I was selected and invited to present this family of univariate/multivariate time series models at R/Finance 2020 (in Chicago, IL). However, the COVID-19 pandemics decided differently. It’s still a work in progress; comments, remarks, pull requests are welcome as usual (in nnetsauce). But the general philosophy of model construction in this framework is already set and will be explained in this post. An R example, with less details about the implementation (so, R users read on) can also be found in this post: Bayesian forecasting for uni/multivariate time series.

How does this family of time series models work?

By doing what’s shown on the graph before, I make some relatively strong hypotheses:

- I assume that there’s a bidirectional relationship between the series, as in Vector Autoregressive (VAR) models, i.e each series is influencing – and is influenced by – the others.
- I assume that all the successive time series sub-models share the same architecture, the same ML model, hence the same hyperparameters, for learning from the lags.

At testing time, for ML prediction, recursive forecasts are implemented so far. One improvement could be to obtain direct forecasts too, in the future.

In the example presented here, the **shared ML model** is a Bayesian Ridge regression. It **could be any other regression model**, such as Random Forests, Gradient Boosting Machines, or else, as long as the method’s implementation contains methods `fit`

and `predict`

. Hence, “infinity” in the title. No hyperparameter tuning is used here, so, some better results could certainly be obtained.

# Installing packages

```
pip install git+https://github.com/Techtonique/nnetsauce.git
```

```
pip install matplotlib==3.1.3
```

# Obtaining predictions from nnetsauce’s MTS

```
import nnetsauce as ns
import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets, metrics
from sklearn import linear_model
# a simple example with 3 made-up multivariate time series
np.random.seed(123)
M = np.random.rand(10, 3)
M[:,0] = 10*M[:,0]
M[:,2] = 25*M[:,2]
M[:,1] = M[:,2]/23 + 0.5
print("Initial series")
print(M)
print("\n")
# using a Bayesian model along with nnetsauce's MTS
regr = linear_model.BayesianRidge()
obj_MTS = ns.MTS(regr, lags = 2, n_hidden_features=5)
obj_MTS.fit(M)
print("mean forecast -----")
preds = obj_MTS.predict(h=10)
print(preds)
print("\n")
# confidence level = 80%
# makes the assumption of Gaussian uncertainty and works
# only if the shared supervised learning model has an argument `return_std`
# in method `predict`
print("predict with credible intervals and confidence level = 80% -----")
preds_80 = obj_MTS.predict(h=10, return_std=True, level=80)
print(preds_80)
print("\n")
# confidence level = 95%
# makes the assumption of Gaussian uncertainty and works
# only if the shared supervised learning model has an argument `return_std`
# in method `predict`
print("predict with credible intervals and confidence level = 95% -----")
preds_95 = obj_MTS.predict(h=10, return_std=True, level=95)
print(preds_95)
print("\n")
```

```
Initial series
[[ 6.96469186 0.74657767 5.67128634]
[ 5.51314769 0.95989833 10.5776615 ]
[ 9.80764198 1.02275207 12.02329754]
[ 3.92117518 1.29244533 18.22624268]
[ 4.38572245 0.9326568 9.95110638]
[ 7.37995406 0.69070843 4.3862939 ]
[ 5.31551374 1.18956626 15.86002396]
[ 8.49431794 1.16415599 15.27558777]
[ 7.22443383 0.89324854 9.04471639]
[ 2.28263231 1.18584361 15.7744031 ]]
mean forecast -----
[[ 6.09958391 1.0445821 12.56586894]
[ 6.09858969 1.04358789 12.56487472]
[ 6.10102293 1.04602113 12.56730796]
[ 6.10101234 1.04601054 12.56729738]
[ 6.10097759 1.04597578 12.56726262]
[ 6.10097809 1.04597629 12.56726312]
[ 6.10097859 1.04597678 12.56726362]
[ 6.10097857 1.04597677 12.5672636 ]
[ 6.10097857 1.04597676 12.5672636 ]
[ 6.10097857 1.04597676 12.5672636 ]]
predict with credible intervals and confidence level = 80% -----
{'mean': array([[ 6.09958391, 1.0445821 , 12.56586894],
[ 6.09858969, 1.04358789, 12.56487472],
[ 6.10102293, 1.04602113, 12.56730796],
[ 6.10101234, 1.04601054, 12.56729738],
[ 6.10097759, 1.04597578, 12.56726262],
[ 6.10097809, 1.04597629, 12.56726312],
[ 6.10097859, 1.04597678, 12.56726362],
[ 6.10097857, 1.04597677, 12.5672636 ],
[ 6.10097857, 1.04597676, 12.5672636 ],
[ 6.10097857, 1.04597676, 12.5672636 ]]), 'std': array([[4.28508739, 4.28508739, 4.28508739],
[4.28508739, 4.28508739, 4.28508739],
[4.28508739, 4.28508739, 4.28508739],
[4.2850874 , 4.2850874 , 4.2850874 ],
[4.28508736, 4.28508736, 4.28508736],
[4.2850873 , 4.2850873 , 4.2850873 ],
[4.28509003, 4.28509003, 4.28509003],
[4.28509202, 4.28509202, 4.28509202],
[4.28630194, 4.28630194, 4.28630194],
[4.28696269, 4.28696269, 4.28696269]]), 'lower': array([[ 0.60802345, -4.44697835, 7.07430848],
[ 0.60702924, -4.44797257, 7.07331427],
[ 0.60946248, -4.44553933, 7.07574751],
[ 0.60945189, -4.44554992, 7.07573692],
[ 0.60941718, -4.44558463, 7.07570221],
[ 0.60941775, -4.44558405, 7.07570278],
[ 0.60941475, -4.44558705, 7.07569978],
[ 0.60941219, -4.44558962, 7.07569722],
[ 0.6078616 , -4.4471402 , 7.07414663],
[ 0.60701482, -4.44798698, 7.07329985]]), 'upper': array([[11.59114437, 6.53614256, 18.0574294 ],
[11.59015015, 6.53514835, 18.05643518],
[11.59258339, 6.53758159, 18.05886842],
[11.5925728 , 6.537571 , 18.05885783],
[11.592538 , 6.53753619, 18.05882303],
[11.59253843, 6.53753663, 18.05882347],
[11.59254242, 6.53754061, 18.05882745],
[11.59254496, 6.53754315, 18.05882999],
[11.59409553, 6.53909373, 18.06038056],
[11.59494231, 6.53994051, 18.06122734]])}
predict with credible intervals and confidence level = 95% -----
{'mean': array([[ 6.09958391, 1.0445821 , 12.56586894],
[ 6.09858969, 1.04358789, 12.56487472],
[ 6.10102293, 1.04602113, 12.56730796],
[ 6.10101234, 1.04601054, 12.56729738],
[ 6.10097759, 1.04597578, 12.56726262],
[ 6.10097809, 1.04597629, 12.56726312],
[ 6.10097859, 1.04597678, 12.56726362],
[ 6.10097857, 1.04597677, 12.5672636 ],
[ 6.10097857, 1.04597676, 12.5672636 ],
[ 6.10097857, 1.04597676, 12.5672636 ]]), 'std': array([[4.28508739, 4.28508739, 4.28508739],
[4.28508739, 4.28508739, 4.28508739],
[4.28508739, 4.28508739, 4.28508739],
[4.2850874 , 4.2850874 , 4.2850874 ],
[4.28508736, 4.28508736, 4.28508736],
[4.2850873 , 4.2850873 , 4.2850873 ],
[4.28509003, 4.28509003, 4.28509003],
[4.28509202, 4.28509202, 4.28509202],
[4.28630194, 4.28630194, 4.28630194],
[4.28696269, 4.28696269, 4.28696269]]), 'lower': array([[-2.29903305, -7.35403486, 4.16725198],
[-2.30002727, -7.35502907, 4.16625776],
[-2.29759403, -7.35259583, 4.168691 ],
[-2.29760462, -7.35260643, 4.16868041],
[-2.2976393 , -7.35264111, 4.16864573],
[-2.29763869, -7.3526405 , 4.16864634],
[-2.29764354, -7.35264535, 4.16864149],
[-2.29764745, -7.35264926, 4.16863758],
[-2.30001887, -7.35502067, 4.16626616],
[-2.30131391, -7.35631571, 4.16497112]]), 'upper': array([[14.49820087, 9.44319907, 20.9644859 ],
[14.49720666, 9.44220485, 20.96349169],
[14.4996399 , 9.44463809, 20.96592493],
[14.49962931, 9.4446275 , 20.96591434],
[14.49959448, 9.44459267, 20.96587951],
[14.49959488, 9.44459307, 20.96587991],
[14.49960071, 9.4445989 , 20.96588574],
[14.4996046 , 9.44460279, 20.96588963],
[14.501976 , 9.44697419, 20.96826103],
[14.50327104, 9.44826923, 20.96955607]])}
```

# Visualizing credible (because Bayesian, “credible”) intervals

```
j0 = 1 # choice of time series index
x = np.linspace(0, 9, 10)
y_est = preds_95['mean'][:,j0]
y_est_upper = preds_95['upper'][:,j0]
y_est_lower = preds_95['lower'][:,j0]
y_est_upper2 = preds_80['upper'][:,j0]
y_est_lower2 = preds_80['lower'][:,j0]
fig, ax = plt.subplots()
ax.plot(x, y_est, '-')
ax.fill_between(x, y_est_lower, y_est_upper, alpha=0.4)
ax.fill_between(x, y_est_lower2, y_est_upper2, alpha=0.2)
```