# Architecture for multivariate multi-time-series model where some features are TS specific and some features are global

I'm looking to build a time series model (using a TCN or a LSTM) with $$N$$ different series, each of which has $$P$$ series-specific features $$\mathbf{X}$$. My input array is of dimension $$N \times t \times P$$, where $$t$$ is the number of time steps.

I've also got features $$G$$, which are constant across all time series. For concreteness, imagine I'm predicting city-level ice cream sales with weather data, and I also want to use GDP growth as a predictor. GDP growth is national. A simple approach could be to augment $$\mathbf{X}$$ with $$G$$, adding 1 to the dimension of $$P$$. Then my forecast output for the next period would be $$N \times 1 \times P+1$$, which is no good because there is a GDP forecast for each city, when in reality GDP growth is common across cities (when measured nationally). I suppose that I want two outputs -- one of shape $$N \times 1 \times P$$, and the other of shape $$1 \times 1 \times 1$$, a scalar (if $$G$$ is of dimension $$t \times 1$$).

Here's a dummy example in which time is a global variable, but it is constant across all series. (Let's just assume for the moment that time isn't exogenous, but rather something to include in a multivariate forecast).

import numpy as np
import matplotlib.pyplot as plt
from keras.models import Model
from keras.layers import Input, Conv1D, Dense

time = np.array(range(100))
brk = np.array((time>40) & (time < 60)).reshape(100,1)
B = np.array([5, -5]).reshape(1,2)
np.dot(brk, B)
y = np.c_[np.sin(time), np.sin(time)] + np.random.normal(scale = .2, size=(100,2))+ np.dot(brk, B)

plt.plot(time, y[:,0])
plt.plot(time, y[:,1])

# Temporal convolutional network
n_filters = 2
filter_width = 3
dilation_rates = [2**i for i in range(5)]
inp = Input(shape=(None, 2))
x = inp
for dilation_rate in dilation_rates:
x = Conv1D(filters=n_filters,
kernel_size=filter_width,
activation = "relu",
dilation_rate=dilation_rate)(x)
x = Dense(2)(x)

model = Model(inputs = inp, outputs = x)
model.summary()

def shift5(arr, num, fill_value=np.nan):
result = np.empty_like(arr)
if num > 0:
result[:num] = fill_value
result[num:] = arr[:-num]
elif num < 0:
result[num:] = fill_value
result[:num] = arr[-num:]
else:
result = arr
return result

X = y.reshape(2,100,1)
X = np.concatenate([X, np.concatenate([time.reshape(100,1),time.reshape(100,1)], axis = 1).reshape(2,100, 1)],
axis = 2)
X_tr = X[:,:95,:]
X_te = X[:,5:,:]

history = model.fit(X_tr, X_te,
batch_size=2,
epochs=10,
verbose = 1)


How would I modify this architecture to have two inputs and two outputs, with both input and output having local and global components? 