MUNI
ECON
SUPTECH WORKSHOP III
Neural networks
• SUPTECH WORKSHOP III • Neural networks
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Black-Litterman model
Let r = (ri, r2,..., rn)T, n e N be asset returns. Assume r ^ N(/i, £). In turn, assume random mean returns li ~ 7V(7t,t£) where tt is determined by the investor. The investor may formulate linear views, such as
Piljll + pi2\±2 + ... + Pinion = qi+Si
where ei ~ 7V(0, of), with of controlling for confidence. Collecting views into a matrix gives PM ~ N(/i, fi) where fi is a diagonal matrix of (of, cr|5of). Then
to = ((rS)"1 + P^n-1?)-^^)-1^ + P^q) S^L = ((rE)"1 + PT^-1P)-1
• SUPTECH WORKSHOP III • Neural networks
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Neurons
I Ji :i-
• SUPTECH WORKSHOP III • Neural networks
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Artificial Neural networks (ANNs)
■ multiple real-valued inputs X = (Xu ...,Xr)T
■ single output Y.
■ The connection is indicated by weight $ .
■ The output is obtained by computing the activation value U as the sum of X, with their respective weights in the vector
(3 = (/?!,/3r), and a bias term f30:
r
1=1
■ the result is passed through an activation function /,
Y = f(U) = f(f30 + X-Tf3)
• SUPTECH WORKSHOP III • Neural networks
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Activation functions
sign(x) i+e-* tanh(x)
-4    -2     0       2      4       -4    -2     0       2       1       -4    -2     0       2 4
• SUPTECH WORKSHOP III • Neural networks
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Single-layer perceptron
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Learning methods
■ Supervised learning: minimizing error function (regression).
■ Unsupervised learning: finding features without teacher (clustering).
■ Reinforcement learning: It differs from supervised learning in that labelled input/output pairs need not be presented, and sub-optimal actions need not be explicitly corrected. Instead the focus is finding a balance between exploration (of uncharted territory) and exploitation (of current knowledge).
• SUPTECH WORKSHOP III • Neural networks
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Feed-forward Neural Network (FNN)
■ neurons are connected only from input to output
■ no connections within a layer
■ hidden layer - neurons not input and output
■ most common - multi-layer perceptron (MLP)
• SUPTECH WORKSHOP III • Neural networks
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One hidden layer
Parallel MLPs
Input
o
Network
Output
Xl
o
X-2
o
o
• SUPTECH WORKSHOP III • Neural networks
Unfolded recurrent neural network
Input
Network
o
X-2
o
o
• SUPTECH WORKSHOP III • Neural networks
Recurrent neural network
Input
Network
xt
o
1 ^	r-s-* A		
} *			
Output
Vt
• SUPTECH WORKSHOP III • Neural networks
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