A quantum approach for time series data based on graph and Schrödinger equations methods

Transcription

1 Modern Physics Letters A Vol. 33, No. 35 (2018) (23 pages) c World Scientific Publishing Company DOI: /S A quantum approach for time series data based on graph and Schrödinger equations methods Pritpal Singh Smt. Chandaben Mohanbhai Patel Institute of Computer Applications, Charusat Campus, Changa, Anand , Gujarat, India pritpalsingh.mca@charusat.ac.in Gaurav Dhiman and Amandeep Kaur Computer Science and Engineering Department, Thapar Institute of Engineering & Technology, Patiala , Punjab, India gaurav.dhiman@thapar.edu kaur.amandeep@thapar.edu Received 10 July 2018 Revised 17 August 2018 Accepted 11 September 2018 Published 24 October 2018 The supremacy of quantum approach is able to solve the problems which are not practically feasible on classical machines. It suggests a significant speed up of the simulations and decreases the chance of error rates. This paper introduces a new quantum model for time series data which depends on the appropriate length of intervals. To provide effective solution of this problem, this study suggests a new graph-based quantum approach. This technique is useful in discretization and representation of logical relationships. Then, we divide these logical relations into various groups to obtain efficient results. The proposed model is verified and validated with various approaches. Experimental results signify that the proposed model is more precise than existing competing models. Keywords: Time series; Quantum; Cluster; Particles. PACS Nos.: Ac, Lx 1. Introduction Various applications of the fuzzy set theory 1 can be found in these papers This theory also has the capability to deal with uncertainties involved in time series events and in optimization process For example, sudden rise and fall of Corresponding author

2 P. Singh, G. Dhiman & A. Kaur daily temperature, sudden increase and decrease of daily stock index price, sudden increase and decrease of rainfall amount indicate the uncertain nature of events, which cannot be described accurately. 15 In fuzzy-based modeling approach, the time series values are represented as fuzzy variables, and employed in modeling and simulation using a concept which is called as the fuzzy time series (FTS). Based on this concept, Song and Chissom 16 initially introduced their first model in 1991 to deal with the uncertainty and imprecise knowledge contained in time series data. Later, Song and Chissom 17 modified their model to obtain higher forecasting accuracy in time series forecasting. Further improvement in Song and Chissom s model 17 was done by Chen 18 in Forecasting result obtained from this model 18 was far better than the previous existing models that were proposed by Song and Chissom. 17,19,20 Recently, many studies provided some improvements in Chen s 18 model in terms of fuzzification, 21 establishment of fuzzy logic relationships (FLRs), 22 and defuzzification to improve the predictive skill of the FTS modeling approach. Therefore, researchers tried to make advancement in these studies using different approaches However, accuracy of FTS modeling approach is mostly depended on defining the effective length of intervals. Various experiments of the researchers in this domain have evinced the significance of defining the effective length of intervals. In FTS modeling approach, this length of intervals is defined by partitioning the universe of discourse into several equal or unequal length of intervals. Cheng et al. 33 demonstrated this effect of partitioning the universe of discourse into several equal or unequal length of intervals in their experimental results. However, most of the researchers use equal length of intervals in time series forecasting. For example, researchers partitioned the universe of discourse into equal length of intervals for the formulation of FLRs. To resolve this issue, Singh and Borah 37 proposed algorithms, which initially partition the universe of discourse into two different parts, then generate the two different sets of intervals having unequal length of intervals. Huarng and Yu 38 proposed a ratio-based interval length determination technique. Their empirical analyses suggested that the ratio-based approach is more robust to represent the observations instead of equal length intervals. Wang and Liu 39 also used the unequal length of intervals in forecasting the time series data set. Recent applications of FTS modeling approaches can be found in these references. 7,40 42 This study presents a new quantum model to deal with the forecasting problems of time series data set based on the FTS modeling approach. The main inducement of this research initiates from the study of drawbacks of various existing FTS models. It is obvious from the literature review (as presented in the previous section) that the performance of the FTS modeling approach predominantly depends on determination of effective length of intervals. 43 For the fuzzification of historical time series data, these effective length of intervals also play a significant role. Previously, most of the existing FTS models 21,44 maintained the fixed length of intervals without any proper justification. To determine the effective length of intervals, historical time series data set should be discretized very sensitively. In these papers, 36,37,

3 A new quantum fuzzy time series forecasting model researchers proposed different time series data clustering algorithms. These clustering algorithms are successfully applied in FTS modeling approach to determine the effective lengths of intervals. However, these algorithms 36,37,40 are simply based on the statistical analyses of the historical time series data set. It cannot consider the relationships existing among the values of historical time series data set. Therefore, to study such kind of relationships among the time series values, it has been decided to use graph-based approach that would provide the facility of data abstraction as well as data clustering. Therefore, in this study, a new quantum graph-based clustering technique has been proposed, which would have such kind of facilities. We have entitled this technique as Node-Based Time Series Data Clustering Technique (NBTSDCT). This technique can represent the individual time series values in terms of nodes using parent child relationship, by providing the facility of data abstraction. Henceforth, this technique makes grouping of nodes with some pre-defined rules, by providing the facility of data clustering. This technique is able to generate the appropriate number of clusters. These clusters are further used for defining the effective length of intervals. Based on these intervals, adopted historical time series data sets are fuzzified. These fuzzified time series values are used to establish the fuzzy logic relationship groups (FLRGs) from various FLRs. Finally, these FLRGs are used to obtain the forecasting results based on the centroid defuzzification method Organization of the paper This paper is organized as follows. In Sec. 2, the theory of FTS followed by problem definition is discussed. In Sec. 3, a new quantum-based time series data clustering technique is introduced. Description of the proposed model is provided in Sec. 4. Experimental results are discussed in Sec. 5. The time complexity of the proposed models is discussed in Sec. 6. In Sec. 7, conclusions are discussed. 2. Preliminaries and Problem Definition In this section, a few basic concepts of FTS is reviewed from this paper. 17 In FTS modeling approach, each time series value is represented by the fuzzy linguistic variable. Definition 1 (FTS). 17 Let M(t) (t =..., 0, 1, 2,...) R, and can be considered as the universe of discourse on which fuzzy sets µ i (t) (i = 1, 2,...) be defined. Let G(t) be a collection of µ i (t) (i = 1, 2,...). Then, G(t) is called a FTS on M(t) (t =..., 0, 1, 2,...). Definition 2 (FLR). 17 Consider that G(t 1) = A i and G(t) = A j, where G(t) is assumed to be caused by G(t 1). The relationship between G(t) and G(t 1) is termed as a FLR between A i and A j, which is defined as A i A j, (1)

4 P. Singh, G. Dhiman & A. Kaur where A i and A j are termed as left-hand side (LHS) and right-hand side (RHS) of the FLR A i A j, respectively. Definition 3 (FLRG). 17 Consider the following m-numbers of FLRs, as A i A k1, A i A k2, A i A km. A FLRG can be defined by considering the same fuzzy linguistic variable on the LHS of FLRs. Hence, the FLRs as shown above can be grouped into a FLRG, as follows: A i A k1, A k2,..., A km. (2) In this FLRG, LHS relation is termed as previous state, whereas RHS relation is termed as current state. 3. A Quantum Graph-Based Approach for Time Series Data Clustering For simplicity, we consider a search agent in two-dimensional search space and solve the P(x + 1) = C h /N using Schrödinger equation as follows: q i = φ Q ij + (1 φ) Q hi, 0 < φ < 1, (3) D(x) = P (x) = 1 L e 2/L, (4) x P (x)dx = e 2/L, (5) where L represents the search scope of each particle. Now, obtain the position of particles using Monte Carlo method as y = q ± L ln(1/v), v U(0, 1), (6) 2 where v is a random number in range lies in range of [0, 1]. Now, compute the distance between particles as L = 2 q y, (7) y = q ± α q y ln(1/v), (8) where α is the parameter of the algorithm. In this study, we present a new quantum-based time series data clustering technique, which can be considered as a class of graph-based approach. In this approach, each node is represented by a time series value, and these nodes are connected by

5 A new quantum fuzzy time series forecasting model Algorithm 1 Finding Root Node Algorithm (FRNA) 1: Input the historical time series data set, as S = {X 1, X 2,..., X N }. (9) Here, each X i (i = 1, 2,..., N) represents the historical time series value. 2: Calculate the range, R g, as R g = MAX value MIN value. (10) Here, MAX value and MIN value represent maximum and minimum values of the sample S, respectively. 3: Calculate the standard deviation (SD), of the sample, S. 4: Calculate width, W, as W = Rg SD N. (11) 5: Compute the universe of discourse, U, of the sample, S as U = [L bound, U bound ]. (12) Here, L bound = MIN value W and U bound = MAX value + W. 6: Calculate mid-point, X mid, as X mid = L bound + U bound 2 7: Assign the X mid as a root node (T root ): y, y [0.5, 1]. (13) T root = X mid. (14) lines or arcs, which are called edges. Hence, this approach of clustering is referred as NBTSDCT. The technique automatically generates the clusters without any supervision. The NBTSDCT represents the data set in the form of a tree by which the representation and visualization of data set is very easy. The proposed NBTSDCT comprises four parts, viz., Finding Root Node Algorithm (FRNA), Tree Construction Algorithm (TCA), Node Insertion Algorithm (NIA) and Node Clustering Algorithm (NCA). All these algorithms are presented next. Explanation for the FRNA: This algorithm is presented as Algorithm 1. To demonstrate the applicability of this technique, the New York Stock Exchange (NYSE) data set, 45 as shown in Table 1, is employed. This algorithm inputs the data set, and helps to find the root node. A numerical example for this algorithm is presented in Appendix A. Explanation for the TCA: This algorithm is presented as Algorithm 2. With the help of input data set and root node, this algorithm helps to represent a tree. An example for this algorithm is presented in Appendix A

6 P. Singh, G. Dhiman & A. Kaur Table 1. Data set of NYSE. Date (mm/dd/yyyy) Actual price 1/11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /15/ /16/ /17/ /12/ /13/ Algorithm 2 Tree Construction Algorithm (TCA) 1: Input: T root, S 2: for i = 1 to N do 3: NIA(T root, X i ) 4: end for Algorithm 3 Node Insertion Algorithm (NIA) 1: Input: T root, X i 2: if (X i < T root ) 3: if T root.left NULL 4: Call: NIA(T root.left, X i ) 5: else 6: T root.left = NULL 7: end if 8: else if(x i > T root ) 9: if T root.right NULL 10: Call: NIA(T root.right, X i ) 11: else 12: T root.right = NULL 13: end if 14: end if Explanation for the NIA: This algorithm is presented as Algorithm 3. This algorithm helps to insert a time series value into a particular node with its proper position in the tree. Basic rules for this process are presented in Appendix A

7 A new quantum fuzzy time series forecasting model Algorithm 4 Node Clustering Algorithm (NCA) 1: Input: T root 2: if(t root = NULL) 3: No Tree Found 4: return 5: else if T root.right NULL && T root.left NULL 6: if T root is not presented in Cluster 7: mindiffnode = makediff (T root, T root.right, T root.left) 8: makecluster(t root, mindiffnode) 9: if mindiffnode = T root.right 10: if ((T root.right).left NULL) 11: add (T root.right).left child in the Cluster 12: end if 13: if ((T root.right).right NULL) 14: Call: NCA((T root.right).right) 15: end if 16: Call: NCA(T root.left) 17: else if((t root.left).left NULL) 18: Call: NCA((T root.left).left) 19: end if 20: if((t root.left).right NULL) 21: add ((T root.left).right) child in the Cluster 22: end if 23: Call: NCA(T root.right) 24: end if 25: end if 26: else ift root.right NULL && T root.left == NULL 27: ift root is not presented in Cluster 28: makecluster(t root, T root.right) 29: if((t root.right).left NULL) 30: add (T root.right).left child in the Cluster 31: end if 32: if((t root.right).right NULL) 33: Call: NCA((T root.right).right) 34: end if 35: end if 36: else ift root.right == NULL && T root.lef T NULL 37: ift root is not presented in Cluster 38: makecluster(t root, T root.left) 39: if((t root.left).left NULL) 40: Call: NCA((T root.left).left) 41: end if 42: if((t root.left).right NULL) 43: add ((T root.left).right) child in the Cluster 44: end if 45: end if 46: else if T root is not presented in the Cluster 47: makecluster(t root) 48: end if 49: return 50: end if

8 P. Singh, G. Dhiman & A. Kaur Explanation for the NCA: This algorithm is presented as Algorithm 4. This algorithm helps in defining clusters with the help of nodes associated with the tree. Basic steps for making clusters are presented in Appendix A. All the clusters which are evolved from the above technique are shown in Table 2. In this table, lower and upper bounds of the intervals are formed from the minimum and maximum values of the corresponding clusters, respectively. Table 2. Clusters, their corresponding elements, intervals and mid-points. Cluster Corresponding element Initial interval Mid-point C 1 ( , , , ) a 1(0) = [ , ] C 2 ( , ) a 2(0) = [ , ] C 3 ( , , , , ) a 3(0) = [ , ] C 4 ( , ) a 4(0) = [ , ] C 5 ( , , ) a 5(0) = [ , ] C 6 ( , , , ) a 6(0) = [ , ] C 7 ( , ) a 7(0) = [ , ] C 8 ( , , , , a 8(0) = [ , ] , , ) C 9 ( , ) a 9(0) = [ , ] C 10 ( , , ) a 10(0) = [ , ] C 11 ( , , , ) a 11(0) = [ , ] C 12 ( , , , ) a 12(0) = [ , ] C 13 ( , ) a 13(0) = [ , ] The Proposed FTS Model This section presents the steps for modification and detailed descriptions of the proposed model Modification in existing FTS model To deal with the forecasting problems of FTS modeling approach, researchers use the following six basic steps. 46 Step 1. Partition the universe of discourse into various equal length of intervals. Step 2. For each of the evolved intervals, define fuzzy linguistic variables. Step 3. Fuzzify the historical time series data set. Step 4. Represent the fuzzified time series values into various FLRs. Step 5. Define the FLRGs from the FLRs. Step 6. Defuzzify the fuzzified time series values from the FLRGs. In this study, above Step 1 is modified by integrating the NBTSDCT, which helps to define the intervals from the universe of discourse of the historical time series data set. Later, these evolved intervals are used for the fuzzification and defuzzification purposes. Descriptions of the proposed model are provided next

9 A new quantum fuzzy time series forecasting model 4.2. Detailed descriptions of the proposed model In this section, the proposed model is elucidated. Each aspect of the model is explained using the New York Stock Exchange (NYSE) data set 45 from the period 11/1/ /31/1999 (format: mm/dd/yyyy). A sample of this data set is listed in Table 1. Each phase of the model is explained next. Step 1. Provide the boundary for the historical time series data set by defining the universe of discourse, and clustered them using the NBTSDCT. Explanation: The proposed NBTSDCT is initially applied to the NYSE data set to obtain the clusters. From each evolved cluster, corresponding interval is defined. The outcomes of this technique are depicted in Table 2. Step 2. Define fuzzy linguistic variable A i, for each of the defined intervals. It is assumed that if any historical time series data set is distributed among n initial intervals (i.e. a 1 (0), a 2 (0),..., a n (0)), then define n fuzzy linguistic variables A 1, A 2,..., A n. It can be represented by fuzzy sets, as follows: A 1 = 1/a 1 (0) + 0.5/a 2 (0) + 0/a 3 (0) + + 0/a n 1 (0) + 0/a n (0), A 2 = 0.5/a 1 (0) + 1/a 2 (0) + 0.5/a 3 (0) + + 0/a n 1 (0) + 0/a n (0), A 3 = 0/a 1 (0) + 0.5/a 2 (0) + 1/a 3 (0) + + 0/a n 1 (0) + 0/a n (0),. A n = 0/a n (0) + 0/a 2 (0) + 0/a 3 (0) /a n 1 (0) + 1/a n (0). Here, maximum degree of membership of fuzzy set A i occurs at interval a i (0), and 1 i n. Explanation: For the NYSE data set, total 13 intervals are evolved, therefore, total 13 different fuzzy linguistic variables are defined as A 1, A 2,..., A 13. Representation of these fuzzy linguistic variables are shown as follows: A 1 = 1/a 1 (0) + 0.5/a 2 (0) + 0/a 3 (0) + + 0/a n 1 (0) + 0/a 13 (0), A 2 = 0.5/a 1 (0) + 1/a 2 (0) + 0.5/a 3 (0) + + 0/a n 1 (0) + 0/a 13 (0), A 3 = 0/a 1 (0) + 0.5/a 2 (0) + 1/a 3 (0) + + 0/a n 1 (0) + 0/a 13 (0),. A 13 = 0/a 1 (0) + 0/a 2 (0) + 0/a 3 (0) /a n 1 (0) + 1/a 13 (0). Here, maximum degree of membership of fuzzy set A i occurs at interval a i (0), and 1 i 13. (15) (16) Step 3. Fuzzify each of the historical time series values. It is considered that if one day s time series value belongs to the interval a i, then it is fuzzified as A i, where 1 i n

10 P. Singh, G. Dhiman & A. Kaur Table 3. Fuzzified NYSE data set. Actual Actual Actual stock Fuzzified stock Fuzzified stock Fuzzified Date price stock Date price stock Date price stock 1/11/ A 2 11/22/ A 8 12/14/ A 8 2/11/ A 1 11/23/ A 6 12/15/ A 11 3/11/ A 1 11/24/ A 6 12/16/ A 11 4/11/ A 2 11/26/ A 5 12/17/ A 11 5/11/ A 2 11/29/ A 5 12/20/ A 8 8/11/ A 2 11/30/ A 4 12/21/ A 8 9/11/ A 2 1/12/ A 6 12/22/ A 8 10/11/ A 1 2/12/ A 7 12/23/ A 12 11/11/ A 1 3/12/ A 11 12/27/ A 12 12/11/ A 2 6/12/ A 8 12/28/ A 12 11/15/ A 2 7/12/ A 8 12/29/ A 13 11/16/ A 5 8/12/ A 8 12/30/ A 12 11/17/ A 4 9/12/ A 8 12/31/ A 13 11/18/ A 7 10/12/ A 8 11/19/ A 6 12/13/ A 8 Explanation: In order to fuzzify each of the historical time series values, it is required to obtain the degree of membership value of each observation associated with each A j (j = 1, 2,..., n), for each day. Since, the maximum membership value of one day s observation will be at interval a i (0) and 1 i n, therefore, corresponding observation is fuzzified, as: A i. For example, the stock exchange value for the day 11/26/1999 is , which belongs to the interval a 5 (0). Hence, the stock exchange value is fuzzified, as: A 5. All these fuzzified NYSE values are shown in Table 3. Step 4. Define the FLRs between the fuzzified time series data set. Explanation: Based on the definition of FLR, we can establish an FLR between two consecutive fuzzified NYSE values. For example, in Table 3, fuzzified NYSE values for the days 4/11/1999 and 5/11/1999 are A 2 and A 2, respectively. So, an FLR can be established between A 2 and A 2, as: A 2 A 2. In this way, each of the FLRs are obtained from the fuzzified NYSE data sets. All these FLRs are listed in Table 4. Step 5. Evolve the FLRGs from the FLRs. Explanation: From Definition (3), an FLRG in the form of A i A m, A n (i.e. Group i) indicates that it has the following FLRs: A i A m, A i A n. In Table 4, the FLRG in the form of A 5 A 4, A 5 (i.e. Group 4) has the following FLRs as A 5 A 4, A 5 A

11 A new quantum fuzzy time series forecasting model Table 4. FLRs of NYSE data set. FLR FLR FLR A 2 A 1 A 6 A 8 A 8 A 8 A 1 A 1 A 8 A 6 A 8 A 8 A 1 A 2 A 6 A 6 A 8 A 11 A 2 A 2 A 6 A 5 A 11 A 11 A 2 A 2 A 5 A 5 A 11 A 11 A 2 A 2 A 5 A 4 A 11 A 8 A 2 A 1 A 4 A 6 A 8 A 8 A 1 A 1 A 6 A 7 A 8 A 8 A 1 A 2 A 7 A 11 A 8 A 12 A 2 A 2 A 11 A 8 A 12 A 12 A 2 A 5 A 8 A 8 A 12 A 12 A 5 A 4 A 8 A 8 A 12 A 13 A 4 A 7 A 8 A 8 A 13 A 12 A 7 A 6 A 8 A 8 A 12 A 13 Table 5. Group FLRGs of NYSE data set. FLRG Group 1 A 1 A 1, A 2 Group 2 A 2 A 1, A 2, A 5 Group 3 A 4 A 6, A 7 Group 4 A 5 A 4, A 5 Group 5 A 6 A 5, A 6, A 7, A 8 Group 6 A 7 A 6, A 11 Group 7 A 8 A 6, A 8, A 11, A 12 Group 8 A 11 A 8, A 11 Group 9 A 12 A 12, A 13 Group 10 A 13 A 12 A list of FLRGs obtained from the NYSE data set is shown in Table 5. For the convenience in simulation, repeated FLRs are not considered in the FLRGs. Step 6. Defuzzify the historical fuzzified time series data set, and obtain the forecasted values. Following two principles are defined to defuzzify the fuzzified time series data set. Based on the application of technique, it is categorized, as: Principles 1 and 2. Steps involved in Principle 1, are explained next. Principle 1: If the required forecasting day is D(t), then the corresponding fuzzified value for the day D(t 1) is required, where t represents the current day, which we want to forecast. Principle 1 is applicable only if there are more

12 P. Singh, G. Dhiman & A. Kaur Table 6. Actual vs. forecasted NYSE price. Year Actual NYSE Forecasted NYSE 1/11/ /11/ /11/ /11/ /11/ /27/ /28/ /29/ /30/ /31/ RMSE 67.10% than one fuzzified values available in the current state. The steps under Principle 1 are explained next. Step 1. Derive the fuzzy linguistic variable for the day D(t 1), as: A i (i = 1, 2, 3..., n). Step 2. Based on this derived fuzzy linguistic variable A i, obtain the FLRG whose previous state is A i (i = 1, 2, 3..., n), and the current state is A k, A s,..., A n, i.e. the FLRG is in the form of A i A k, A s,..., A n. Step 3. Obtain the intervals where these fuzzy linguistic variables A k, A s,..., A n, are associated with maximum degree of memberships. Assume that these corresponding intervals be a k, a s,..., a n. It is also assumed that these intervals have the corresponding mid-points, as: C k, C s,..., C n. Step 4. Based on the following formula, calculate the forecasted value for the day D(t), as: Forecast(t) = C k + C s + C n n, (17) where n is the number of mid-points used in the computation. Principle 2: This principle is useful if there is only one fuzzified value available in the current state. Steps involved in Principle 2, are explained next. Step 1. Derive the fuzzy linguistic variable for the day D(t 1), as: A i (i = 1, 2, 3..., n). Step 2. Based on this derived fuzzy linguistic variable A i, obtain the FLRG whose previous state is A i (i = 1, 2, 3..., n), and the current state is A j (j = 1, 2, 3..., n), i.e. the FLRG is in the form of A i A j. Step 3. Obtain the interval where the fuzzy linguistic variable A j, is associated with maximum degree of membership. Assume that this corresponding interval be a j. It is also assumed that this interval has the corresponding mid-point as C j. Hence, this mid-point C j is the forecasted value for the day D(t)

13 A new quantum fuzzy time series forecasting model Hence, the proposed FTS model is applied to obtain the forecasted values for the NYSE data set. The performance of the model is evaluated using the RMSE (refer to Appendix B). The RMSE value for the forecasted NYSE data set is presented in Table Experimental Results In this section, various experimental results, statistical analyses along with comparison study results are discussed Empirical analysis for the NYSE data set To assess the performance of the proposed model, its forecasted results are compared with various existing FTS models 18,45,47 49 in terms of the RMSE. From Table 7, it is obvious that the proposed FTS model gets lower RMSE in comparison to the competing models Statistical analysis of the forecasting results Forecasted result obtained for the NYSE data set 45 is analyzed with various statistical parameters, viz., means, root mean square error (RMSE) and Theil s U Statistic (refer to Appendix B). The empirical results are depicted in Table 8. From Table 8, it is clear that the mean of observed values is closed to the mean of forecasted values. Forecasted results in terms of RMSE indicate very small error rate. In Table 8, U value is closer to 0, which indicates the effectiveness of the proposed model. The means, RMSE, and Theil s U Statistic values for the FTS-RBIFA model is also compared with the existing FTS models. 18,45,47 49 It indicates that the proposed models is also statistically significant in comparison to the existing FTS models. 18,45,47 49 Furthermore, the proposed model is applied to forecast the stock index prices (closing) of different historical time series data sets, such as Yahoo Inc., IBM and Oracle Corporation. These data sets are collected from the website: The performance of the proposed hybrid model is evaluated with various statistical parameters, as shown in Table 9. From Table 9, all the statistical parameters signify the effectiveness of the FTS-RBIFA model Comparison between existing models and the proposed model In this subsection, we present the forecasted results for the stock index data set of TAIFEX (from 3/8/ /9/1998). These results are obtained using the FTS-RBIFA model. Forecasting results are compared with the existing models. 18,43,47,50 55 The performance of the forecasting results is evaluated using the AFER (refer to Appendix B). From Table 10, it is obvious that the proposed model outperforms the existing models 18,43,47,50 55 in terms of the AFER

14 P. Singh, G. Dhiman & A. Kaur Table 7. A comparison of the existing models with the proposed FTS-RBIFA model for the NYSE data set. Stock Chen 18 Yu 47 Teoh et al. 45 QPSO 48 QGA 49 Proposed FTS Date index model model model model model model /15/ /16/ /17/ /18/ /19/ /22/ /23/ /24/ /26/ /29/ /30/ /13/ /14/ /15/ /16/ /17/ /20/ /21/ /22/ /23/ /27/ /28/ /29/ /30/ /31/ RMSE 85.28% % % % % 67.90% 5.4. In-sample and out-sample forecasting using large data sample In this subsection, we present the forecasted results using the stock index data set of SBI (from 1/ /30/2016). This data set contains total 4024 time series values. From this data set, final price of SBI stock index data set is tried to be

15 A new quantum fuzzy time series forecasting model Table 8. Statistical comparison of forecasting results obtained from the FTS-RBIFA model and the existing FTS models for the NYSE data set. Statistics Chen 18 Yu 47 Teoh et al. 45 QPSO 48 QGA 49 Ā observed (in dollar) Ā forecasted (in dollar) Forecasted RMSE (in dollar) U Table 9. Evaluation of forecasting results obtained from the FTS-RBIFA model in terms of statistical parameters. Yahoo Inc. IBM Oracle Corp. Statistics 1/6/ /7/2015 1/6/ /7/2015 1/6/ /7/2015 Ā observed (in dollar) Ā forecasted (in dollar) Forecasted RMSE (in dollar) U Table 10. A comparison of the AFER values with different models for forecasting the stock index data set of TAIFEX. Models AFER (TAIFEX) Grey model % Chen high-order model (3rd-order) % BPNN model % Yu model % Chen model % QPSO % QGA % Type-2 model % Fuzzy-neuro model % FTS-GA model % FTS-PSO model % predicted. To check the performance of the model, this data is classified into two parts, as: (a) period from 1/02/ /12/2016 is used for in-sample forecasting, and (b) period from 09/13/ /30/2016 is used for out-sample forecasting. By employing the NBTSDCT, this whole sample is discretized into 200 intervals. Based on these intervals, the proposed FTS-RBIFA Model is applied to obtain the forecasting results. These forecasting results are shown in Table 11. In this table, forecasting results obtained by various existing models 18,43,47,50 55 are also

16 P. Singh, G. Dhiman & A. Kaur Table 11. In-sample and out-sample forecasting using the stock index data set of SBI. Models AFER (in-sample) AFER (out-sample) Grey model % 3.28% Chen high-order model (3rd-order) % 3.29% Chen model % 2.15% Yu model % 2.18% BPNN model % 2.14% QPSO % 2.11% QGA % 2.09% Fuzzy-neuro model % 2.06% Type-2 model % 2.01% FTS-GA model % 1.91% FTS-PSO model % 1.60% presented. The performance of all these forecasting results is evaluated using the AFER (refer to Appendix B). From Table 11, it is obvious that the proposed model outperforms the existing models 18,43,47,50 55 for forecasting the stock index prices of SBI in case of both in-sample and out-sample forecasting. 6. Time Complexity Analysis of the Proposed Models This study consists of two models, viz., FTS model and FTS-RBIFA model. We will initially discuss the time complexity of the proposed FTS model followed by the FTS-RBIFA model. Hence, the time complexity of the proposed FTS model is as follows. (1) The proposed FTS model mainly consists of two parts, viz., NBTSDCT and the basic steps involved in the Chen 18 model. Hence, the time complexity of the proposed model will be the overall summation of time complexities of NBTSDCT and the basic steps involved in the Chen 18 model. Initially, the time complexity of the NBTSDCT is computed, which mainly consists of four main parts as: FRNA, TCA, NIA, and NCA. Time complexities of constituent part of NBTS- DCT, overall time complexity of NBTSDCT, time complexity of steps involved in the Chen 18 model, and overall time complexity combining NBTSDCT and Chen 18 model (i.e. the proposed model) are given as follows. The time complexity of the FRNA (O FRNA ) is as follows: O FRNA = O(N) log(n). (18) Here, N is the sample size. The time complexity of the TCA (O TCA ) is as follows: Here, N is the sample size. O TCA = O(N) log(n). (19)

17 A new quantum fuzzy time series forecasting model The time complexity of the NIA (O NIA ) is as follows: O NIA = O(log(N)). (20) Here, N is the sample size. The time complexity of the NCA (O NCA ) is as follows: O NCA = O(N). (21) Here, N is the sample size. Using Eqs. (18) (21), overall time complexity of the NBTSDCT (O NBTSDCT ) is as follows: O NBTSDCT = O FRNA + O TCA + O NIA + O NCA = O(N) log(n) + O(N) log(n) + O(log(N)) + O(N) = O(N log N). (22) The time complexity of the Chen 18 model (O chen ) is as follows: O chen = O(C). (23) Here, C is the total number of clusters. Using Eqs. (22) and (23), overall time complexity of the proposed FTS model (O FTS ) is the summation of time complexities of the NBTSDCT and the Chen 18 model, which is as follows: 7. Conclusions O FTS = (O NBTSDCT ) + (O chen ) = O(N log N) + O(C) = O(C + N log N). (24) Forecasting accuracy of the FTS modeling approach depends on determination of length of intervals. Therefore, in this study, the authors have presented a quantum technique that can determine the intervals from the historical time series data set. This technique is called the NBSCDT. In the FTS modeling approach, initially this NBSCDT is applied to obtain the clusters. Based on these evolved clusters, corresponding intervals are defined which are unequal in length. Using these intervals, various fuzzy linguistic variables are defined, which are used to fuzzify the historical time series data set. Based on these fuzzified time series data set, various FLRGs are defined, which are further employed for the defuzzification operation. This model is entitled as the FTS model. The proposed FTS model is verified with various financial time series data sets. Experimental results show that forecasting accuracy of the FTS model is not only better, but also more precise than the various existing competing models

18 P. Singh, G. Dhiman & A. Kaur Forecasting accuracy is not the only way to compare the models; however, time complexity analysis also plays a significant role. In this paper, time complexities of the proposed model are evaluated, and it is found that running time of the FTS model grows close to the linear. However, the most time consuming part in the proposed FTS model is the application of NBTSDCT (which is used for clustering time series data) for obtaining the forecasted values. The proposed FTS model has a limitation that it can only be applied in onefactor time series data sets. Researchers can improve the performance of this model by developing more robust time series data clustering algorithms. Appendix A. Numerical Example for the NBTSDCT The NBTSDCT consists of four main algorithms, as: (a) Finding Root Node Algorithm (FRNA), (b) Tree Construction Algorithm (TCA), (c) Node Insertion Algorithm (NIA), and (d) Node Clustering Algorithm (NCA). Each step of these algorithms is explained using the NYSE data set, as shown in Table 1. A.1. Finding Root Node Algorithm (FRNA) Calculations for the FRNA are as follows. Step 1. Initially, we input the NYSE as S = { , , ,..., }. Step 2. The value of R g for the considered data set can be obtained from Eq. (10), as R g = = Step 3. The value of SD for the considered data set is SD = Step 4. The value of W can be obtained from Eq. (11), as W = = Step 5. The universe of discourse, U, can be defined from Eq. (12), as U = [ , ]. Step 6. The X mid can be determined from Eq. (13), as X mid = = Step 7. The X mid value obtained from Step 6 can be considered as T root, which is

19 A new quantum fuzzy time series forecasting model Fig. A.1. A symbolic representation of a tree formed by Algorithms 2 and 3. A.2. Tree Construction Algorithm (TCA) A graph-based tree is designed based on the input sample, S, and Troot. The Troot value is obtained from the Algorithm 1 (i.e. FRNA), which is A.3. Node Insertion Algorithm (NIA) After finding Troot value, we check any value which is less than Troot will be placed on left side of the Troot, otherwise place on right side of the Troot. In Fig. A.2, stock value is less than the Troot, therefore it is placed on the left side. This node is represented as Troot.LEFT. Similarly, stock exchange value is greater than the Troot, therefore it placed on the right side. This node is represented as Troot.RIGHT. This process is repeated, until all the values are processed. Fig. A.2. A tree of the input data set. In this tree, Troot =

20 P. Singh, G. Dhiman & A. Kaur Fig. A.3. Clusters of the input data set. The output of this algorithms is depicted in Fig. A.2. Hence, to insert a value in a node follows the following rules. Initially determine Troot. Any value less than the Troot, then insert it into the left of Troot. This is represented as Troot.LEFT. Any value greater than the Troot, then insert it into the right of Troot. This is represented as Troot.RIGHT. Each node contains at least one unique value. The values in the left side of a subtree are always less than the value in its parent node (i.e. Troot.LEFT). The values in the right side of a subtree are always greater than the value in its parent node (i.e. Troot.LEFT). The values in the right side of a subtree are always greater than the value in its parent node (i.e. Troot.RIGHT). The values in the left side of a subtree are always less than the value in its parent node (i.e. Troot.RIGHT). Duplicate values in the nodes are not allowed. A.4. Node Clustering Algorithm (NCA) Make the clusters from the tree, as shown in Fig. A.2. For this purpose, Algorithm 4 (i.e. NCA) is used. Output of this algorithm is depicted in Fig. A.3. A numerical computation of this algorithm is as follows: Step 1. A symbolic representation of a tree, formed by Algorithms 2 and 3, is shown in Fig. A.1. In this tree, initially we check that Troot exists or not. If Troot exists, then check that Troot has left (Troot.LEFT) and right (Troot.RIGHT) children both. If both children exist, then compute the difference between

21 A new quantum fuzzy time series forecasting model values of the T root and T root.right, and T root and T root.left. Make a cluster with corresponding child (either T root.left or T root.right) and T root, which have the minimum difference. In Fig. A.3, T root = , it has both T root.left = and T root.right = Now, the mindiffnode = T root.right T root has the minimum difference, which is 611. Hence, make the cluster with T root.right and T root, and represent this cluster as C 1 (see Fig. A.3). Step 2. Now, according to Algorithm 4, T root.right has (T root.right).right child with value , so we add this value to the cluster C 1. Note that if the mindiffnode = T root T root.left has the minimum difference, then we will make the cluster with T root.left and T root. Again, according to Algorithm 4, if T root.left has (T root.left).right with certain value, then we add this value to this cluster. Step 3. If only one child (either T root.left or T root.right) exists for each T root, then make the cluster with either T root and T root.left or T root and T root.right. Step 4. Repeat Steps 1 3, until all the nodes of the tree are processed. Appendix B. List of Statistical Parameters (1) The mean can be defined as n i=1 Ā = Actual i. (B.1) d (2) The root mean square error (RMSE) can be defined as d i=1 RMSE = (Forecasted i Actual i ) 2. (B.2) d (3) The average forecasting error rate (AFER) can be defined as AFER = 1 n Forecasted i Actual i n A i 100%. i=1 (4) The formula used to calculate Theil s U statistic is d (B.3) i=1 U = (Actual i Forecasted i ) 2 d d. (B.4) i=1 Actual2 i + i=1 Forecasted2 i Here, each Forecasted i and Actual i is the forecasted and final price of day i respectively, d is the total number of days to be forecasted. In Eq. (B.1), {A 1, A 2,..., A n } are the observed values of the final price, and Ā is the mean value of these observations. Similarly, mean for predicted time series data set are computed. For a good forecasting, the observed means should be close to the predicted means. In Eq. (B.2), a small RMSE value indicates good forecasting. Similarly, in

22 P. Singh, G. Dhiman & A. Kaur Eq. (B.3), a small AFER value indicates good forecasting. In Eq. (B.4), U is bound between 0 and 1, with values closer to 0 indicating good forecasting accuracy. Acknowledgments This research is supported by the Department of Science and Technology (DST)- SERB, Government of India, under Grant EEQ/2016/ References 1. L. A. Zadeh, Inform. Control 8, 338 (1965). 2. M. Andrecut, Mod. Phys. Lett. B 13, 33 (1999). 3. P. Singh and G. Dhiman, A fuzzy-lp approach in time series forecasting, in Pattern Recognition and Machine Intelligence, eds. B. U. Shankar, K. Ghosh, D. P. Mandal, S. S. Ray, D. Zhang and S. K. Pal (Springer International Publishing, 2017), pp Z. Tian, L.-M. Jia, H.-H. Dong, Z.-D. Zhang and Y.-D. Ye, Mod. Phys. Lett. B 29, (2015). 5. P. Singh, K. Rabadiya and G. Dhiman, Mod. Phys. Lett. B 32, (2018). 6. P. Singh and G. Dhiman, Appl. Soft Comput. 72, 121 (2018). 7. H. Singh and B. S. Khehra, Mod. Phys. Lett. B 32, (2018). 8. D. Li, Y. Fu and L. Yang, Mod. Phys. Lett. B 31, (2017). 9. J. P. Shi, Mod. Phys. Lett. B 31, (2017). 10. R. K. Chandrawat, R. Kumar, B. P. Garg, G. Dhiman and S. Kumar, An analysis of modeling and optimization production cost through fuzzy linear programming problem with symmetric and right angle triangular fuzzy number, in Proc. Sixth Int. Conf. on Soft Computing for Problem Solving (Springer, 2017), pp G. Dhiman and V. Kumar, Adv. Eng. Softw. 114, 48 (2017). 12. G. Dhiman and V. Kumar, Knowl.-Based Syst. 150, 175 (2018). 13. P. Singh and G. Dhiman, J. Comput. Sci. 27, 370 (2018). 14. G. Dhiman and V. Kumar, Knowl.-Based Syst. 159, 20 (2018). 15. P. Singh, Applications of Soft Computing in Time Series Forecasting: Simulation and Modeling Techniques, Studies in Fuzziness and Soft Computing, Vol. 330 (Springer, 2015). 16. Q. Song and B. S. Chissom, Forecasting enrollments with fuzzy time series: Part I, The Annual Meeting of the Mid-South Educational Research Association, Q. Song and B. S. Chissom, Fuzzy Sets Syst. 54, 1 (1993). 18. S. M. Chen, Fuzzy Sets Syst. 81, 311 (1996). 19. Q. Song and B. S. Chissom, Fuzzy Sets Syst. 54, 1 (1993). 20. Q. Song and B. S. Chissom, Fuzzy Sets Syst. 62, 1 (1994). 21. J. R. Hwang, S. M. Chen and C. H. Lee, Fuzzy Sets Syst. 100, 217 (1998). 22. K.-H. Huarng and T. H.-K. Yu, Int. J. Innov. Comput. I. 8, 7415 (2012). 23. G. Dhiman and A. Kaur, Spotted hyena optimizer for solving engineering design problems, in Int. Conf. Machine Learning and Data Science (MLDS), December 2017, pp G. Dhiman and V. Kumar, Spotted hyena optimizer for solving complex and nonlinear constrained engineering problems, in Harmony Search and Nature Inspired Optimization Algorithms (Springer, 2019), pp G. Dhiman and A. Kaur, A hybrid algorithm based on particle swarm and spotted hyena optimizer for global optimization, Advances in Intelligent Systems and Computing (Springer, 2018), in press

23 A new quantum fuzzy time series forecasting model 26. A. Kaur and G. Dhiman, A review on search-based tools and techniques to identify bad code smells in object-oriented systems, in Harmony Search and Nature Inspired Optimization Algorithms (Springer, 2019), pp G. Dhiman and A. Kaur, Designs 2(3) (2018). 28. H.-T. Liu and M.-L. Wei, Expert Syst. Appl. 37, 6310 (2010). 29. S.-M. Chen and K. Tanuwijaya, Expert Syst. Appl. 38, (2011). 30. Y.-L. Huang, S.-J. Horng, T.-W. Kao, R.-S. Run, J.-L. Lai, R.-J. Chen, I.-H. Kuo and M. K. Khan, Int. J. Innov. Comput. I. 7, 4027 (2011). 31. L. Wang, X. Liu and W. Pedrycz, Expert Syst. Appl. 40, 5673 (2013). 32. W. Lu, X. Chen, W. Pedrycz, X. Liu and J. Yang, Int. J. Approx. Reason. 57, 1 (2015). 33. C. Cheng, J. Chang and C. Yeh, Technol. Forecast. Soc. 73, 524 (2006). 34. H.-K. Yu, Physica A 346, 657 (2005). 35. S.-T. Li and Y.-C. Cheng, Comput. Math. Appl. 53, 1904 (2007). 36. P. Singh, Neural Comput. Appl., 1 (2016). 37. P. Singh and B. Borah, Knowl.-Based Syst. 46, 12 (2013). 38. K. Huarng and T. H.-K. Yu, IEEE T. Syst. Man Cy. B 36, 328 (2006). 39. W. Wang and X. Liu, Inform. Sci. 294, 78 (2015). 40. P. Singh and B. Borah, Eng. Appl. Artif. Intell. 26, 2443 (2013). 41. P. Singh, Geosci. Front. 42. P. Singh, Int. J. Mach. Learn. Cyb. 9, 491 (2018). 43. P. Singh and B. Borah, Int. J. Approx. Reason. 55, 812 (2014). 44. K. Huarng, Fuzzy Sets Syst. 123, 369 (2001). 45. H. J. Teoh, C.-H. Cheng, H.-H. Chu and J.-S. Chen, Data Knowl. Eng. 67, 103 (2008). 46. P. Singh, Int. J. Mach. Learn. Cyb. 1 (2015). 47. H.-K. Yu, Physica A 349, 609 (2005). 48. S. Yang et al., in Congress on Evolutionary Computation (CEC 2004) (IEEE, 2004), Vol. 1, pp A. Narayanan and M. Moore, Quantum-inspired genetic algorithms, in Proc. IEEE Int. Conf. Evolutionary Computation (IEEE, 1996), pp S.-M. Chen, Cybernet. Syst. 33, 1 (2002). 51. K. Huarng and H. Yu, Physica A 353, 445 (2005). 52. A. Samvedi and V. Jain, Eng. Appl. Artif. Intell. 53. P. Singh and B. Borah, Stoch. Env. Res. Risk A. 27, 1585 (2013). 54. P. Singh and B. Borah, Knowl. Inf. Syst. 38, 669 (2014). 55. S.-M. Chen and N.-Y. Chung, Int. J. Intell. Syst. 21, 485 (2006)