Linear Kriging Essay

The Kriging weights w_i1,w_i2,w_i3,…w_ik can be used to estimate the fair value of VA contracts x ⃑_i by the following formula, y ̂_i=∑_(j=1)^k▒〖w_ij∙y_j 〗.
The kriging weights can be calculated by the following linear equations, Where the is a control variable, which is used to make sure that ∑_(j=1)^k▒w_ij =1.

V_rs=α+exp⁡(-3/β D(z ⃑_r,z ⃑_s,λ)),r,s=1,2,3,…,k,
D_ij=α+exp⁡(-3/β D(x ⃑_i,z ⃑_j,λ)),j=1,2,3,…,k,

The D(.) function is the distance function mentioned in the clustering section,α≥0 and β≥0 are two parameters. The above linear equation system has a unique solution because D(z ⃑_r,z ⃑_s,λ)>0 for all r,s=1,2,3,…,k,.

Then the fair value of whole portfolio X=(x ⃑_1,x ⃑_2,x ⃑_3,…,x ⃑_n)can be estimated by the following formula,
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The only difference is how the representative contracts are selected. The k-centroids μ ⃑_1,μ ⃑_2,μ ⃑_3,…,μ ⃑_kare obtained by k-prototype algorithm. Then, for the MAS, the representative contracts are selected from the whole portfolio. For the MWS method, the representative contracts are selected from each subset of the portfolio X=(x ⃑_1,x ⃑_2,x ⃑_3,…,x ⃑_n). For example, 100000 contracts are divided into 20 subsets, each subset has 5000 contracts. For the MAS method, we obtain 5 centroids in the subset, then choose 5 closest contracts from the whole portfolio X=(x ⃑_1,x ⃑_2,x ⃑_3,…,x ⃑_n ) that contains 100000 contracts. For the MWS method, we obtain 5 centroids in the subset, then choose 5 closest contracts from the 5000 contracts in the same subset as representative contracts. The MWS method can save a lot of time comparing with the MAS method by avoiding a lot of distance calculations in the nearest neighbor mapping closest contract step. However, the closest contracts in the MAS method usually have shorter distance to the centroids. The 5000 contracts in each subset are also included in the whole portfolio X=(x ⃑_1,x ⃑_2,x ⃑_3,…,x ⃑_n), and it is possible that exists contracts with shorter distance to the centroids from other subsets. This impact on accuracy is tested in Table 4 and Table 5. For example, in Figure 2, if the centroid is denoted as C and the closest contract in the same subset is B. The contract with shortest distance to the centroid is denoted as A, but it is not divided into same subset. As a result, the B is the representative contract by the MWS method, and the A is the representative contract by the MAS method. The closer total distance in MAS increase the accuracy of the result. However, the duplication produced in MAS decrease the accuracy. It is necessary to analyze which one has the stronger influence on the

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