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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
from keras.optimizers import Adam

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, 
               padding='causal',
               activation = "relu",
               dilation_rate=dilation_rate)(x)
x = Dense(2)(x)


model = Model(inputs = inp, outputs = x)
model.compile(optimizer = Adam(), loss='mean_squared_error')
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?

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Stacked LSTM is one option in this scenario

enter image description here

This assumes that First two LSTMs have different frequencies and City has static features (Like lat/long, one-hot-encoded value etc). If City is also time-series like series of population , mean income; it will be an LSTM as well.

Code example for stacked LSTM : https://machinelearningmastery.com/stacked-long-short-term-memory-networks/

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