A linear combination is a combination of several variables (or vectors) such that no variable (or vector) is multiplied by either itself or another: they may be multiplied by constants, and are combined by simple addition or subtraction.
A linear combination of x, y and z can always be written ax + by + cz, where a, b and c are constants. Note that a and b might be positive numbers, but either or both of them could also be negative or zero.
Importance of Linear Combinations in Statistics
Linear combinations are often the first choice for a statistical model because they are:
- Simple to conceptualize,
- Relatively easy to work with,
- Easy to do calculations for, compared to some of the more complex modeling equations.
They also provide a fairly accurate model for many real life situations.
Suppose you calculated the population mean of every variable in your linear combination. Then the population mean for your linear combination will be just the linear combination of the component variables. So if your variables x, y and z each had population means l, m, and n, the population mean of the linear combination
ax + by + cz would be just
al + bm + cn.
Finding the variance and covariance of linear combinations is only slightly more complicated, and involves calculating a double sum over the variables.
Belk, James. Linear Combinations and Span. Math 213 Class Notes. Retrieved from http://faculty.bard.edu/belk/math213s14/LinearCombinationsAndSpanRevised.pdf on August 7, 2018.
Korch, Michal. Teaching, Linear Algebra WNE UW 3: Vector Spaces and Linear Combinations. Published Online October 10, 2015. Retrieved from https://www.mimuw.edu.pl/~m_korch/3-vector-spaces-linear-combinations/ on August 7, 2018.
Penn State University. Lesson 2: Linear Combinations of Random Variables. From STAT 505: Applied Multivariate Statistical Analysis. Retrieved from https://onlinecourses.science.psu.edu/stat505/node/11/ on August 7, 2018.
Weisstein, Eric W. “Linear Combination.” From MathWorld–A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/LinearCombination.html on August 7, 2018.
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