1 Introduction
2 Backgrounds and notations
2.1 Matrix-variate Gaussian distribution
2.2 Matrix-variate Student-t distribution
3 Multivariate Gaussian and Student-t process regression models
3.1 Multivariate Gaussian process regression (MV-GPR)
3.1.1 Kernel
3.1.2 Parameter estimation
3.1.3 Comparison with the existing methods
3.2 Multivariate Student-t process regression (MV-TPR)
4 Experiments
4.1 Simulated example
4.1.1 Evaluation of parameter estimation
Noise level \(\sigma ^2_n\) | 0.1 | 0.05 | 0.01 |
---|---|---|---|
\({\hat{\sigma }}^2_n\) | 0.041 | 0.041 | 0.041 |
\({\hat{s}}_f^2\) | 0.235 | 0.235 | 0.238 |
\({\hat{\ell }}^2\) | 0.080 | 0.079 | 0.078 |
\({\hat{\varphi }}_{11}\) | 0.284 | 0.280 | 0.271 |
\({\hat{\varphi }}_{22}\) | 0.282 | 0.279 | 0.276 |
\({\hat{\phi }}_{12}\) | 0.645 | 0.645 | 0.632 |
Noise level \(\sigma ^2_n\) | 0.1 | 0.05 | 0.01 |
---|---|---|---|
\({\hat{\sigma }}^2_n\) | 0.053 | 0.058 | 0.061 |
\({\hat{s}}_f^2\) | 0.181 | 0.178 | 0.163 |
\({\hat{\ell }}^2\) | 0.083 | 0.075 | 0.071 |
\({\hat{\varphi }}_{11}\) | 0.015 | 0.015 | 0.014 |
\({\hat{\varphi }}_{22}\) | 0.013 | 0.013 | 0.013 |
\({\hat{\phi }}_{12}\) | 0.007 | 0.007 | 0.007 |
\({\hat{\nu }}\) | 0.193 | 0.190 | 0.179 |
4.2 Evaluation of prediction accuracy
Output 1 (\(y_1\)) | Output 2 (\(y_2\)) | |||||||
---|---|---|---|---|---|---|---|---|
MV-GPR | MV-TPR | GPR | TPR | MV-GPR | MV-TPR | GPR | TPR | |
Training | 0.877 | 0.884 | 0.880 | 0.875 | 0.862 | 0.849 | 0.864 | 0.862 |
Test | 1.453 | 1.309 | 1.869 | 1.869 | 1.282 | 1.145 | 1.478 | 1.480 |
Output 1 (\(y_1\)) | Output 2 (\(y_2\)) | |||||||
---|---|---|---|---|---|---|---|---|
MV-GPR | MV-TPR | GPR | TPR | MV-GPR | MV-TPR | GPR | TPR | |
Training | 0.927 | 0.919 | 0.892 | 0.890 | 0.871 | 0.850 | 0.852 | 0.850 |
Test | 1.496 | 1.332 | 1.870 | 1.867 | 1.315 | 1.156 | 1.520 | 1.521 |
4.3 Real-data examples
4.3.1 Bike rent prediction
MV-GPR | MV-TPR | GPR | TPR | |
---|---|---|---|---|
(a) MSE | ||||
Casual | 0.411 | 0.334 | 0.424 | 0.397 |
Registered | 0.982 | 0.903 | 1.134 | 1.111 |
MMO | 0.982 | 0.903 | 1.134 | 1.111 |
(b) MAE | ||||
Casual | 0.558 | 0.488 | 0.540 | 0.546 |
Registered | 0.897 | 0.855 | 0.916 | 0.907 |
MMO | 0.897 | 0.855 | 0.916 | 0.907 |
4.3.2 Air quality prediction
MV-GPR | MV-TPR | GPR | TPR | |
---|---|---|---|---|
(a) MSE | ||||
PT08S1CO | 0.091 | 0.065 | 0.079 | 0.074 |
PT08S2NMHC | \(8.16\times 10^{-5}\) | \(3.42\times 10^{-5}\) | \(1.91\times 10^{-7}\) | \(7.32\times 10^{-8}\) |
PT08S3NOx | 0.036 | 0.027 | 0.022 | 0.025 |
PT08S4NO2 | 0.015 | 0.014 | 0.010 | 0.009 |
PT08S5O3 | 0.092 | 0.073 | 0.060 | 0.067 |
MMO | 0.092 | 0.073 | 0.079 | 0.074 |
(b) MAE | ||||
PT08S1CO | 0.240 | 0.204 | 0.212 | 0.223 |
PT08S2NMHC | \(6.39\times 10^{-3}\) | \(1.15\times 10^{-2}\) | \(1.80\times 10^{-4}\) | \(9.26\times 10^{-5}\) |
PT08S3NOx | 0.141 | 0.122 | 0.115 | 0.120 |
PT08S4NO2 | 0.095 | 0.089 | 0.079 | 0.073 |
PT08S5O3 | 0.231 | 0.210 | 0.199 | 0.205 |
MMO | 0.240 | 0.210 | 0.212 | 0.223 |
4.4 Application to stock market investment
4.4.1 Data preparation
4.4.2 Prediction model and strategy
Decision | Condition |
---|---|
Buy | \({\hat{LR}}_i>0\), and \(BS_i>0\) and we have the position of cash |
Sell | \({\hat{LR}}_i<0\), and \(BS_i<0\) and we have the position of share |
Keep | No action is taken for the rest of the option |
4.4.3 Chinese companies in NASDAQ
Ticker | Exchange | Company |
---|---|---|
BIDU | NASDAQ | Baidu, Inc. |
CTRP | NASDAQ | Ctrip.com International, Ltd. |
NTES | NASDAQ | NetEase, Inc. |
4.4.4 Diverse sectors in Dow 30
Ticker | Company | Exchange | Industry | Industry\(^{4}\) (ICB) |
---|---|---|---|---|
DD | DuPont | NYSE | Chemical industry | Basic Materials |
KO | Coca-Cola | NYSE | Beverages | Consumer Goods |
PG | Procter and Gamble | NYSE | Consumer goods | Consumer Goods |
MCD | McDonald’s | NYSE | Fast food | Consumer Goods |
NKE | Nike | NYSE | Apparel | Consumer Services |
DIS | Walt Disney | NYSE | Broadcasting and entertainment | Consumer Services |
HD | The Home Depot | NYSE | Home improvement retailer | Consumer Services |
WMT | Wal-Mart | NYSE | Retail | Consumer Services |
JPM | JPMorgan Chase | NYSE | Banking | Financials |
GS | Goldman Sachs | NYSE | Banking, financial services | Financials |
V | Visa | NYSE | Consumer banking | Financials |
AXP | American Express | NYSE | Consumer finance | Financials |
TRV | Travelers | NYSE | Insurance | Financials |
UNH | UnitedHealth Group | NYSE | Managed health care | Health Care |
JNJ | Johnson & Johnson | NYSE | Pharmaceuticals | Health Care |
MRK | Merck | NYSE | Pharmaceuticals | Health Care |
PFE | Pfizer | NYSE | Pharmaceuticals | Health Care |
BA | Boeing | NYSE | Aerospace and defense | Industrials |
MMM | 3M | NYSE | Conglomerate | Industrials |
GE | General Electric | NYSE | Conglomerate | Industrials |
UTX | United Technologies | NYSE | Conglomerate | Industrials |
CAT | Caterpillar | NYSE | Construction and mining equipment | Industrials |
CVX | Chevron | NYSE | Oil and gas | Oil and Gas |
XOM | ExxonMobil | NYSE | Oil and gas | Oil and Gas |
CSCO | Cisco Systems | NASDAQ | Computer networking | Technology |
IBM | IBM | NYSE | Computers and technology | Technology |
AAPL | Apple | NASDAQ | Consumer electronics | Technology |
INTC | Intel | NASDAQ | Semiconductors | Technology |
MSFT | Microsoft | NASDAQ | Software | Technology |
VZ | Verizon | NYSE | Telecommunication | Telecommunications |
Ticker | Industry | Buy&Sell strategy | Buy&Hold strategy | ||||||
---|---|---|---|---|---|---|---|---|---|
MV-GPR | MV-TPR | GPR | TPR | Stock | INDU | NDX | SPX | ||
CVX | Oil and Gas | 3rd | 4th | 2nd | 1st | 8th | 7th | 5th | 6th |
XOM | Oil and Gas | 4th | 2nd | 3rd | 1st | 8th | 7th | 5th | 6th |
MMM | Industrials | 2nd | 3rd | 1st | 4th | 5th | 8th | 6th | 7th |
BA | Industrials | 1st | 2nd | 3rd | 4th | 8th | 7th | 5th | 6th |
CAT | Industrials | 3rd | 4th | 2nd | 1st | 8th | 7th | 5th | 6th |
GE | Industrials | 2nd | 4th | 3rd | 1st | 8th | 7th | 5th | 6th |
UTX | Industrials | 2nd | 4th | 3rd | 1st | 8th | 7th | 5th | 6th |
KO | Consumer Goods | 2nd | 1st | 3rd | 4th | 6th | 8th | 5th | 7th |
MCD | Consumer Goods | 2nd | 4th | 1st | 3rd | 8th | 7th | 5th | 6th |
PG | Consumer Goods | 3rd | 4th | 1st | 2nd | 5th | 8th | 6th | 7th |
JNJ | Health Care | 3rd | 2nd | 1st | 4th | 6th | 8th | 5th | 7th |
MRK | Health Care | 3rd | 2nd | 4th | 1st | 8th | 7th | 5th | 6th |
PFE | Health Care | 4th | 1st | 3rd | 2nd | 8th | 7th | 5th | 6th |
UNH | Health Care | 2nd | 3rd | 1st | 4th | 5th | 8th | 6th | 7th |
HD | Consumer Services | 1st | 4th | 3rd | 2nd | 5th | 8th | 6th | 7th |
NKE | Consumer Services | 2nd | 3rd | 4th | 1st | 5th | 8th | 6th | 7th |
WMT | Consumer Services | 1st | 4th | 3rd | 2nd | 5th | 8th | 6th | 7th |
DIS | Consumer Services | 3rd | 2nd | 1st | 4th | 5th | 8th | 6th | 7th |
AXP | Financials | 2nd | 4th | 1st | 3rd | 8th | 7th | 5th | 6th |
GS | Financials | 2nd | 1st | 3rd | 4th | 5th | 8th | 6th | 7th |
JPM | Financials | 2nd | 4th | 1st | 3rd | 6th | 8th | 5th | 7th |
TRV | Financials | 2nd | 3rd | 1st | 4th | 5th | 8th | 6th | 7th |
V | Financials | 1st | 4th | 3rd | 2nd | 5th | 8th | 6th | 7th |
AAPL | Technology | 4th | 2nd | 3rd | 1st | 5th | 8th | 6th | 7th |
CSCO | Technology | 2nd | 1st | 3rd | 4th | 5th | 8th | 6th | 7th |
IBM | Technology | 4th | 1st | 2nd | 3rd | 8th | 7th | 5th | 6th |
INTC | Technology | 3rd | 4th | 2nd | 1st | 5th | 8th | 6th | 7th |
MSFT | Technology | 2nd | 4th | 1st | 3rd | 5th | 8th | 6th | 7th |
Industry Portfolio | Buy&Sell strategy | Buy&Hold strategy | ||||||
---|---|---|---|---|---|---|---|---|
MV-GPR | MV-TPR | GPR | TPR | Stock | INDU | NDX | SPX | |
Oil and Gas | 4th | 3rd | 2nd | 1st | 8th | 7th | 5th | 6th |
Industrials | 2nd | 4th | 3rd | 1st | 8th | 7th | 5th | 6th |
Consumer Goods | 1st | 4th | 2nd | 3rd | 7th | 8th | 5th | 6th |
Health Care | 4th | 1st | 3rd | 2nd | 6th | 8th | 5th | 7th |
Consumer Services | 1st | 4th | 3rd | 2nd | 5th | 8th | 6th | 7th |
Financials | 1st | 4th | 2nd | 3rd | 5th | 8th | 6th | 7th |
Technology | 4th | 3rd | 1st | 2nd | 5th | 8th | 6th | 7th |