52 CORRELATION-DRIVEN NONPARAMETRIC LEARNING
Algorithm 8.1: CORrelation-driven Nonparametric expert: CORN (w, ρ).
Input: w: Window size; ρ: Correlation coefficient threshold; x
t
1
: Historical
market sequence; t: Index of current period.
Output: b
t+1
: Expert’s portfolio for period t +1.
begin
Initialize C
t
=∅, b
t+1
=
1
m
,...,
1
m
;
if t ≤ w then
return b
t+1
;
end
for i = w +1,w+2,...,t do
if corrcoef (x
i−1
i−w
, x
t
t−w+1
) ≥ ρ then
C
t
= C
t
∪i;
end
end
if C
t
=∅then
Search for an optimal portfolio: b
t+1
= arg max
b∈
m
i∈C
t
(b ·x
i
);
end
return b
t+1
;
end
period t, we propose to learn an optimal portfolio following the idea of BCRP (Cover
1991), which maximizes the expected multiplicative return over the sequence of
similar price relatives, that is,
b
t+1
(w, ρ) = arg max
b∈
m
i∈C
t
(w,ρ)
(b ·x
i
), (8.1)
where
m
represents an m-dimensional simplex. In case that C
t
(w, ρ) is empty (espe-
cially for a large ρ value), we will simply adopt uniform portfolio
1
m
,...,
1
m
. Note
that the correlation-similar set usually contains a large number of correlated price
relatives. If one similar price relative vector has occurred frequently in history, it will
also appear multiple times in the correlation-similar set. In other words, Equation 8.1
has more or less considered the occurrence/confidence of the correlated price relative
vectors, which would avoid simply taking an extreme case in history.
We further combine all experts according to their historical performance S
t
(w, ρ)
and a probability distribution function q(w,ρ). Specifically, CORN combines experts’
portfolios and calculates the final portfolio for period t +1as
b
t+1
=
w,ρ
q(w,ρ)S
t
(w, ρ)b
t+1
(w, ρ)
w,ρ
q(w,ρ)S
t
(w, ρ)
, (8.2)
where b
t+1
(w, ρ) represents the portfolio computed by expert E(w, ρ) and S
t
(w, ρ)
represents its historical performance. For an individual expert, the higher its historical
return, the higher its weight assigned in the final portfolio.
T&F Cat #K23731 — K23731_C008 — page 52 — 9/28/2015 — 21:18
ALGORITHMS 53
After releasing the price relative vector of x
t+1
, CORN updates the cumulative
wealth,
S
t+1
= S
t
×(b
t+1
·x
t+1
).
For the underlying experts, CORN updates their cumulative wealth,
S
t+1
(w, ρ) = S
t
(w, ρ) ×(b
t+1
(w, ρ) ·x
t+1
),
where S
t
(w, ρ) represents the cumulative wealth achieved by expert E(w, ρ) till
period t.
Therefore, it is straightforward that the cumulative wealth achieved by the pro-
posed CORN strategy after n periods is equivalent to a q-weighted sum of all experts’
returns,
S
n
=
w,ρ
q(w,ρ)S
n
(w, ρ). (8.3)
Clearly, the final cumulative return is affected by all underlying experts, and the
portions of contributions made by each expert are determined by the predefined
distribution q(w,ρ) and expert’s performance S
n
(w, ρ).
Ideally, indexed by (w, ρ), we can choose CORN experts such that they cover all
possible parametersettings,thus eliminating their effects.However,the computational
cost of such a combination is inhibitively high. To boost the efficiency, we can choose
finite discrete dimensions of the parameters, that is, a specified number of (w, ρ)
combinations.
The selection of experts also trades off an individual expert’s performance and its
computational time. First, Equation 8.3 clearly shows that each expert contributes to
the final cumulative wealth by its performance; thus, choosing a worse expert may
lower the final performance. Second, the mixture’s computation time is generally the
summation of all experts’individual time. In other words, choosing too many experts,
which cost too much time, may affect its practical scalability.
In this study, we first adopted uniform combination, which chooses a uniform
distribution of q(w,ρ), and named it “CORN uniform combination” (CORN-U).
Algorithm 8.2 shows the details of the proposed CORN-U algorithm. In particular,
we assign the same weights to all CORN experts, although the weights can be adjusted
if we have more information. Moreover, CORN-U only considers P = 1 and chooses
a specific value of ρ.
The above uniform combination algorithm may include some poor experts, lead-
ing to the degradation of overall performance. To overcome such limitations, the
second algorithm, “CORN top-K combination” (CORN-K), combines only the top
K best experts. Algorithm 8.3 illustrates the proposed CORN-K algorithm. In partic-
ular, it chooses the top K experts with the highest historical returns and uniformly
combines them. That is, the strategy assigns the set of top K experts a uniform distri-
bution q(w,ρ) =
1
K
, while the weights assigned to other experts are simply set to 0.
Moreover, for the proposed CORN-K algorithm, we define P ≥ 1 associated experts,
each of which has a different ρ value.
T&F Cat #K23731 — K23731_C008 — page 53 — 9/28/2015 — 21:18
54 CORRELATION-DRIVEN NONPARAMETRIC LEARNING
Algorithm 8.2: Online portfolio selection with CORN uniform algorithm
(CORN-U).
Input: W : Maximum window size; ρ: Correlation coefficient threshold;
x
n
1
= (x
1
,...,x
n
): Historical market sequence.
Output: S
n
: Final cumulative wealth.
begin
Initialize S
0
and W experts: S
0
= 1, b
1
=
1
m
1, q(w,ρ) =
1
W
,
S
0
(w, ρ) = 1, b
1
(w, ρ) =
1
m
1,w = 1,...,W;
for t = 1 to n do
Rebalance the portfolio to b
t
;
Receive current price relatives: x
t
;
Update the cumulative wealth: S
t
= S
t−1
×(b
t
·x
t
);
Update the experts’ cumulative wealth:
S
t
(w, ρ) = S
t−1
(w, ρ) ×(b
t
(w, ρ) ·x
t
);
Update next portfolio:
begin
for w = 1 to W do
CORN expert (Algorithm 8.1) finds a portfolio:
b
t+1
(w, ρ) = CORN(w, ρ)
end
Combine experts’ portfolios:
b
t+1
=
w
q(w,ρ)S
t
(w, ρ)b
t+1
(w, ρ)
w
q(w,ρ)S
t
(w, ρ)
end
end
end
Remarks on Aggregation: Note that the aggregation or combination rule
described in Equation 8.2 is a special case of the general concept of exponential
weighting. For a learning parameter η > 0, put
b
t+1
=
w,ρ
q(w,ρ)e
η log S
t
(w,ρ)b
t+1
(w,ρ)
w,ρ
q(w,ρ)e
η log S
t
(w,ρ)
.
If η = 1, then one gets the rule in Equation 8.2. The proof of universal consistency of
B
H
, B
K
, and B
NN
works without any difficulties if η ≤ 1. However, the exponential
results are superior if η is much larger than 1, but there is no theoretical support for this
phenomenon. The large η corresponds to CORN-K with K = 1 (i.e., this rule is the
T&F Cat #K23731 — K23731_C008 — page 54 — 9/28/2015 — 21:18
ALGORITHMS 55
Algorithm 8.3: Online portfolio selection with CORN top-K algorithm
(CORN-K).
Input: W : Maximum window size; P : The number of correlation coefficient
thresholds; K: The number for top K experts; x
n
1
= (x
1
,...,x
n
):
historical market sequence.
Output: S
n
: Final cumulative wealth.
begin
Initialize S
0
and W ×P experts: S
0
= 1, P =
0,
1
P
,...,
P −1
P
,
q(w,ρ) =
1
W ×P
, S
0
(w, ρ) = 1,w = 1,...,W,ρ ∈ P;
for t = 1 to n do
Rebalance the portfolio to b
t
;
Receive current price relatives: x
t
;
Update the cumulative wealth: S
t
= S
t−1
×(b
t
·x
t
);
Update the experts’ cumulative wealth:
S
t
(w, ρ) = S
t−1
(w, ρ) ×(b
t
(w, ρ) ·x
t
);
Update experts’ weights:
begin
Select top K experts K ={(w, ρ)} w.r.t. S
t
(w, ρ) ;
Set weights for the top K experts: q(w,ρ) =
1
K
,(w,ρ) ∈ K;
Set zero weights for other experts: q(w,ρ) = 0,(w,ρ) ∈ K;
end
Update next portfolio:
begin
for w = 1 to W do
for ρ ∈ P do
CORN expert (Algorithm 8.1) finds a portfolio:
b
t+1
(w, ρ) = CORN(w, ρ)
end
end
Combine top K experts’ portfolios:
b
t+1
=
w,ρ
q(w,ρ)S
t
(w, ρ)b
t+1
(w, ρ)
w,ρ
q(w,ρ)S
t
(w, ρ)
end
end
end
follow the winner rule, which experts called follow the leader in the machine-learning
literature) (Cesa-Bianchi and Lugosi 2006). It would be nice to prove or disprove that
the follow the leader aggregation results in universally consistent strategies (i.e.,
asymptotically it is of growth optimal for any stationary and ergodic market process).
T&F Cat #K23731 — K23731_C008 — page 55 — 9/28/2015 — 21:18
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.
T&F Cat #K23731 — K23731_C008 — page 56 — 9/28/2015 — 21:18
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.