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RNNs for Timeseries Analysis

This document presents a comprehensive overview of recurrent neural networks (RNNs) and their applications in time series analysis, including fundamental concepts such as feedforward networks, backpropagation, and long short-term memory (LSTM) architectures. It discusses the optimization problems involved in machine learning, introduces gated recurrent units (GRUs) as a simplified alternative to LSTMs, and highlights the use of Keras for building neural network models. Practical applications of these techniques span various fields, including language modeling, speech recognition, and time series forecasting.

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RNNs for Timeseries Analysis 
 www.data4sci.com github.com/DataForScience/RNN
www.data4sci.com@bgoncalves The views and opinions expressed in this tutorial are those of the authors and do not necessarily reflect the official policy or position of my employer. The examples provided with this tutorial were chosen for their didactic value and are not mean to be representative of my day to day work. Disclaimer
www.data4sci.com@bgoncalves References
www.data4sci.com@bgoncalves How the Brain “Works” (Cartoon version)
www.data4sci.com@bgoncalves How the Brain “Works” (Cartoon version) • Each neuron receives input from other neurons • 1011 neurons, each with with 104 weights • Weights can be positive or negative • Weights adapt during the learning process • “neurons that fire together wire together” (Hebb) • Different areas perform different functions using same structure (Modularity)
www.data4sci.com@bgoncalves How the Brain “Works” (Cartoon version) Inputs Outputf(Inputs)
www.data4sci.com@bgoncalves Optimization Problem • (Machine) Learning can be thought of as an optimization problem. • Optimization Problems have 3 distinct pieces: • The constraints • The function to optimize • The optimization algorithm. Neural Network Prediction Error Gradient Descent
www.data4sci.com@bgoncalves Artificial Neuron x1 x2 x3 xN w 1j w2j w3j wN j zj wT x aj (z) w 0j 1 Inputs Weights Output Activation function Bias
www.data4sci.com@bgoncalves Activation Function - Sigmoid (z) = 1 1 + e z • Non-Linear function • Differentiable • non-decreasing • Compute new sets of features • Each layer builds up a more abstract
 representation of the data • Perhaps the most common http://github.com/bmtgoncalves/Neural-Networks
www.data4sci.com@bgoncalves Activation Function - tanh (z) = ez e z ez + e z • Non-Linear function • Differentiable • non-decreasing • Compute new sets of features • Each layer builds up a more abstract
 representation of the data http://github.com/bmtgoncalves/Neural-Networks
www.data4sci.com@bgoncalves Forward Propagation • The output of a perceptron is determined by a sequence of steps: • obtain the inputs • multiply the inputs by the respective weights • calculate output using the activation function • To create a multi-layer perceptron, you can simply use the output of one layer as the input to the next one.
 x1 x2 x3 xN w 1j w2j w3j wN j aj w 0j 1 wT x 1 w 0k w 1k w2k w3k wNk ak wT a a1 a2 aN
www.data4sci.com@bgoncalves Backward Propagation of Errors (BackProp) • BackProp operates in two phases: • Forward propagate the inputs and calculate the deltas • Update the weights • The error at the output is a weighted average difference between predicted output and the observed one. • For inner layers there is no “real output”!
www.data4sci.com@bgoncalves The Cross Entropy is complementary to sigmoid activation in the output layer and improves its stability. Loss Functions • For learning to occur, we must quantify how far off we are from the desired output. There are two common ways of doing this: • Quadratic error function: • Cross Entropy E = 1 N X n |yn an| 2 J = 1 N X n h yT n log an + (1 yn) T log (1 an) i
www.data4sci.com@bgoncalves Gradient Descent • Find the gradient for each training batch • Take a step downhill along the direction of the gradient 
 
 • where is the step size. • Repeat until “convergence”. H ✓mn ✓mn ↵ @H @✓mn @H @✓mn ↵
www.data4sci.com@bgoncalves
www.data4sci.com@bgoncalves Feed Forward Networks h t Output h t = f (xt) xt Input
www.data4sci.com@bgoncalves Feed Forward Networks h t xt Information
 Flow Input Output h t = f (xt)
www.data4sci.com@bgoncalves h t = f (xt, h t−1) Recurrent Neural Network (RNN) h t Output h t Output Previous Output Information
 Flow h t−1 h t = f (xt) xt Input
www.data4sci.com@bgoncalves Recurrent Neural Network (RNN) xt h t h t−1 h t xt+ 1 h t+ 1 h t+ 1 xt−1 h t−1 h t−2 • Each output depends (implicitly) on all previous outputs. • Input sequences generate output sequences (seq2seq)
www.data4sci.com@bgoncalves Recurrent Neural Network (RNN) h t h t h t−1 xt tanh h t = tanh (Wh t−1 + Uxt) Concatenate both inputs.
www.data4sci.com@bgoncalves Timeseries • Temporal sequence of data points • Consecutive points are strongly correlated • Common in statistics, signal processing, econometrics, mathematical finance, earthquake prediction, etc • Numeric (real or discrete) or symbolic data • Supervised Learning problem: Xt = f (Xt−1, ⋯, Xt−n )
github.com/DataForScience/RNN
www.data4sci.com@bgoncalves Long-Short Term Memory (LSTM) xt h t ct−1 ct xt+ 1 h t+ 1 ct+ 1 xt−1 h t−1 ct−2 • What if we want to keep explicit information about previous states (memory)? • How much information is kept, can be controlled through gates. h t−2 h t−1 h t h t+ 1 • LSTMs were first introduced in 1997 by Hochreiter and Schmidhuber
www.data4sci.com@bgoncalves σ σ f g × σ ×i o Long-Short Term Memory (LSTM) h t h t h t−1 xt ct−1 ct g = tanh (Wg h t−1 + Ug xt) ct = (ct−1 ⊗ f) + (g ⊗ i) h t = tanh (ct) ⊗ o +× × + 1− Element wise addition Element wise multiplication 1 minus the input tanh i = σ (Wih t−1 + Uixt) f = σ (Wf h t−1 + Uf xt) o = σ (Wo h t−1 + Uo xt) tanh
www.data4sci.com@bgoncalves σ σ f g × σ ×i o Long-Short Term Memory (LSTM) h t h t h t−1 xt ct−1 ct g = tanh (Wg h t−1 + Ug xt) ct = (ct−1 ⊗ f) + (g ⊗ i) h t = tanh (ct) ⊗ o +× × + 1− Element wise addition Element wise multiplication 1 minus the input tanh i = σ (Wih t−1 + Uixt) f = σ (Wf h t−1 + Uf xt) o = σ (Wo h t−1 + Uo xt) Forget gate:
 How much of the previous state should be kept? tanh
www.data4sci.com@bgoncalves σ σ f g × σ ×i o Long-Short Term Memory (LSTM) h t h t h t−1 xt ct−1 ct g = tanh (Wg h t−1 + Ug xt) ct = (ct−1 ⊗ f) + (g ⊗ i) h t = tanh (ct) ⊗ o +× × + 1− Element wise addition Element wise multiplication 1 minus the input tanh i = σ (Wih t−1 + Uixt) f = σ (Wf h t−1 + Uf xt) o = σ (Wo h t−1 + Uo xt) Input gate:
 How much of the previous output should be remembered? tanh
www.data4sci.com@bgoncalves σ σ f g × σ ×i o Long-Short Term Memory (LSTM) h t h t h t−1 xt ct−1 ct g = tanh (Wg h t−1 + Ug xt) ct = (ct−1 ⊗ f) + (g ⊗ i) h t = tanh (ct) ⊗ o +× × + 1− Element wise addition Element wise multiplication 1 minus the input tanh i = σ (Wih t−1 + Uixt) f = σ (Wf h t−1 + Uf xt) o = σ (Wo h t−1 + Uo xt) Output gate:
 How much of the previous output should contribute? All gates use the same inputs and activation functions, but different weights tanh
www.data4sci.com@bgoncalves σ σ f g × σ ×i o Long-Short Term Memory (LSTM) h t h t h t−1 xt ct−1 ct g = tanh (Wg h t−1 + Ug xt) ct = (ct−1 ⊗ f) + (g ⊗ i) h t = tanh (ct) ⊗ o +× × + 1− Element wise addition Element wise multiplication 1 minus the input tanh i = σ (Wih t−1 + Uixt) f = σ (Wf h t−1 + Uf xt) o = σ (Wo h t−1 + Uo xt) Output gate:
 How much of the previous output should contribute? tanh
www.data4sci.com@bgoncalves σ σ f g × σ ×i o Long-Short Term Memory (LSTM) h t h t h t−1 xt ct−1 ct g = tanh (Wg h t−1 + Ug xt) ct = (ct−1 ⊗ f) + (g ⊗ i) h t = tanh (ct) ⊗ o +× × + 1− Element wise addition Element wise multiplication 1 minus the input tanh i = σ (Wih t−1 + Uixt) f = σ (Wf h t−1 + Uf xt) o = σ (Wo h t−1 + Uo xt) State:
 Update the current state tanh
www.data4sci.com@bgoncalves σ σ f g × σ ×i o Long-Short Term Memory (LSTM) h t h t h t−1 xt ct−1 ct g = tanh (Wg h t−1 + Ug xt) ct = (ct−1 ⊗ f) + (g ⊗ i) h t = tanh (ct) ⊗ o +× × + 1− Element wise addition Element wise multiplication 1 minus the input tanh i = σ (Wih t−1 + Uixt) f = σ (Wf h t−1 + Uf xt) o = σ (Wo h t−1 + Uo xt) Output:
 Combine all available information. tanh
www.data4sci.com@bgoncalves Using LSTMs inputs W1 LSTM W2 Neuron inputs W1 LSTM inputs W1 LSTM inputs W1 LSTM inputs W1 LSTM inputs W1 LSTM inputs W1 LSTM inputs W1 LSTM Sequence Length #features #LSTMcells
github.com/DataForScience/RNN
www.data4sci.com@bgoncalves Applications • Language Modeling and Prediction • Speech Recognition • Machine Translation • Part-of-Speech Tagging • Sentiment Analysis • Summarization • Time series forecasting
www.data4sci.com@bgoncalves Neural Networks? h t h t h t−1 xt ct−1 ct
www.data4sci.com@bgoncalves Or legos? https://keras.io
www.data4sci.com@bgoncalves Keras • Open Source neural network library written in Python • TensorFlow, Microsoft Cognitive Toolkit or Theano backends • Enables fast experimentation • Created and maintained by François Chollet, a Google engineer. • Implements Layers, Objective/Loss functions, Activation functions, Optimizers, etc… https://keras.io
www.data4sci.com@bgoncalves Keras • keras.models.Sequential(layers=None, name=None)- is the workhorse. You use it to build a model layer by layer. Returns the object that we will use to build the model • keras.layers • Dense(units, activation=None, use_bias=True) - None means linear activation. Other options are, ’tanh’, ’sigmoid’, ’softmax’, ’relu’, etc. • Dropout(rate, seed=None) • Activation(activation) - Same as the activation option to Dense, can also be used to pass TensorFlow or Theano operations directly. • SimpleRNN(units, input_shape, activation='tanh', use_bias=True, dropout=0.0, return_sequences=False) • GRU(units, input_shape, activation='tanh', use_bias=True, dropout=0.0, return_sequences=False) https://keras.io
www.data4sci.com@bgoncalves Keras • model = Sequential() • model.add(layer) - Add a layer to the top of the model • model.compile(optimizer, loss) - We have to compile the model before we can use it • optimizer - ‘adam’, ‘sgd’, ‘rmsprop’, etc… • loss - ‘mean_squared_error’, ‘categorical_crossentropy’, ‘kullback_leibler_divergence’, etc… • model.fit(x=None, y=None, batch_size=None, epochs=1, verbose=1, validation_split=0.0, validation_data=None, shuffle=True) • model.predict(x, batch_size=32, verbose=0) - fit/predict interface https://keras.io
www.data4sci.com@bgoncalves Gated Recurrent Unit (GRU) • Introduced in 2014 by K. Cho • Meant to solve the Vanishing Gradient Problem • Can be considered as a simplification of LSTMs • Similar performance to LSTM in some applications, better performance for smaller datasets.
www.data4sci.com@bgoncalves σ tanh × σ × r cz Gated Recurrent Unit (GRU) h t h t h t−1 xt +× z = σ (Wzh t−1 + Uzxt) r = σ (Wrh t−1 + Urxt) c = tanh (Wc (h t−1 ⊗ r) + Ucxt) h t = (z ⊗ c) + ((1 − z) ⊗ h t−1) 1− × + 1− Element wise addition Element wise multiplication 1 minus the input
www.data4sci.com@bgoncalves σ tanh × σ × r cz Gated Recurrent Unit (GRU) h t h t h t−1 xt +× z = σ (Wzh t−1 + Uzxt) r = σ (Wrh t−1 + Urxt) c = tanh (Wc (h t−1 ⊗ r) + Ucxt) h t = (z ⊗ c) + ((1 − z) ⊗ h t−1) 1− × + 1− Element wise addition Element wise multiplication 1 minus the input Update gate:
 How much of the previous state should be kept?
www.data4sci.com@bgoncalves σ tanh × σ × r cz Gated Recurrent Unit (GRU) h t h t h t−1 xt +× z = σ (Wzh t−1 + Uzxt) r = σ (Wrh t−1 + Urxt) c = tanh (Wc (h t−1 ⊗ r) + Ucxt) h t = (z ⊗ c) + ((1 − z) ⊗ h t−1) 1− × + 1− Element wise addition Element wise multiplication 1 minus the input Reset gate:
 How much of the previous output should be removed?
www.data4sci.com@bgoncalves σ tanh × σ × r cz Gated Recurrent Unit (GRU) h t h t h t−1 xt +× z = σ (Wzh t−1 + Uzxt) r = σ (Wrh t−1 + Urxt) c = tanh (Wc (h t−1 ⊗ r) + Ucxt) h t = (z ⊗ c) + ((1 − z) ⊗ h t−1) 1− × + 1− Element wise addition Element wise multiplication 1 minus the input Current memory:
 What information do we remember right now?
www.data4sci.com@bgoncalves σ tanh × σ × r cz Gated Recurrent Unit (GRU) h t h t h t−1 xt +× z = σ (Wzh t−1 + Uzxt) r = σ (Wrh t−1 + Urxt) c = tanh (Wc (h t−1 ⊗ r) + Ucxt) h t = (z ⊗ c) + ((1 − z) ⊗ h t−1) 1− × + 1− Element wise addition Element wise multiplication 1 minus the input Output:
 Combine all available information.

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