1 Introduction
2 Non-parametric estimators of the regression function
2.1 Kernel estimators
Uniform |
\(K(x) = {\frac{1}{2}}\,I(-1 < x < 1) \)
|
Triangular | K(x) = (1 − |x|) I(− 1 < x < 1) |
Biweight |
\(K(x) = {\frac{15}{16}}(1 - u^2)\,I(-1 < x < 1) \)
|
Gaussian |
\(K(x) = {\frac{1}{\sqrt{2\pi}}}\exp{-u^2/2}\)
|
2.2 Support vector machines
3 The modification of the data space
4 Problems with the typical kernel prediction
t
|
x
t
|
\(\widetilde{f}_{NW}(x_{t-12})\)
| absolute error |
---|---|---|---|
I 60 | 417 | 389 | 28 |
II 60 | 391 | 377 | 14 |
III 60 | 419 | 448 | 29 |
IV 60 | 461 | 442 | 19 |
V 60 | 472 | 451 | 21 |
VI 60 | 535 | 514 | 21 |
VII 60 | 622 | 0 |
622
|
VIII 60 | 606 | 0 |
606
|
IX 60 | 508 | 501 | 7 |
X 60 | 461 | 449 | 12 |
XI 60 | 390 | 390 | 0 |
XII 60 | 432 | 447 | 15 |
5 HASKE algorithm
5.1 Background
5.2 Definition
t
|
x
t
| NW(x
t−12) |
\({\rm HASKE}_\mu(x_{t-12})\)
|
\({\rm HASKE}_{\mu, \alpha}(x_{t-12})\)
|
---|---|---|---|---|
I 60 | 417 | 389 | 382 | 411 |
II 60 | 391 | 377 | 367 | 395 |
III 60 | 419 | 448 | 421 | 453 |
IV 60 | 461 | 442 | 412 | 444 |
V 60 | 472 | 451 | 437 | 471 |
VI 60 | 535 | 514 | 493 | 531 |
VII 60 | 622 | 0 | 553 | 595 |
VIII 60 | 606 | 0 | 555 | 598 |
IX 60 | 508 | 501 | 487 | 524 |
X 60 | 461 | 448 | 421 | 454 |
XI 60 | 390 | 390 | 383 | 412 |
XII 60 | 432 | 447 | 420 | 452 |
RMSE | 275.26 | 37.39 | 17.18 |
-
For i = 1 to \(i = {\frac{\mu_{max} - 1}{\Updelta\mu}}\) observe the prediction error on the phase tune set rmseph.
-
Select the minimal value of the phase prediction error rmseph. The argument μph is the argument of the minimal rmseph value.
6 The HKSVR estimator
6.1 Background
6.2 Definition
7 Time series prediction
7.1 HASKE results
Decomposition | SF | GM | NW |
HASKE
| ||||
---|---|---|---|---|---|---|---|---|
Trend | Exp. | Lin. | ||||||
Model | Add. | Mult. | Add. | Mult. | ||||
M series | 13.36 | 23.23 | 28.47 | 33.68 | 31.24 | 58.53 | 42.77 |
8.74
|
N series | 21.99 | 50.69 | 33.65 | 49.22 | 43.81 | 75.30 | 49.09 |
4.6
|
G series | 40.32 | 26.60 | 64.63 | 68.52 | 139.01 | 475.64 | 275.26 |
17.18
|
E series | 72.06 | 77.72 | 72.56 | 77.57 | 33.66 | 301.30 |
33.37
| 36.76 |
7.2 The HKSVR results
nth Maximal harmonic | Prediction horizon p
| |||||||||
---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
2 | −91 | −40 | −124 | −117 | −53 | −149 | −25 | 25 | 3 | 1 |
3 | −1 | 23 | 6 | 14 | 4 | 14 | 91 | 45 | 19 | 16 |
4 | −298 | 13 | 40 | 25 | 11 | 69 | 48 | 19 | −8 | −41 |
5 | −46 | 112 | −19 | −17 | −11 | −10 | −24 | −27 | −20 | −47 |
6 | −44 | −24 | 3 | 17 | 26 | 21 | 40 | 0 | −146 | −11 |
7 | −13 | 4 | −34 | −28 | 6 | −2 | −5 | −4 | 0 | 0 |
8 | −2 | 3 | 0 | −11 | 266 | 208 | 189 | 136 | 104 | 102 |
9 | −33 | −49 | 403 | 347 | −6 | 0 | 0 | 0 | 0 | 0 |
10 | 114 | 152 | −143 | 0 | 0 | −85 | 0 | 0 | 0 | 0 |
Horizon | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
Avg | −45.9 | 21.7 | 14.6 | 25.6 | 26.9 | 7.4 | 35.0 | 21.7 | −5.4 | 2.3 |
SD | 109.6 | 67.6 | 158.0 | 127.7 | 92.4 | 98.7 | 69.2 | 47.7 | 63.9 | 42.9 |
p
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
ρWIG20
| −2.4 |
3.1
|
10.8
|
5.0
|
3.4
|
13.4
|
2.0
|
2.2
| −11.9 |
18.9
|
ρrWIG20
| −4.9 | −1.6 | −1.3 | −1.8 | −2.6 |
3.6
|
23.9
| −2.1 |
4.9
| −4.3 |
7.3 ρ coefficient normalization and results interpretation
-
Q(ρ) = 0 is the asymptotic worst value and corresponds to ρ = −∞
-
Q(ρ) = 1 is the asymptotic best value and corresponds to ρ = ∞
-
Q(ρ) ∈ (−∞, 0.5) corresponds to prediction worsening
-
Q(ρ) ∈ (0.5, ∞) corresponds to prediction improvement
-
Q(ρ) = 0.5 means that there is no improvement and it corresponds to ρ = 0
-
1 − Q(−ρ) > Q(ρ), for ρ > 0, that for small |ρ| values the worsening has stronger influence on the Q value that the improvement (for example: Q(0.01) = 0.51 and Q(−0.01) = 0.3).
p
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
Q (WIG20) | 0.29 |
0.59
|
0.75
|
0.62
|
0.59
|
0.79
|
0.57
|
0.57
| 0.01 |
0.87
|
Q (rWIG20) | 0.12 | 0.36 | 0.39 | 0.34 | 0.27 |
0.60
|
0.92
| 0.32 |
0.62
| 0.15 |
p
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
h
2
| 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.91 | 0.91 |
h
3
| 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 |
h
4
| 0.94 | 0.94 | 0.97 | 0.97 | 0.97 | 0.97 | 0.97 | 0.97 | 0.97 | 0.97 |
p
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
h
2
| 0.01 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.02 | 0.02 | 0.01 | 0.03 |
h
3
| 0.01 | 0.01 | 0.02 | 0.01 | 0.01 | 0.01 | 0.02 | 0.01 | 0.01 | 0.01 |
h
4
| 0.01 | 0.02 | 0.01 | 0.02 | 0.01 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 |