统计学习笔记(4) 线性回归(1)

Basic Introduction

In this chapter, we review some of the key ideas underlying the linear regression model, as well as the least squares approach that is most commonly used to fit this model.

Basic form:

“≈” means “is approximately modeled as”, to estimate the parameters, by far the most common approach involves minimizing the least squares criterion. Let training samples be (x1,y1),...,(xn,yn), then define the residual sum of squares (RSS) as

And we get

About the result, in the ideal case, that is with enough samples, we can get a classification result called population regression line.

Population regression line: the best fitting result

Least squared regression line: with limited samples

To evaluatehow well our estimation meets the true values, we use standard error, e.g. to estimate the mean value. The variance of sample mean  is

where σ is the standard deviation of each of the realization yi (i=1,2,...,n, y1,...yn are uncorrelated). n is the number of samples.

Definition of standard deviation:

Suppose X is a random variable with mean value

It is the square root of variance. About the computation of standard deviation:

Let x1, ..., xN be samples, then we calculate the standard deviation with (here is the number of limited samples):

Note: for limited samples, we use N-1 to divide, which is the common case, for unlimited samples in the ideal case, we use N to divide.

In the following formulation:

                                                                            
                                                       ,
Here SE means standard error. Standard error is standard deviation divided by sqrt(n), it means the accuracy of results while the standard deviation means the accuracy of data.

An estimation of σ is residual standard error. We use residual standard error (RSE) to estimate it.

The 95% confidence interval for β1 and β0:

  

As an example, we can use

to get the probability that β1 is zero, or the non-existence of the relationship between X(predictor) and Y(response).

Assessing the Accuracy of the Model
There are 3 criterions:

1. Residual Standard Error

     

2. R^2 Statistic

    

TSS measures the total variance in the response Y, and can be thought of as the amount of variability inherent in the response before the regression is performed. In contrast, RSS measures the amount of variability that is left unexplained after performing the regression. TSS − RSS measures the amount of variability in the response that is explained (or removed) by performing the regression. An R^2 statistic that is close to 1 indicates that a large proportion of the variability in the response has been explained by the regression. On the other hand, we can use

to assess the fit of the linear model. In simple linear model,R^2=Cor(X,Y)^2.

R^2 is normalized, when the actual line is steeper, TSS is larger, and because of

RSS is also larger.

Multiple Linear Regression

we can use least squares to get

we also use F-statistic to

An explanation of the above expression is

                                                                            

The RSS represents the variability left unexplained, TSS is the total variability, as we have already estimated p variables and there are n variables as a whole, so the variance of RSS is n-p-1, TSS-RSS is p.

Subtopic0: Hypothesis Testing in Single and Multiple Linear Regression

                   

                                                          

 

The zero in Q剩 should be changed to 1.

Subtopic1:whether each of the predictors is useful in predicting the response.

To indicate which one of

                                                                  

is true. If F-statistic is close to 1, then H0 is true. If F-statistic is greater than 1, then Ha is true.

It turns out that the answer depends on the values of n and p. When n is large, an F-statistic that is just a little larger than 1 might still provide evidence against H0. In contrast, a larger F-statistic is needed to reject H0 if n is small.

When H0 is true and the errors i have a normal distribution, the F-statistic follows an F-distribution. For any given value of n and p, any statistical software package can be used to compute the p-value associated with the F-statistic using this distribution. Based on this p-value, we can determine whether or not to reject H0.

Here the p-value is defined as the probability, under the assumption of hypothesis H, of obtaining a result equal to or more extreme than what was actually observed. Here the reason that a smaller p can indicate the existence of the relationship between at least one of p parameters and the result is that onlyWhen H0 is true and the errors i have a normal distribution, the F-statistic follows an F-distribution. And when p is small, the hypothesis under consideration may not adequately explain the observation.The smaller p-value is, the more suspectable H0 is.

 

If we only want to estimate a proportion of the parameters, for example. q parameters,

We fit a second model that uses all the variables except those last q. Suppose that the residual sum of squares for that model is RSS0,The dimension of freedom of RSS0 is q larger than that of RSS.

                                                                                       

When q=1, it satisfies t-distribution. So it reports the partial effect of adding that variable to the model.

F-statistic results in p-value.

Note: But we cannot only estimate the p-value for each predictor. We should also estimate the overall F-statistic. for even if H0 is true, there is only a 5% chance that the F-statistic will result in a p-value below 0.05, regardless of the number of predictors or the number of observations.
Subtopic2: Importance of variables
To make sure of the importance of the predictors, we can try:

Method1: Forward Selection.We begin with the null model, then fit p simple linear regressions and add to the null model the variable that results in the lowest RSS.

Method2: Backward Selection. We start with all variables in the model, and backwardremove the variable with the largest p-value.

Method3: Mixed Selection.

Process: We start with no variables in the model, we add the variable that provides the best fit,

Stop: If at any point the p-value for one of the variables in the model rises above a certain threshold, then we remove that variable from the model. We continue to perform these forward and backward steps until all variables in the model have a sufficiently low p-value, and all variables outside the model would have a large p-value if added to the model.
Comment: Backward selection cannot be used if p > n, while forward selection can always be used. Forward selection is a greedy approach, and might include variables early that later become redundant. Mixed selection can remedy this.
Subtopic3: Model fitting quality

Two criterion: R²(the fraction of variance explained) and RSE. By adding predictors, if the RSE is greatly reduced, then the predictor is useful.But models with more variables can have higher RSE if the decrease in RSS is small relative to the increase in p.

In addition to looking at the RSE and R2 statistics just discussed, it can be useful to plot the data.

Graphical summaries can reveal problems with a model that are not visible from numerical statistics. For example, Figure below displays a three-dimensional plot of TV and radio versus sales. We see that some observations lie above and some observations lie below the least squares regression plane. In particular, the linear model seems to overestimate sales for instances in which most of the advertising money was spent exclusively on either TV or radio. It underestimates sales for instances where the budget was split between the two media. This pronounced non-linear pattern cannot be modeled accurately using linear regression. It suggests a synergy or interaction effect between the advertising media, whereby combining the media together results in a bigger boost to sales than using any single medium. In Section 3.3.2, we will discuss extending the linear model to accommodate such synergistic effects through the use of interaction terms.

Qualitative Predictors
The predictors can take on 2 values, just like binary, the response is quantitative. 

Further introduction to the linear model

Two basic assumptions:

1. The relationship between the predictors and response are additive and linear.

2. The effect of changes in a predictor Xj on the response Y is independent of the values of the other predictors.

There are some flaws in the above models. e.g. a synergy effect. That is, equal increases in both predictors can contribute more to the increase in response than unbalanced increases among predictors. For example, we can modify

                                                                                                                   

to

                                                                                                    

Non-linear relationships:

For example, if the linear model has low R^2 value, then we change Y=a+bX1 to Y=a+bX1+cX1^2 and represent it as Y=a+bX1+cX2 where X2=X1^2, and we can also use standard linear regression software to estimate a, b and c.

Potential problems:

1. Non-linearity

See the residual plot to see whether there's discernible pattern, if there is, then the model should not be linear. e.g. Left: obvious discernible pattern. Right: Not obvious.

2. Correlation of Error Terms

An important assumption of the linear regression model is that the error terms, 1, 2, . . . , n, are uncorrelated. i represents different samples. Then the estimated standard errors will tend to underestimate the true standard error. Also plot the residual as a function of time.

In the top panel, we see the residuals from a linear regression fit to data generated with uncorrelated errors. There is no evidence of a time-related trend in the residuals. In contrast, the residuals in the bottom panel are from a data set in which adjacent errors had a correlation of 0.9.
3. Non-constant Variance of Error Terms

Cause: It is often the case that the variances of the error terms are non-constant. For instance, the variances of the error terms may increase with the value of the response.

Solution: When faced with this problem, one possible solution is to transform the response Y using a concave function such as log Y or √Y . Such a transformation results in a greater amount of shrinkage of the larger responses. Sometimes we have a good idea of the variance of each response.

                   

Another example, if different observations have different variances, we can fit our model by weighted least squares, with weights proportional to the inverse weighted variances.

4. Outliers

We can plot the studentized residuals: It is normalized by standard errors.

5. High leverage Points

We just saw that outliers are observations for which the response yi is unusual given the predictor xi. In contrast, observations with high leverage have an unusual value for xi.

Phenomenon:

Left: Observation 41 is a high leverage point, while 20 is not. The red line is the fit to all the data, and the blue line is the fit with observation 41 removed. Center: The red observation is not unusual in terms of its X1 value or its X2 value, but still falls outside the bulk of the data, and hence has high leverage. Right: Observation 41 has a high leverage and a high residual.

Introduction:

we observe that removing the high leverage observation has a much more substantial impact on the least squares line than removing the outlier. It is cause for concern if the least squares line is heavily affected by just a couple of observations.

Detection:

In order to quantify an observation’s leverage, we compute the leverage statistic:

If a given observation has a leverage statistic that greatly exceeds (p+1)/n, then we may suspect that the corresponding point has high leverage.

6. Collinearity

Compared with the left image, the right one is colinearity. The two predictors are correlated with each other.

Difficulties arise from colinearity are

Left: Contour of RSS associated with different possiblecoefficientestimates for the regression of balance (response) on limit (predictor 1) and age (predictor 2).

The result is a small change in the data could cause the pair of coefficient values that yield the smallest RSS—that is, the least squares estimates—to move anywhere along this valley. collinearity reduces the accuracy of the estimates of the regression coefficients, it causes the standard error for ^βj to grow.

The probability of correctly power detecting a non-zero coefficient—is reduced by collinearity. A simple way to detect collinearity is to look at the correlation matrix of the predictors.

Multicollinearity-cannot be detected bylooking at the correlation matrices
Unfortunately, not all collinearity problems can be detected by inspection of the correlation matrix: it is possible for collinearity to exist between three or more variables even if no pair of variables has a particularly high correlation. We call this situation multicollinearity.

Instead of inspecting the correlation matrix, a better way to assess multi-collinearity collinearity is to compute the variance inflation factor (VIF).

The smallest possible value for VIF is 1, which indicates the complete absence of collinearity. A VIF value that exceeds 5 or 10 indicates a problematic amount of colinearity.

For each predictor,

is R^2 from a regression of Xj(act as response) onto all of the other predictors.

Solution
The first is to drop one of the problematic variables from the regression. The second solution is to combine the collinear variables together into a single predictor.