7
Jul 18

Euclidean space geometry: vector operations

Euclidean space geometry: vector operations

The combination of these words may sound frightening. In fact, if you want to succeed with matrix algebra, you need to start drawing inspiration from geometry as early as possible.

Sum of vectors

Definition. The set of all n-dimensional vectors x=(x_1,...,x_n) with x_{i}\in R is denoted R^n and is called a Euclidean space.

R^2 is a plane. The space we live in is R^3. Our intuition doesn't work in dimensions higher than 3 but most facts we observe in real life on the plane and in the 3-dimensional space have direct analogs in higher dimensions. Keep in mind that x=(x_1,...,x_n) can be called a vector or a point in R^{n}, depending on the context. When we think of it as a vector, we associate with it an arrow that starts at the origin (0,...,0) and ends at the point (x_1,...,x_n).

Careful inspection shows that the sum of two vectors x+y=(x_1+y_1,...,x_n+y_n) is found using the parallelogram rule in Figure 1. The rule itself comes from physics: if two forces are applied to a point, their resultant force is found by the parallelogram rule. Whatever works in real life is guaranteed to work in Math.

Figure 1. Sum of vectors

Figure 1. Sum of vectors

Exercise 1. Let e_1=(1,0) be the unit vector of the x axis and e_2=(0,1) the unit vector of the y axis. Find the sum e_1+e_2. Generalization: if x runs over the whole x axis and y runs over the whole y axis, what is the set of resulting sums x+y? Further generalization: on the plane take two straight lines L_1 and L_2 that pass through the origin and are not parallel to one another. If x runs over L_1 and y runs over L_2, what is the set of resulting sums x+y?

This seemingly innocuous exercise leads to profound ideas, to be considered later. The answer for the last question is that the sums x+y will cover the whole R^{2}. This fact is written like this: \{x+y:x\in L_1,\ y\in L_2\}=R^2. Note that I intentionally use words that emphasize movement and geometry: "runs over" and "covers the whole". One of the differences between elementary and higher mathematics is that the former deals with fixed elements and the latter with sets within which there are movement and change.

Multiplication of a vector by a number

If a is a number and x=(x_1,...,x_n) is a vector, we put ax=(ax_1,...,ax_n) (scaling or multiplication by a number). Scaling x by a positive number a means lengthening it in case a>1 and shortening in case a<1. Scaling by a negative number means, additionally, reverting the direction of x,  see Figure 2.

Scaling a vector

Figure 2. Scaling a vector

 

Exercise 2. Take a nonzero vector x=(x_1,x_2) and find the set of all products ax, where a runs over R.

The answer is that \left\{ ax:a\in R\right\} is the straight line drawn through the vector x or, alternatively, the straight line drawn through the origin and point x.

The expression ax+by is called a linear combination of vectors x,y with coefficients a,b\in R. To find ax+by, you first scale x and y and then add the results.

Exercise 3. Let x,y be two nonparallel vectors on the plane. Describe the set \left\{ ax+by:a,b\in R\right\} verbally and geometrically.

Verbally, this is the set of all linear combinations ax+by that result when a,b run over R. Geometrically, this will be the whole plane R^2.

26
Dec 16

Multiple regression through the prism of dummy variables

Agresti and Franklin on p.658 say: The indicator variable for a particular category is binary. It equals 1 if the observation falls into that category and it equals 0 otherwise. I say: For most students, this is not clear enough.

Problem statement

Figure 1. Residential power consumption in 2014 and 2015. Source: http://www.eia.gov/electricity/data.cfm

Residential power consumption in the US has a seasonal pattern. Heating in winter and cooling in summer cause the differences. We want to capture the dependence of residential power consumption PowerC on the season.

 Visual approach to dummy variables

Seasons of the year are categorical variables. We have to replace them with quantitative variables, to be able to use in any mathematical procedure that involves arithmetic operations. To this end, we define a dummy variable (indicator) D_{win} for winter such that it equals 1 in winter and 0 in any other period of the year. The dummies D_{spr},\ D_{sum},\ D_{aut} for spring, summer and autumn are defined similarly. We provide two visualizations assuming monthly observations.

Table 1. Tabular visualization of dummies
Month D_{win} D_{spr} D_{sum} D_{aut} D_{win}+D_{spr}+ D_{sum}+D_{aut}
December 1 0 0 0 1
January 1 0 0 0 1
February 1 0 0 0 1
March 0 1 0 0 1
April 0 1 0 0 1
May 0 1 0 0 1
June 0 0 1 0 1
July 0 0 1 0 1
August 0 0 1 0 1
September 0 0 0 1 1
October 0 0 0 1 1
November 0 0 0 1 1

Figure 2. Graphical visualization of D_spr

The first idea may be wrong

The first thing that comes to mind is to regress PowerC on dummies as in

(1) PowerC=a+bD_{win}+cD_{spr}+dD_{sum}+eD_{aut}+error.

Not so fast. To see the problem, let us rewrite (1) as

(2) PowerC=a\times 1+bD_{win}+cD_{spr}+dD_{sum}+eD_{aut}+error.

This shows that, in addition to the four dummies, there is a fifth variable, which equals 1 across all observations. Let us denote it T (for Trivial). Table 1 shows that

(3) T=D_{win}+D_{spr}+ D_{sum}+D_{aut}.

This makes the next definition relevant. Regressors x_1,...,x_k are called linearly dependent if one of them, say, x_1, can be expressed as a linear combination of the others: x_1=a_2x_2+...+a_kx_k.  In case (3), all coefficients a_i are unities, so we have linear dependence. Using (3), let us replace T in (2). The resulting equation is rearranged as

(4) PowerC=(a+b)D_{win}+(a+c)D_{spr}+(a+d)D_{sum}+(a+e)D_{aut}+error.

Now we see what the problem is. When regressors are linearly dependent, the model is not uniquely specified. (1) and (4) are two different representations of the same model.

What is the way out?

If regressors are linearly dependent, drop them one after another until you get linearly independent ones. For example, dropping the winter dummy, we get

(5) PowerC=a+cD_{spr}+dD_{sum}+eD_{aut}+error.

Here is the estimation result for the two-year data in Figure 1:

PowerC=128176-27380D_{spr}+5450D_{sum}-22225D_{aut}.

This means that:

PowerC=128176 in winter, PowerC=128176-27380 in spring,

PowerC=128176+5450 in summer, and PowerC=128176-22225 in autumn.

It is revealing that cooling requires more power than heating. However, the summer coefficient is not significant. Here is the Excel file with the data and estimation result.

The category that has been dropped is called a base (or reference) category. Thus, the intercept in (5) measures power consumption in winter. The dummy coefficients in (5) measure deviations of power consumption in respective seasons from that in winter.

Here is the question I ask my students

We want to see how beer consumption BeerC depends on gender and income Inc. Let M and F denote the dummies for males and females, resp. Correct the following model and interpret the resulting coefficients:

BeerC=a+bM+cF+dM^2+eF^2+fFM+(h+iM)Inc.

Final remark

When a researcher includes all categories plus the trivial regressor, he/she falls into what is called a dummy trap. The problem of linear dependence among regressors is usually discussed under the heading of multiple regression. But since the trivial regressor is present in simple regression too, it might be a good idea to discuss it earlier.

Linear dependence/independence of regressors is an exact condition for existence of the OLS estimator. That is, if regressors are linearly dependent, then the OLS estimator doesn't exist, in which case the question about its further properties doesn't make sense. If, on the other hand, regressors are linearly independent, then the OLS estimator exists, and further properties can be studied, such as unbiasedness, variance and efficiency.

13
Oct 16

Properties of means

Properties of means, covariances and variances are bread and butter of professionals. Here we consider the bread - the means

Properties of means: as simple as playing with tables

Definition of a random variable. When my Brazilian students asked for an intuitive definition of a random variable, I said: It is a function whose values are unpredictable. Therefore it is prohibited to work with their values and allowed to work only with their various means. For proofs we need a more technical definition: it is a table values+probabilities of type Table 1.

Table 1.  Random variable definition
Values of X Probabilities
x_1 p_1
... ...
x_n p_n

Note: The complete form of writing {p_i} is P(X = {x_i}).

Mean (or expected value) value definitionEX = x_1p_1 + ... + x_np_n = \sum\limits_{i = 1}^nx_ip_i. In words, this is a weighted sum of values, where the weights p_i reflect the importance of corresponding x_i.

Note: The expected value is a function whose argument is a complex object (it is described by Table 1) and the value is simple: EX is just a number. And it is not a product of E and X! See how different means fit this definition.

Definition of a linear combination. See here the financial motivation. Suppose that X,Y are two discrete random variables with the same probability distribution {p_1},...,{p_n}. Let a,b be real numbers. The random variable aX + bY is called a linear combination of X,Y with coefficients a,b. Its special cases are aX (X scaled by a) and X + Y (a sum of X and Y). The detailed definition is given by Table 2.

Table 2.  Linear operations definition
Values of X Values of Y Probabilities aX X + Y aX + bY
x_1 {y_1} p_1 a{x_1} {x_1} + {y_1} a{x_1} + b{y_1}
...  ... ...  ...  ...  ...
x_n {y_n} p_n a{x_n} {x_n} + {y_n} a{x_n} + b{y_n}

Note: The situation when the probability distributions are different is reduced to the case when they are the same, see my book.

Property 1. Linearity of means. For any random variables X,Y and any numbers a,b one has

(1) E(aX + bY) = aEX + bEY.

Proof. This is one of those straightforward proofs when knowing the definitions and starting with the left-hand side is enough to arrive at the result. Using the definitions in Table 2, the mean of the linear combination is
E(aX + bY)= (a{x_1} + b{y_1}){p_1} + ... + (a{x_n} + b{y_n}){p_n}

(distributing probabilities)
= a{x_1}{p_1} + b{y_1}{p_1} + ... + a{x_n}{p_n} + b{y_n}{p_n}

(grouping by variables)
= (a{x_1}{p_1} + ... + a{x_n}{p_n}) + (b{y_1}{p_1} + ... + b{y_n}{p_n})

(pulling out constants)
= a({x_1}{p_1} + ... + {x_n}{p_n}) + b({y_1}{p_1} + ... + {y_n}{p_n})=aEX+bEY.

See applications: one, and two, and three.

Generalization to the case of a linear combination of n variables:

E({a_1}{X_1} + ... + {a_n}{X_n}) = {a_1}E{X_1} + ... + {a_n}E{X_n}.

Special cases. a) Letting a = b = 1 in (1) we get E(X + Y) = EX + EY. This is called additivity. See an application. b) Letting in (1) b = 0 we get E(aX) = aEX. This property is called homogeneity of degree 1 (you can pull the constant out of the expected value sign). Ask your students to deduce linearity from homogeneity and additivity.

Property 2. Expected value of a constant. Everybody knows what a constant is. Ask your students what is a constant in terms of Table 1. The mean of a constant is that constant, because a constant doesn't change, rain or shine: Ec = c{p_1} + ... + c{p_n} = c({p_1} + ... + {p_n}) = 1 (we have used the completeness axiom). In particular, it follows that E(EX)=EX.

Property 3. The expectation operator preserves order: if x_i\ge y_i for all i, then EX\ge EY. In particular, the mean of a nonnegative random variable is nonnegative: if x_i\ge 0 for all i, then EX\ge 0.

Indeed, using the fact that all probabilities are nonnegative, we get EX = x_1p_1 + ... + x_np_n\ge y_1p_1 + ... + y_np_n=EY.

Property 4. For independent variables, we have EXY=(EX)(EY) (multiplicativity), which has important implications on its own.

The best thing about the above properties is that, although we proved them under simplified assumptions, they are always true. We keep in mind that the expectation operator E is the device used by Mother Nature to measure the average, and most of the time she keeps hidden from us both the probabilities and the average EX.