Definition 1 The two-dimension transaction table is a data set items associate frequency pattern table, which is made of item rows and item columns, saving the combination count information for a transaction database. A transaction is a collection of multiple items in database. And the arbitrary combination of two items in the transaction is called basic associate frequency pattern unit.

From a transaction of database, a sequence of items is considered as a queue. So the each previous item and the following each item in the item queue constitutes the rows and columns of two-dimensional table, the value between relevant row and column in the table is the frequency count. Since a trade database is composed of many transactions, it also includes all the items of transaction expressing the rows and columns of two-dimen- sional table, showing all the transaction combination counts between every two items in trade database.

For the simple example, a transaction T₁ is a sequence of items collection {abcef}, the two-dimension table of frequency item pattern count is created in Figure 1. The row is made of item a, b, c ,e f, and the column is also made of item a, b, c, e, f. The frequency count value of each two item combination of {a, b}, {a, c}, {a, e}, {a, f}, {b, c}, {b, e}, {b, f}, {c, e}, {c, f}, {e, f} is recorded as 1 in the table, others is recorded as zero in the table. The number (Num) 0, 1, 2, 3, 4 is marked as sequent item a, b, c, e, f.

Definition 2 Hash-T is a hash table, which combines the candidate item frequency pattern with hash table from transaction database.

The candidate items are the whole transaction items in the trade database. Firstly, the database is thoroughly scanned, and the candidate items are counted by frequency, as is called Support degree (Sup). Then all the items are sorted by the sequence. At the same time, the hash table is used for exactly defining candidate items sequence by Sequence Number (SN) and Item Name (Name). The example of Hash-T, from a transaction database including three transaction items, is as Figure 2.

From Figure 2, transaction database is composed as three transactions, including {abcef}, {acg} and {ei}. All the items are scanned and the frequency items are counted, then the Hash-T is constituted by the title Num, SN, Name and Sup, noting as number, address, item name and support degree.

Definition 3 FP-Table is a combination table based on improved FP-Growth algorithm, integrating sorted Hash-T and two-dimension table.

Given Figure 2 example, firstly, a two dimension table is created by frequency count of all the item pairs in the transaction database. Then Hash-T is sorted by support degree, so the sequence of candidate items is a, c, e, b, f, g and i. And the items of a, c, e are kept as mining items according to the support degree value of 2 (the support degree value is defined as great and equal 2). At last, the two dimension table is converted as FP-Table, which is shown as Figure 3.

Lemma 1 The item pair in the FP-Table is the minimal combination unit. And the frequency combination item depends on the frequency count of item.

Proof Firstly, given a transaction including two items {A₁A₂}, the frequency item and the frequency count are scanned, noted as 1 A ₂: 1>;

Then, given two transaction including collection as {A₁A₂}, {A₁A₂A₃}, the frequency items and the frequency count are also scanned, note as 1 A ₂: 2>, 1 A ₃: 1>, 2 A ₃: 1>, 1 A ₂ A ₃: 1>. Above frequency count value of combination item 1 A ₂ A ₃: 1> is 1, just due to the minimal frequency count value of item pair 1 A ₃: 1> and 2 A ₃: 1> is 1;

Next, given n transaction including collection as {A₁A₂}, {A₁A₂A₃}, ∙∙∙, {A₁A₂A₃∙∙∙A_n}, the frequency items are scanned and noted as 1 A ₂: n>, 1 A ₃: n-1>, ∙∙∙, 1 A _n: 1>, 2 A ₃: n-1>, ∙∙∙, 2 A _n: 1>, ∙∙∙, 1 A ₂ A ₃: n-1>, ∙∙∙, 1 A ₂ A ₃ A ₄: n-2>, ∙∙∙, 1 A ₂ A ₃ ∙∙∙A _{n -1} A _n: 1>. The frequency count value of combination item of 1 A ₂ A ₃ ∙∙∙A _{n -1} A _n: 1> is 1, due to the minimal frequency count value of item pair 1 A ₂: n>, 1 A ₃: n-1>, 1 A ₄: n-2>,∙∙∙, 1 A _n: 1> is 1.

FP-Table algorithm is implemented as two steps. The first step is the creation process of FP-Table, and the next step is the mining process of FP-Table.

Building FP-Table Algorithm: given a trade database, once the transaction database is scanned, the candidate frequency items set is set into hash set table Hash-T, including SN, Name and Sup. At the same time, the frequency counts between all item pairs are recorded in two-dimension table. Then, the sort process is implemented in Hash-T according to the descending order of support degree. The items, which support degree is less than threshold value, are removed from Hash-T, and the item number is reordered again, Hash-T is updated as result. So the FP-Table is created by Hash-T index of items, and the transaction frequency count value of index item in Hash-T is set into FP-Table, obtaining from two-dimension table. The algorithm is shown as Figure 4.

Mining FP-Table Algorithm: the combination items and frequency counts are given according to the Hash-T item index from FP-Table. Firstly, the item pairs and support degree are created, and then more items are combined with a set of transaction according the lemma 1. After all the frequency items are combined recursively, the mining process of FP-Table is completed. The algorithm is shown as Figure 5.

Given transaction mining in electricity market, the transaction database is proposed for electricity transaction as a case study. The items of electricity transaction are derived from mobile electricity trade system platform. Items are classified as power generation trade (a), direct trading between power users and power generation enterprises (b), electricity price (c), inter provincial Trading (d), bilateral negotiation (e), centralized matchmaking (f), medium and long-term transactions (g), month transactions (i), year transactions (j), and delivery of electricity trading (l). The Hash-T table and two-dimension table are built as Figure 6.

For the Hash-T table, all the items are sorted by the support degree (Sup). Then the items which support that degree is less than threshold are removed from Hash-T table. Next, the FP-Table is constituted as the following step: first, the sorted items are recorded in the dynamic array; second, the frequency count of item in Hash-T is selected and filled into FP-table. Since the FP-Table is composed by row and column, the frequency count in FP-Table is expressed as associated relationship among item pairs. But the sparse data value of frequency count in FP-Table is the majority; the dynamic array technology is used to deal with the effective count value. The FP-Table is built as Figure 7.

From the FP-Table, the frequency item in electricity market is recursively proceeded to pattern mining according to lemma 1. For example, as to item g in sorted Hash-T and FP-Table, the frequency count value between item pairs of g and a is 5, the frequency count value between item pairs of g and c is 5, and the frequency count value between item pairs of g and e is 4. So the frequency item pattern mining process is described as the following: once the frequency count value of g and e is 4, all the frequency count of item combination with e is only 4, that is, the frequency pattern of g has {a g: 5}, {c g: 5}, {e g: 4}, {a c g: 5}, {a e g: 4}, {c e g: 4}, {a c e g: 4} according to the pattern count condition of .

All the items of frequency pattern in electricity trade database are proceeded by the above method. And the experiment proves that the algorithm is efficient and robust.