A Comprehensive Price Prediction System Based on Inverse Multiquadrics Radial Basis Function for Portfolio Selection

Price prediction plays a crucial role in portfolio selection (PS). However, most price prediction strategies only make a single prediction and do not have efficient mechanisms to make a comprehensive price prediction. Here, we propose a comprehensive price prediction (CPP) system based on inverse multiquadrics (IMQ) radial basis function. First, the novel radial basis function (RBF) system based on IMQ function rather than traditional Gaussian (GA) function is proposed and centers on multiple price prediction strategies, aiming at improving the efficiency and robustness of price prediction. Under the novel RBF system, we then create a portfolio update strategy based on kernel and trace operator. To assess the system performance, extensive experiments are performed based on 4 data sets from different real-world financial markets. Interestingly, the experimental results reveal that the novel RBF system effectively realizes the integration of different strategies and CPP system outperforms other systems in investing performance and risk control, even considering a certain degree of transaction costs. Besides, CPP can calculate quickly, making it applicable for large-scale and time-limited financial market.


Introduction
The target of PS is to achieve some long-term financial goals by constructing an effective investment strategy that can reasonably allocate wealth among a set of assets. There are two main theories about PS. One is the mean-variance theory proved that MQ performed best in dealing with the interpolation problem of scatter data, MQ was conditionally positive definite. Therefore, a more application-oriented IMQ was proposed. The outstanding advantages of IMQ are good global feature, strict positive definite and stable eigenvalue [14] [18]. For example, Abbasbandy [14] pointed out that IMQ could be used to approximate the unknown analytic function to get a more stable and accurate solution in solving the global optimization problem. And Tanbay [19] pointed out that compared with GA in the solution of the neutron diffusion equation, IMQ had a more stable performance and could obtain highly numerical solution. Therefore, we incorporate IMQ rather than traditional GA in our novel RBF system to increase efficiency and robustness.
In this paper, a comprehensive price prediction (CPP) system based on IMQ radial basis function is constructed. The system firstly uses IMQ basis function to construct a novel RBF system. Then, combining with the novel RBF system, a portfolio update strategy based on kernel and trace operator is constructed. Now let's consider H different price prediction strategies. This paper mainly focuses on three strategies, namely EMA, L 1 -median and PP. Firstly, CPP selects the best-performing strategy according to investing performance of all strategies within the recent window and given it the largest influence in future price prediction. Secondly, CPP exploits the similarity between the best-performing strategy and other price prediction strategies to calculate the influence of other strategies. This system effectively integrates the advantages of all price prediction strategies and innovative measuring the influence by investing performance. In genneral, this paper's main contributions are as follows: 1) Propose a novel RBF system based on IMQ radial basis function and centered on multiple price predictions, which form a comprehensive price prediction.
2) Propose a comprehensive combination of aggressive strategies and moderate strategies to achieve a better balance between returns and risks.
3) Propose a portfolio update strategy based on kernel and trace operator.
The rest parts of this paper are presented as follows. Section 2 describes the relevant problem setting and related work about PS. The CPP system is introduced and described in detail in Section 3. Experiments on 4 benchmark data sets are carried out to assess CPP in Section 4. Finally, conclusions are presented in Section 5.

The Relevant Problem Setting
In this paper, d assets with a time span of n periods in financial market are considered. For the sake of understanding, let's think of a period as a day. The asset prices of the tth period is presented by the close prices vector At the beginning of each trading period, wealth needs to be allocated across a range of assets. In this paper, when investing, the proportion of each asset in total weath is recorded as the portfolio vector. Suppose there are d assets and their portfolio vector in the tth period is where d ∆ is a d-dimensional simplex. A non-negative constraint indicates that non-short-selling and the equality constraint indicates that self-financing, which means it is not allowed to borrow money and all of the wealth is reinvested.
Since all the wealth in the previous periods is invested over next period, the cumulative wealth (CW) increases at a multiple rate, i.e.
Τ v s is the increasing factor. So after n periods, the final cumulative wealth is where 0 W is the initial wealth. In this paper, for the convenience of calculation, it is assumed that 0 1 W = . Then the final cumulative wealth n W is The ultimate purpose of PS system is to maximize the final cumulative wealth n W by constructing a set of the portfolio vectors { } 1 From the above equation, this is equivalent to maximizing the increasing factor t t Τ v s . Note that this optimization problem does not require statistical assumption about the changes in asset price.

Related Work
In this subsection, some classical prediction strategies are introduced to help us understand how to build a PS system.
The UBAH strategy [5], which is generally used as a market strategy to generate a market index, is to start with an equal distribution of wealth among d assets Both OLMAR [10] and RMR [8], which keep a moderate attitude, use the mean reversion phenomenon to predict the future asset prices. OLMAR points out that the future asset prices would recover to historical moving average and proposed the exponential moving averages (EMA). The EMA exploits all historical price information to achieve price prediction. The specific representation of EMA is as follow: where t EMA represents the previous EMA and 1 is a d-dimensional vector with components of 1. 0 1 α < < is a decaying factor and t s is the real price relative on the tth period.
When expanding t EMA , then (8) as a result, EMA really makes full use of all historical prices and gives lager weight to more recent price information. Unlike OLMAR, RMR no longer uses the simple mean, but instead exploits the robustness of L 1 -median [26] [27] to predict the future asset prices. Statistically speaking, the L 1 -median has a more attractive property than the simple mean because it's breakdown point is 0.5, meaning that when 50% of the points in the data set are pollution values, the L 1 -median can take values that exceed all boundaries. A higher the breakdown point means a more stable estimator, and the breakdown point of the simple mean is 0. The corresponding future price relative of RMR is ( ) where ( ) where ⋅ represents the Euclidean norm. OLMAR and RMR exploit the same optimization approach to update strategies as follows: EMA and L 1 -median essentially exploits the principle of mean reversion. They are cautious in their price prediction. However, there are plenty of evidence in real financial markets that irrational investment can keep prices trends. Therefore, the importance of trend-following strategies should not be ignored. In real financial markets, most investors profit from rising prices. So they are more concerned about recent maximum prices. PPT system suggests using the PPs from different asset prices within a time window [6].
and , 1 P t+ s  can also be understood as the growth potential of the assets.
EMA and L 1 -median belong to the trend-reversing, both of which are conservative and moderate investment strategies. In contrast, PP is an active and aggressive strategy as it belongs to the trend-following. Depending on the financial environment, sometimes aggressive strategies are needed to achieve high returns, while sometimes moderate strategies are needed to avoid risks. All of these motivate us to construct a comprehensive price prediction system that can effectively integrate the advantages of different strategies.

Novel RBF System Based on IMQ Function
The classical expression of RBF system is as follow: where x and y are input and output respectively, Besides GA, IMQ is another radial basis function that cannot be ignored. Here, a novel RBF system based on IMQ radial basis function is constructed, that is There is a theoretical basis for this improvement. GA and IMQ are essentially the same and both are positive definite functions [28]. However, in practical application, IMQ performance is more stable and better than GA [14] [19].
As for the formula in this paper, H price prediction strategies are denoted as where d ∆ is defined as Equation (2) in Equation (15), , represents the increasing factor of the hth price prediction strategy of the ( ) t k − th period and t k − s is the actual price relative generated by Equation (1). The method is firstly , , [31]. Then it ex- s  is select the best-performing strategy, where the best-performing strategy means that it can get the highest return even in the worst financial environment. The general process is that the smallest increasing factors of each strategy is selected within a time window, and then the largest increasing factor is selected from a set composed of the smallest increasing factors. This approach ensures that we get the best price prediction strategy in the worst trading environment, which is the key to improving the overall robustness of the system.
In the novel RBF system mentioned in Equation (14), all qualified portfolios calculated by Equation (15) serve as centers of the novel RBF system, and the best strategy , 1 t * + s  serves as fixed inputs. The specific form of novel RBFs in Equation (14) is transformed into where 1 t + ∆v represents the updated increment of the portfolio on the ( ) 1 t + th period and The reason for this representation of will be explained in the later Section 3.3. From the above formula, it can be seen that the system quantifies the similarity degree between , 1 The proposed novel RBF system in Equation (17) is different from in Equation (13) in many different ways which are mainly reflected in the problem set-ting, data characteristics and the selection of radial basis functions.
2) Although the inputs of those two RBF systems are fixed, , 1 t * + s  is determined by the recent investing performance of all prediction strategies. This means that each price prediction strategy is likely to be an input.
3) The basis functions of those two RBF systems are different. The system in Equation (13) uses Gaussian radial basis function, while Equation (17) uses the IMQ radial basis function.
4) The objective of Equation (13) is to fit y . So the back-propagation methods can be used to solve the problem [29] [30]. However, the objective of Equation (17) aims to maximize the generalized increasing factor and the solution method is different from Equation (13).

Comprehensive Price Prediction System
In order to apply the theory to practice, the next step is to construct a PS model using the proposed novel RBF system. As described in Section 2.1, in order to obtain better investing performance, the increasing factor where tr is the trace operator, V is an 3-dimensional vector with component v , Ψ is a diagonal matrix with ψ as diagonal elements, Compared with the classical form of the increasing factor the reason for this simplification is that ˆt v is fixed and ˆ0 t Therefore, the optimization goal is now switching from to the update in-

Solution Algorithm
In this subsection, the solution algorithm of CPP is introduced in detail, which has briefly concluded in Proposition 1. It is worth noting that our solution is suboptimal, not only because there is a certain bias in estimating the future with historical data but also to avoid over-fitting. So it's not necessary to get the optimal solution.
, we make Proof. The first step is to prove that 1 t + u satisfies all constraints in Equation (19), that is where 3= U u1 , On the one hand, the right side of the Equation (23) The second equation is derived from the idempotent of On the other hand, the left side of the Equation (23) can be converted to ( ) The inequality is derived from Cauchy-Schwarz inequality.
Take Equation (23), Equation (25) and Equation (26) into consideration, we have ( ) Hence, we can deduce ˆε < u . This contradicts ˆε ≤ u . It is proved that the optimization problem in Equation (19) obtain the maximum at This shows that According to Equation (25) and Equation (29), we obtain ε * > u . This contradicts the constraint ε * < u .
To sum up, we can get that d d where ε  is a mapping projected onto the ε -Euclidean ball. Compared with Equation (17), in order to satisfy the constraint conditions in Equation (19), 1 t + u adds two operators on the basis of 1 t + ∆v . Thus, this is also explains why the update increment of the portfolio is represented as Equation (17).
Then, the portfolio 1t + v on the next period is shown as follows: The complete CPP system is outlined in Algorithm 1. CPP is a fast algorithm because it only uses ordinary matrix calculation without any iterative calculation, which significantly reduces the operation time.

Data Sets and Comparison Approaches
In this subsection, in order to comprehensively assess the performance of sys-  [32], SSPO [33] and AICTR [34], to compete with CPP that we proposed.
A detailed descriptions of these systems are as follow.
1) RMR: RMR uses L 1 -median to make price prediction as describled in Section 2.2.
2) OLMAR: OLMAR uses the moving averages to make price prediction, which are described in Section 2.2.
3) PPT: PPT is an aggressive strategy that uses the maximum values of different assets to make price prediction as mentioned in Section 2.2. 1 . In general, the parameters of CPP are determined based on the results of final cumulative wealth (CW), operating in the same way as previous studies. The calculation of final CW is described in detail in Section 2.1. First, we set the window size 5 ω = , which is widely used and consistent with other systems.
Secondly, we change one parameter to fix the other parameters for the experiments. Since ε is the updating strength, it is roughly estimated to be larger value, while 2 h σ is a parameter used to evaluate the difference between two portfolios, it should be a small value. On the one hand, we firstly set 5 ω = , 1400 ε = , and then set 2 h σ change between 0.0007 and 1.0012. According to the results in Figure 1, the investing performance of CPP around 2 0.0008 h σ = is stable and good. On the other hand, we firstly set 5 ω = , 2 0.0008 h σ = and then make ε change between 1200 and 1700. The results are shown in Figure   2 and we know that CPP is stable and good around 1400 ε = . Therefore, the parameters of CPP are set as: 2 0.0008

Experimental Results
In this paper, a scheme containing seven evaluation indicators are designed to assess the performance of different systems and achieve the most excellent results. These seven indicators can be roughly divided into three categories, namely investing performance, risk metrics and application issues. Investing performance M. M. Zheng Applied Mathematics   includes CW, mean excess return (MER) and α Factors. Risk metrics consist of sharpe ratio (SR) and information ratio (IR). As for application issues, we chose transaction cost and running times to assess them. Those indications will be discussed in the following subsection.  16, 9.59), respectively. In addition, DJIA is a challenging data set because many systems do not perform well in this data set, such as PPT, and OLMAR. But CPP can reach a value of 4.74, which is 53.90% higher than PPT. These results shows that CPP is an effective PS system and can accumulate more wealth in the real financial market.

Investing
In order to show the superiority of the system CPP, the CWs of each system on DJIA is plotted in Figure 3. By observing the Figure 3, the excellent investing performance of CPP can be shown more intuitively.
2) MER: Return is a financial term that describes the proportion of wealth by a PS system gained or lost over one investing period. In this paper, the daily return r present the daily returns of attended PS system and the market baseline on the tth period, respectively. Note that, the UBAH system is defined as the market baseline.
The MER results from various PS systems are described in Table 2 tively. In addition, a small MER gap is likely to produce a lager CW gap in the long-term. Therefore, the above results demonstrate that CPP can achieve an outstanding investing performance.
3) α Factor: MER measures the investing performance of a PS system without considering market risks. However, in the real financial market, the volatility of the market will undoubtedly affect the performance of assets. Capital asset pricing model(CAPM) [36] points out that the expected return sources of PS systems can be divided into two parts: the first part comes from the market return, and the second comes from the inherent excess return, also called α Factor [37] [38]. Therefore, α Factor able to evaluate the investing performance of different PS systems:  Table  3. CPP achieved the highest α on three data sets and ranked second on the NYSE(N). For example, CPP (0.0042, 0.0002) achieves a higher α , compared show that CPP is still able to achieve higher inherent excess return in the face of market volatility. In addition, the statistical t-test is used to determine whether α is significantly lager than 0, proving that the inherent excess return is not achieved by luck. The results of p-value presented in Table 3   The results are presented in Figure 4. CPP achieves the highest CW on all data sets when the transaction cost rate γ fluctuates between 0% and 0.15%. In addition, even when γ reaches a high value ( 0.15% 0.5% γ ≤ ≤ ), CPP still outperforms other PS systems on three data sets. Therefore, it shows that CPP can bear moderate transaction costs and can be applied in real world financial markets.
2) Running Times: Running Time is an important indicator to judge whether a system can be applied in a large-scale and time-limiting environment, such as High-Frequency Trading (HFT) [42]. We use a regular computer equipped an and HS300, respectively. Therefore, CPP has good computational efficiency and can be applied in large-scale financial markets.

Conclusion
In this paper, we proposed a new CPP system based on IMQ radial basis function with an integration of three different aggressive and moderate strategies for effective and robust PS. Instead of using a traditional GA function, here we chose a more stable and accurate function that is IMQ for the novel RBF system, which centers on multiple strategies. With regard to portfolio update, different from the traditional increasing factor, we propose a generalized growth factor based on a kernel and trace operation. And CPP is fast and can be applied in larger scale and limited time financial environment. Extensive experiments are performed on 4 worldwide benchmark data sets to indicate that CPP can effectively integrate the advantages of different strategies and it was proved to be effective in PS. On the one hand, in most cases CPP outperforms other commonly used systems in performance indicators CW, MER and α Factor. On the other hand, CPP achieves the highest SR and IR compared with other systems.
The results show that CPP has not only excellent investing performance but also good risk control ability. In addition, CPP can withstand reasonable transaction costs and fast operation, which have to be considered in the real financial market. In conclusion, CPP is an efficient and robust PS system and deserves further investigation. Of course, the CPP system has its own shortcomings. On one hand, the problem of reducing transaction cost was not considered at the initial stage of modeling. On other hand, this paper considers only the three strategies. In the future, we can improve the performance of the system from these two aspects.

Acknowledgements
Thanks to Jinan University for the resources provided, and to tutors for their valuable advice on this manuscript. This research is supported by the National

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.