Base Shifting, Splicing and Deflating in Index Numbers | Quantitative Aptitude for CA Foundation PDF Download

Base shifting

• Base shifting is used when the base period is changed, for example, when the base year for a price index is updated to reflect changes in the economy. To shift the base period, we divide all the index values by the value of the index in the old base period and multiply by the value of the index in the new base period. 
• This has the effect of re-scaling the index so that it reflects changes relative to the new base period. For example, if the CPI in the old base period is 100 and in the new base period is 120, we can shift the base by dividing all the index values by 100 and multiplying by 120.

Splicing

• Splicing is used when two or more index series are joined together to form a continuous series. This is typically done when the base period of one series is different from that of another series, or when there is a break in the data due to changes in methodology or coverage. To splice two series together, we need to adjust the values of one series so that they are on the same base as the other series. 
• This can be done by multiplying the values of the series by a scaling factor that reflects the difference in the base periods. For example, if one series has a base period of 1990 and another series has a base period of 2000, we can adjust the values of the second series by multiplying them by 100/90.

Deflating

• Deflating is used to remove the effect of price changes on economic variables so that we can compare them over time in real terms. To deflate an index, we divide it by a price index, such as the CPI, that reflects changes in the general level of prices. This has the effect of removing the influence of price changes on the variable, so that we can compare it over time in constant dollars. 
• For example, if we want to compare the value of Gross Domestic Product (GDP) in real terms over time, we can divide the nominal GDP by the GDP deflator, which is a price index that reflects changes in the prices of all goods and services produced in the economy.

Base Shifting, Splicing and Deflating in algebra

Base shifting, splicing, and deflating are three techniques used in numerical analysis and linear algebra. These techniques are used to manipulate and analyze data and are particularly useful in solving problems related to matrices, vectors, and linear equations.

Base Shifting: Base shifting is a technique used to convert a polynomial from one base to another. This technique is particularly useful in polynomial interpolation, where we want to approximate a function using a polynomial of a given degree.

Suppose we have a polynomial f(x) of degree n, defined as:

• f(x) = a_0 + a_1x + a_2x² + … + a_nxⁿ

We can rewrite this polynomial in terms of a new base (x – c), where c is some constant, as follows:

• f(x) = b_0 + b_1(x – c) + b_2(x – c)² + … + b_n(x – c)ⁿ

The coefficients b_i can be calculated using a recursive formula known as the Horner’s method. Once we have the polynomial expressed in terms of the new base, we can evaluate it at any point x using the standard polynomial evaluation algorithm. Base shifting is also used in numerical integration, where we want to approximate the value of a definite integral using a finite number of function evaluations. In this case, we can shift the integration limits to a new range (a, b), where a and b are constants, and then apply a numerical integration method such as the trapezoidal rule or Simpson’s rule.

Splicing: Splicing is a technique used to combine two or more matrices or vectors into a single entity. This technique is useful in solving systems of linear equations, where we want to express the system in matrix form and then apply matrix algebra to find the solution. Suppose we have two matrices A and B, each of size n x m, and we want to splice them into a single matrix C of size n x 2m. We can do this by concatenating the columns of A and B, as follows:

• C = [A | B]

where | denotes the horizontal concatenation operator. The resulting matrix C has the same number of rows as A and B, but twice as many columns. Splicing can also be used to combine two or more vectors into a single vector. Suppose we have two vectors u and v, each of size n, and we want to splice them into a single vector w of size 2n. We can do this by concatenating the elements of u and v, as follows:

• w = [u, v]

where, denotes the vertical concatenation operator. The resulting vector w has twice as many elements as u and v.

Deflating: Deflating is a technique used to reduce the degree of a polynomial by one. This technique is useful in polynomial root-finding algorithms, where we want to find the roots of a polynomial of a given degree.
Suppose we have a polynomial f(x) of degree n, defined as:

• f(x) = a_0 + a_1x + a_2x² + … + a_nxⁿ

and we have found one of its roots, say x = r. We can use the deflation technique to reduce the degree of the polynomial by one, as follows:

• g(x) = (x – r) * q(x)

where q(x) is a polynomial of degree n-1. We can find the coefficients of q(x) by dividing f(x) by (x – r) using polynomial long division. The resulting polynomial g(x) has one less root than f(x), and we can repeat the deflation process on g(x) to find the remaining roots.

Deflation can also be used in polynomial interpolation, where we want to approximate a function using a polynomial of a given degree. Suppose we have a set of n+1 data points (x_i, y_i), and we want to find a polynomial of degree n that passes through all the points. We can use the Lagrange interpolation formula to find the coefficients of the polynomial, but this can lead to numerical instability for large values of n. To overcome this problem, we can use the deflation technique to find a sequence of polynomials of decreasing degree that approximate the function with increasing accuracy.