LSTM vs Transformer for Time Series Prediction

Stacked LSTM vs Transformer for Time Series Prediction

1. Stacked LSTM

LSTM is a type of Recurrent Neural Network (RNN)...

Architecture

Input sequence → LSTM layer 1 → LSTM layer 2 → Dense → Output

Mathematics

LSTM uses gates to control information flow:

• Forget gate \(f_t\):
  \[ f_t = σ(W_f · [h_{t-1}, x_t] + b_f) \]
• Input gate \(i_t\) and candidate state \(C̃_t\):
  \[ i_t = σ(W_i · [h_{t-1}, x_t] + b_i) \] \[ C̃_t = tanh(W_C · [h_{t-1}, x_t] + b_C) \]
• Cell state update \(C_t\):
  \[ C_t = f_t ⊙ C_{t-1} + i_t ⊙ C̃_t \]
• Output gate \(o_t\) and hidden state \(h_t\):
  \[ o_t = σ(W_o · [h_{t-1}, x_t] + b_o) \] \[ h_t = o_t ⊙ tanh(C_t) \]

Where \(x_t\) is the input, \(h_{t-1}\) is the previous hidden state, σ is the sigmoid activation, and ⊙ is element-wise multiplication.

2. Transformer-based Time Series Predictor

Transformers use self-attention to model dependencies across all timesteps simultaneously. They are highly effective for long sequences and allow parallel computation.

Architecture

Input sequence → Positional Encoding → Transformer Encoder → Feedforward → Output

Mathematics

1. Compute queries (Q), keys (K), values (V):
   \[ Q = X W_Q, \quad K = X W_K, \quad V = X W_V \]
2. Compute attention scores:
   \[ Attention(Q,K,V) = softmax((Q K^T)/\sqrt{d_k}) V \]
3. Multi-head attention:
   \[ MultiHead(Q,K,V) = Concat(head1,...,headh) W_O \]
4. Feedforward & LayerNorm:
   \[ FFN(x) = ReLU(x W_1 + b_1) W_2 + b_2 \]

3. Key Differences

Feature	Stacked LSTM	Transformer-based Predictor
Temporal modeling	Sequential, step-by-step	Global, all timesteps at once
Parallelization	Hard to parallelize	Fully parallelizable
Long-range dependencies	Hard for very long sequences	Excellent via attention
Memory	Hidden states carry info	Attention scores model relationships
Training speed	Slower	Faster for long sequences
Complexity	Simpler, fewer parameters	Higher parameters, more data needed
Use-case	Short-medium sequences	Long sequences, complex patterns

Summary

• Stacked LSTM: Good for short-term sequential prediction like daily stock prices. Simpler and requires less data.
• Transformer: State-of-the-art for long-term forecasting, captures global dependencies, and allows parallel training. Needs more computation and data.

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LSTM vs Encoder-only Transformer

LSTM = reads sequence step-by-step (memory-based)

Transformer = looks at entire sequence at once (attention-based)

How They Process Data

LSTM (Sequential)

x₁ → x₂ → x₃ → x₄ → x₅
h₁ → h₂ → h₃ → h₄ → h₅

Transformer (Parallel)

[x₁, x₂, x₃, x₄, x₅] → processed together

Summary

Feature	LSTM	Transformer
Processing	Sequential	Parallel
Dependencies	Weak for long sequences	Strong
Speed	Slow	Fast
Core idea	Memory	Attention

LSTM = memory over time

Transformer = attention over relationships