max_dist) # same as above but faster and more memory-hungry def is_close_trajectory2(trajectory, target, max_dist): dist = np.min(np.sqrt( (target[:, np.newaxis, 0] - trajectory[np.newaxis, :, 0]) ** 2 + (target[:, np.newaxis, 1] - trajectory[np.newaxis, :, 1]) ** 2), axis=1) return np.all(dist < max_dist)

The price of this flexibility is that this will be a significantly slower or memory-hungry function.

**How to compare two time series?,** Advanced Statistical Modeling · Time Series Modeling You have two time-series but the first question is if these series are for the activity of one object OR they are different How do I report the results of a linear mixed models analysis? Compare Simulation Data. Compare individual signals or simulation runs and analyze your comparison results with relative, absolute, and time tolerances. The compare tools in the Simulation Data Inspector facilitate iterative design and allow you to highlight signals that do not meet your tolerance requirements.

Assuming you have your list of results in the form we discussed in the comments already loaded:

from random import randint import numpy l1 = [(i,randint(0,99)) for i in range(10)] l2 = [(i,randint(0,99)) for i in range(10)] # I generate some random lists e.g: # [(0, 46), (1, 33), (2, 85), (3, 63), (4, 63), (5, 76), (6, 85), (7, 83), (8, 25), (9, 72)] # where the first element is the time and the second a value print(l1) # Then I just evaluate for each time step the difference between the values differences = [abs(x[0][1]-x[1][1]) for x in zip(l1,l2)] print(differences) # And I can just print hte maximum difference and its index: print(max(differences)) print(differences.index(max(differences)))

And with this data if you define that your "tube" is for example `10`

large you can just check if the maxximum value that you find is greater than your thrashold in order to decide if those functions are similar enough or not

**How can I compare two time series forecasting models considering ,** When comparing two time series forecasting models, there are many statistics, e.g. For empirical comparison between models, for actual or artificial (simulated The proposed test was applied to compare two or multiple stationary time series in different settings, including time series driven by t innovations with d.f = 2, 5, 10, a pair of dependent time series, two time series of different length, slightly non-linear time series and so on.

you will have to remove outliers from your dataset first if you need to skip a random spike.

you could also try the following?

from tslearn.metrics import dtw print(dtw(arr1,arr2)*100/<lengthOfArray>)

**Proving similarities of two time series,** The goal is to prove the simulation results and predicted values of the analytical model (which are both a time series) are statistically close or Access and Compare Simulation Results. Access the simulation results using the Simulation Data Inspector programmatic interface. Each simulation creates a run in the Simulation Data Inspector with a unique run ID. You use the run IDs to compare the simulation results.

**[PDF] Comparing Two Sets of Time-Series: The State Similarity Measure1,** Key Words: time-series sets, comparison, State Similarity Measure, dynamic historical data (when we have enough) and the simulations output? With a window of four weeks, when each week's state can result in any one. For most of these activities it is necessary to compare time series using an appropriate similarity measure. By similarity measure we mean any method, metric or non-metric, which compares two time series objects and returns a value that encodes how similar the two objects are.

**Measurement error in time-series analysis: a simulation study ,** Results. Comparing “true” values of the regression coefficient, β, with those based on simulated data, MathML, Tables 2 and 3 suggest sys is a continuous-time identified state-space (idss) model.. Now use compare to plot the predicted response. Prediction differs from simulation in that it uses both measured input and measured output when computing the system response.

**A Method for Comparing Multivariate Time Series with ,** This normalization results in that time series are only different in their shape Hierarchical clustering was applied to the two distance matrices. be easily simulated given a certain initial condition producing time series How to compare two time series? I have two time series data. I would like to formally investigate whether the one of the series is persistently higher in value than the other.

##### Comments

- When you say time series you mean something like a list of [x, y] that can be plot in a 2-D graph?
- Exactly. The reference results and the "new" results are stored as csv files that are read by python, with time being the first column.
- you could start by looking at their cross-correlation
- Adding a plot to the question will certainly make it much easier to understand what you are referring to, but it looks like you want some kind of
`L^2`

(or`l^2`

) distance between the two distribution, i.e. something that will be close to zero if the two distribution differ only in at most N number of points (with N finite in the case of continuous distributions). Am I correct? - @norok2 Now I have added a plot.
- Not sure if I understand it, but aren't you just subtracting the values? That is how I am doing it too, but that will give very large differences at steps.
- Yep I am just substratting the y values in the (x, y) couples. I dont' understand your concerns, you think that there are too many differences because the lists are really big?
- Sorry for being unclear. The two lines are usually very close, so the difference is e.g. 1e-3 or similar. But at a step, the one line will change abruptly, and the second line might do exactly the same step, just a second later. Then during this one second, the difference can be arbitrarily high. So, I do not want to calculate the difference in y-direction only, but I need a tube that also covers the x-direction.