56 CORRELATION-DRIVEN NONPARAMETRIC LEARNING
8.4 Analysis
In this section, we first analyze CORN’s universal consistency with respect to the
class of all ergodic process.
Theorem 8.1 The portfolio scheme CORN is universal with respect to the class of
all ergodic processes such that E{|log X
j
|} < ∞, for j = 1,...,m.
Proof The proof can be found in Appendix B.1.1.
In the CORN expert learning procedure, there are two key parameters: the cor-
relation coefficient threshold ρ and the window size w. Below, we analyze how they
affect the algorithms.
As shown in the motivating example, the correlation coefficient threshold ρ is
critical to a correlation-similar set. If ρ is negative, the correlation-similar set would
contain some negatively correlated price relative vectors or irrelevant vectors. On
the other hand, if ρ is too large, for example, ρ ≥ 0.5, the correlation-similar set
would neglect some positively correlated vectors. Since the correlation-similar set is
crucial in selecting optimal portfolios, it would harm the learning performance if it
either contains negatively correlated vectors/irrelevant vectors or discards positively
correlated vectors. Empirically, we found that the optimal ρ value is often dataset
dependent, but often close to 0, which will be verified in Section 13.3.1. Moreover,
we note that CORN would degrade to a special case when ρ → 1. As ρ → 1, fewer
market windows are highly positively correlated to the latest window. In the extreme
case of ρ = 1, C
t
(w, ρ) becomes almost empty, which thus reduces to the uniform
CRP strategy.
∗
Another key parameter for the CORN expert learning process is window size.
Since the calculation of correlation coefficient treats market windows as a vector, the
window size does not have a significant impact on the final portfolio. When certain
experts give verybadperformance,thefinal result tends to be relatively stable sincethe
proposed combination methods (viz., CORN-U and CORN-K) will reduce the impact
of these experts and thus provide a stable portfolio. We will numerically analyze the
effect of window size in Section 13.3.1, which shows that the proposed combination
can effectively smoothen the performance curve.
The simplicity and effectiveness of CORN raise a fundamental question: Is it
reasonable to select a portfolio using only market price information?” While our
goal is not to resolve the philosophical debates between fundamental and techni-
cal analysts, we believe this work goes a long way to provide empirical evidence
endorsing the effectiveness of technical analysis. Moreover, note that the success of
CORN depends on three basic assumptions that form the basis of most technical anal-
ysis methods, including: (i) market action discounts everything; (ii) price moves in
trends; and (iii) history tends to repeat itself. The first point assumes that stock prices
at any given time reflect everything that has or could affect a company, including
fundamental factors. And the second and third points directly lead to our proposed
∗
This is not a general case, which depends on the initial portfolio and default values if a similarity set
is empty.
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