This paper has applied the matrix Gaussian distribution of the likelihood function of the complete data set to reduce time complexity of multi-output relevance vector regression from O V M 3 to O V 3 + M 3 , where V and M are the number of output dimensions and basis functions respectively and V < M. Our experimental results demonstrate that the proposed method is more competitive and faster than the existing methods like Thayananthan et al. (2008). Its computational efficiency and accuracy can be attributed to the different model specifications of the likelihood of the data, as the existing method expresses the likelihood of the training data as the product of Gaussian distributions whereas the proposed method expresses it as the matrix Gaussian distribution.

We propose an optimal intraday trading algorithm to reduce overall transaction costs through absorbing price shocks when an online portfolio selection (OPS) rebalances its portfolio. Having considered the real-time data of limit order books (LOB), the trading algorithm optimally splits a sizeable market order into a number of consecutive market orders to minimise the overall transaction costs, including both the market impact costs and the proportional transaction costs. The proposed trading algorithm, compatible to any OPS methods, optimises the number of intraday trades as well as finds an optimal intraday trading path. Backtesting results from the historical LOB data of NASDAQ-traded stocks show that the proposed trading algorithm significantly reduces the overall transaction costs in an environment of limited market liquidity.

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The performance of online (sequential) portfolio selection (OPS), which rebalances a portfolio in every period (e.g. daily or weekly) in order to maximise the portfolio's expected terminal wealth in the long run, has been overestimated by the ideal assumption of unlimited market liquidity (i.e. no market impact costs). Therefore, a new transaction cost factor model that considers both market impact costs, estimated from limit order book data, and proportional transaction costs (e.g. brokerage commissions or transaction taxes in a fixed percentage) has been proposed in this paper to measure existing OPS strategies performance in a more practical way as well as to develop a more effective OPS method. Backtesting results from the historical limit order book (LOB) data of NASDAQ-traded stocks show both the performance deterioration of existing OPS methods by the market impact costs and the superiority of our proposed OPS method in the environment of limited market liquidity.

This paper aims to decrease the time complexity of multi-output relevance vector regression from O V M 3 to O V 3 + M 3 , where V is the number of output dimensions, M is the number of basis functions, and V < M . The experimental results demonstrate that the proposed method is more competitive than the existing method, with regard to computation time. MATLAB codes are available at FMRVR ter 6), Thayananthan et al. (2008) uses the Bayes' theorem and the kernel trick to perform regression, but it has the limitation of low computational efficiency. Therefore, a new faster algorithm is proposed in this paper: it uses the matrix normal distribution to model correlated outputs, while the existing algorithm uses the multivariate normal distribution.The contributions of this paper are: • in Section 4, to propose a new algorithm with less time complexity than the existing MRVR algorithm by Thayananthan (2005, Chapter 6), Thayananthan et al. (2008); • in Section 5, to present Monte Carlo simulation results to compare between the existing and the proposed MRVR algorithm in terms of accuracy and computation time.The rest of this paper is organised as follows: Section 2 lists related work. Section 3 and Section 4 describe the existing and proposed algorithms of MRVR, respectively. Section 5 shows the experimental results by using MATLAB, and Section 6 gives the conclusion.