advanced-algorithmic-trading

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p∑
〈u, v〉 = u j v j (19.14)
j=1
Hence for two observations an inner product is defined as:
p∑
〈x i , x k 〉 = x ij x kj (19.15)
j=1
We will not dwell on the details since they are beyond the scope of this book, however it
is possible to show that a linear support vector classifier for a particular observation x can be
represented as a linear combination of inner products:
f(x) = b 0 +
With n a i coefficients, one for each of the training observations.
n∑
a i 〈x, x i 〉 (19.16)
i=1
To estimate the b 0 and a i coefficients we only need to calculate ( n
2)
= n(n − 1)/2 inner
products between all pairs of training observations. In fact, we really only need to calculate the
inner products for the subset of training observations that represent the support vectors. I will
call this subset S . This means that:
a i = 0 if x i /∈ S (19.17)
Hence we can rewrite the representation formula as:
f(x) = b 0 + ∑ i∈S
a i 〈x, x i 〉 (19.18)
It turns out that this is a major advantage for computational efficiency.
This now motivates the extension to SVMs. If we consider the inner product 〈x i , x k 〉 and
replace it with a more general inner product "kernel" function K = K(x i , x k ), we can modify
the SVC representation to use non-linear kernel functions. Hence we can adjust how we calculate
"similarity" between two observations. For instance, to recover the SVC we just take K to be as
follows:
p∑
K(x i , x k ) = x ij x kj (19.19)
j=1
Since this kernel is linear in its features the SVC is known as the linear SVC. We can also
consider polynomial kernels, of degree d: