Machine Learning

Table of Contents

1 Introduction

  • Euclidean distance

1.1 General Concepts

X
predictors, independent variables, features, variables
Y
response, dependent variable
\begin{eqnarray} Y = f(X) + \epsilon \end{eqnarray}
\(\epsilon\)
random error term, 1) independent of X and 2) has mean zero
statistical learning
a set of approaches for estimating \(f\).
reducible error
\(\hat{f}\) is not an accurate perfect estimate of \(f\)
irreducible error
the error term \(\epsilon\) This may due to unmeasured variables
overfitting
follow error too closely
degree of freedom
a quantity that summarizes the flexibility of a curve
indicator variable
\(I(y_i = \hat{y}_i)\) is 0 or 1

1.2 Different models

parametric method
reduce the fitting problem to estimating a set of coefficients. But it will be imprecise if the model choose-d is very different from the true model
non-parametric method
a very large number of observations is required.
regression
a quantitative response
classification
a qualitative response

1.3 The purpose of estimate \(f\)

prediction
\(\hat{f}\) is treated as black box
inference
understand the relationship between X and Y. \(\hat{f}\) cannot be treated as black box.

1.4 Classification

Bayes Classifier
assign each observation to the most likely class, given its predictor values.
Bayes Error Rate
Bayes classifier produces the lowest possible test error rate. The Bayes error rate is analogous to the irreducible error. This means it is the optimal value. So Bayes classifier serves as an unattainable gold standard against which to compare other methods.
Bayes Decision Boundary
determine the prediction
K-Nearest Neighbors
K points in the training set that are closest to \(x_0\), represented by \(N_0\). Use the largest probability.

1.5 Model Accuracy

mean square error (MSE)
\begin{eqnarray} MSE = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{f}(x_i))^2 \end{eqnarray}
training MSE
test MSE

1.5.1 The bias-variance trade-off

variance
the amount by which \(\hat{f}\) would change if we estimated it using a different training data set. More flexible statistical methods have higher variance.
bias
the error that is introduced by approximating a real-life problem. E.g. it is unlikely that any real-life problem has simple linear relationship. More flexible methods result in less bias.

2 Linear Regression

regress Y on X, or Y onto X.

\begin{eqnarray} Y = \beta_0 + \beta_1 X \end{eqnarray}
intercept
\(\beta_0\)
slope
\(\beta_1\)

2.1 Estimate Coefficients

Residual Sum of Squares (RSS)
\begin{eqnarray} RSS = e_1^2 + e_2^2 + \ldots + e_n^2 \\ e_i = y_i - \hat{y}_i \end{eqnarray}

So RSS can also be written as:

\begin{eqnarray} RSS = \sum_{i=1}^n (y_i - \hat{y}_i)^2 \end{eqnarray}

Least square approach chooses \(\beta_0\) and \(\beta_1\) to minimize RSS. The result:

\begin{eqnarray} \hat{\beta_1} & = & \frac{\sum_{i=1}^n (x_i - \bar{x}) (y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2} \\ \hat{\beta_0} & = & \bar{y} - \hat{\beta_1} \bar{x} \end{eqnarray}

2.2 accuracy

2.2.1 accuracy of coefficients measure

\(\hat{\mu}\)
sample mean
\(SE(\hat{\mu})\)
standard error
\begin{eqnarray} Var(\hat{\mu}) = SE(\hat{\mu})^2 = \frac{\sigma^2}{n} \end{eqnarray}
Residual Standard Error (RSE)
\begin{eqnarray} RSE = \sqrt{RSS / (n-2)} \end{eqnarray}
95% confidence interval
a range of values such that with 95% probability, the range will contain the true unknown value of the parameter.

For linear regression, the 95% confidence interval for \(\beta_1\) is

\begin{eqnarray} \hat{\beta_1} +- 2 SE(\hat{\beta_1}) \end{eqnarray}
2.2.1.1 Hypothesis Test

Standard error can be used to perform Hypothesis test on coefficients. NULL hypothesis:

H0: There is no relationship between X and Y.

alternative hypothesis:

Ha: There is some relationship between X and Y.

We can reject H0 if \(\beta_1\) is far enough from 0. The "far enough" depends on \(SE(\hat{\beta_1})\). If standard error is small, a small \(\beta_1\) is enough. We compute the t-statistic:

\begin{eqnarray} t = \frac{\hat{\beta_1} - 0}{SE(\hat{\beta_1})} \end{eqnarray}

It measures the number of standard deviations that \(\hat{\beta_1}\) is away from 0.

p-value
the probability of observing any value equal to \(|t|\) or larger, assuming \(\beta_1=0\). Reject null hypothesis if p-value is small enough, often 1%.

2.2.2 accuracy of model

Use:

Residual Standard Error (RSE)
R2 statistic
RSE is measured in terms of units of Y. It is not a clear metrics. R2 is between 0 and 1. Near 1 means a large proportion of the variability in the response has been explained by the regression. Near 0 indicates the regression did not explain the variability very well.
Total Sum of Square (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.
\begin{eqnarray} TSS = \sum (y_i - \bar{y})^2 \\ R^2 = \frac{TSS - RSS}{TSS} \end{eqnarray}
F-statistic
Used when we want to test null hypothesis for all the predictors at once.
\begin{eqnarray} F = \frac{(TSS - RSS) / p}{RSS / (n-p-1)} \end{eqnarray}

in the above formula, \(n\) is the number of observation, \(p\) is number of features.

When there's no relationship between the response and predictors, one would expect the F-statistic to take on a value close to 1. If it is large, say 570, we can reject the NULL hypothesis. Now the hypothesis becomes

H0: β1 = β2 = … = βp = 0

Meaning that there's at least one feature relates to the response.

However, f-statistics is not always clear. When \(n\) is large, an F-statistic a little larger than 1 might still provide evidence against \(H_0\). We can also compute p-value associated with F-statistic. Again, a small p-value reject NULL hypothesis.

2.3 Assumptions

2.3.1 Kris Note

Note that in Kris's note, the assumptions and testing stragtage is pretty different. Three assumptions should be checked:

  • \(E[e_i] = 0\)
  • \(Var[e_i] = \sigma^2\)
  • \(Cov(e_i, e_j) = 0, \forall i \neq j\)

The third one can be detected in autocorrelation function (ACF). Every vertical line crossing the confidence interval (95%) is suspicious.

2.3.1.1 Tests
  • Testing for randomness of the residuals: Runs test.
  • Testing for nonconstant variance of the residuals: Breusch-Pagan test (for linear models).

2.3.2 non-linearity of the response-predictor relationships

2.3.2.1 Detection

plot the residual plots. The residual is \(e_i = y_i - \hat{y}_i\). Plot it versus the predictor \(x_i\), or in case of multiple predictors, versus the fitted response. The shape should not have clear patterns, i.e. it should be a flat horizontal line. If it is "U shape", then it indicates non-linear

2.3.2.2 Solution

try some non-linear transformations of the predictors, such as \(log X\), \(\sqrt{X}\), \(X^2\).

2.3.3 correlation of error terms

The error terms \(\epsilon_i\) should be uncorrelated.

2.3.3.1 Consequence

Confidence and prediction intervals will be narrower. p-values associated with the model will be lower.

2.3.3.2 Detection

Plot the residuals. If the error terms are correlated, we will see tracking in the residuals, i.e. adjacent residuals may have similar values.

2.3.4 non-constant variance of error terms

The linear regression model assumes the error terms have a constant variance, i.e. \(Var(\epsilon_i) = \sigma^2\).

2.3.4.1 Consequence

Standard errors, confidence interval, hypothesis test.

2.3.4.2 Detection

E.g. the residuals increase with the value of response. To confirm it is constant, we can use Breusch-Pagan test.

2.3.4.3 Solution

Transform response Y to concave function such as \(log Y\) or \(\sqrt{Y}\).

2.3.5 Outliers

An outlier is a point for which yi is far from the value predicted by the model.

2.3.5.1 Consequence

An outlier does not have much effect on the least squares fit. But it can cause RSE and R2 to decrease.

2.3.5.2 Detection

Residual plots can be used. But more clearly to judge is to use studentized residuals, computing by dividing each residuals \(e_i\) by its estimated standard error. If the value exceeds 3, it is possibly an outlier.

2.3.5.3 Solution
  • remove the observation
  • think about a missing predictor

2.3.6 High-leverage points

observations with an unusual value for \(x_i\).

2.3.6.1 Consequence

Impact on least squares line.

2.3.6.2 Detection

compute leverage statistic.

\begin{eqnarray} h_i = \frac{1}{n} + \frac{(x_i - \bar{x})^2}{\sum_{i'=1}^n (x_{i'} - \bar{x})^2} \end{eqnarray}

This value is always between \(1/n\) and 1. The average value is \((p+1)/n\). If it greatly exceeds \((p+1)/n\), it is a high leverage point.

2.3.7 collinearity

Two or more predictor variables are closely related to one another.

2.3.7.1 Consequence
  • Difficult to separate out the individual effects of collinear variables on the response.
  • Reduce the accuracy of estimates of the regression coefficients
  • standard error for \(\hat{\beta}_j\) to grow
  • reduce the power of hypothesis test, i.e. may not be able to reject it
2.3.7.2 Detection
  • look at correlation matrix, to see if correlated
  • multicollinearity: three or more variables correlated

In multicollinearity, we need to compute variance inflation factor (VIF). A VIF value that exceeds 5 or 10 indicates a problematic amount of collinearity.

2.4 KNN regression

  1. find the K nearest observations, \(N_0\)
  2. estimate \(f(x_0)\) using the average of all the training responses in \(N_0\)
\begin{eqnarray} \hat{f}(x_0) = \frac{1}{K} \sum_{x_i \in N_0} y_i \end{eqnarray}

3 Classification

3.1 logistic regression

Linear regression will have negative values. So we use logistic regression, the response is between 0 and 1.

logistic function
\begin{eqnarray} p(X) = \frac{e^{\beta_0 + \beta_1 X}}{1 + e^{\beta_0 + \beta_1 X}} \\ \frac{p(X)}{1 - p(X)} = e^{\beta_0 + \beta_1 X} \\ log(\frac{p(X)}{1 - p(X)}) = \beta_0 + \beta_1 X \end{eqnarray}

The left hand side of last formula is called log-odds or logit. So logistic regression model has a logit that is linear in X.

3.1.1 Estimate the coefficients

To fit the model, we use maximum likelihood. the likelihood function:

\begin{eqnarray} l(\beta_0, \beta_1) = \prod_{i:y_i=1} p(x_i) \prod_{i':y_{i'} = 0} (1-p(x_{i'})) \end{eqnarray}

3.1.2 multiple logistic regression

\begin{eqnarray} log(\frac{p(X)}{1-p(X)}) = \beta_0 + \beta_1 X_1 + ... + \beta_p X_p \end{eqnarray}

3.2 Linear Discriminant Analysis (LDA)

Assume the observations within each class are drawn from a multivariate Gaussian distribution. Then plug estimates into Bayes' theorem, in order to perform prediction. Many prior and posterior.

It is an alternative to logistic regression. It is more stable, and popular for more than two response classes. it is closely related to logistic regression.

3.3 Quadratic Discriminant Analysis (QDA)

It has similar assumptions as LDA except:

  • LDA assumes K classes share a common covariance matrix.
  • But QDA assume that each class has its own covariance matrix.

3.3.1 How to choose from LDA or QDA

The bias-variance trade-off.

LDA is better:

  • LDA is a much less flexible classifier than QDA.
  • So it has a much lower variance.
  • It improve the prediction performance.

QDA is better:

  • if the assumption of common covariance matrix is bad, LDA is bias
  • if the training set is very large, the variance of the classifier is not a major concern.

3.4 KNN

4 Re-sampling Methods

model assessment
The process of evaluating a model's performance.
model selection
the process of selecting the proper level of flexibility for a model

4.1 cross-validation

Hold out a subset of training observations from the fitting process, for test.

4.1.1 Validation set approach

Randomly divide the available set of observations into two parts, a training set and a validation set or hold-out set. It has two drawbacks

4.1.2 cross validation

It is a refinement of validation set approach to address the two drawbacks.

4.1.2.1 Leave-One-Out Cross-Validation (LOOCV)

every time, a single observation pair is left out. Calculate the MSE. Repeat these steps for every observation pair, i.e. each one has the chance to be left out. The test MSE is the average

\begin{eqnarray} CV_{(n)} = \frac{1}{n} \sum_{i=1}^n MSE_i \end{eqnarray}

The advantage:

  • it is far less bias
  • perform multiple times will always get the same result, i.e. no randomness

But it can be expensive. But there's an "amazing shortcut".

4.1.2.2 k-Fold Cross-Validation

randomly divide observations into k groups. Every group is left out, as in LOOCV. The test MSE is still the average.

4.2 bootstrap

Can be used to estimate the standard errors of the coefficients from a linear regression fit. It can be easily applied to a wide range of statistical learning methods.

We want to know the standard errors of our estimated coefficients. We sample from the original dataset, with replacement, 1000 samples. For each one, calculate the value. Then use the mean. With replacement means some observations can occur more than once. We can sample 6 items into a sample even if the total number of observations is only 3.

5 Linear Model Selection and Regularization

5.1 Subset Selection

5.1.1 best subset selection

Try all the combinations of the features. Pick the best model from \(2^p\) possibilities.

5.1.2 Stepwise Selection

The best subset selection cannot be used with large \(p\) value.

5.1.2.1 forward stepwise selection

adds predictors to the model, one at a time, until all of the predictors are in the model. At each step, the variable gives the greatest additional improvement to the fit is added.

5.1.2.2 backward stepwise selection

Start from all predictors, iteratively remove the least useful predictor, one at a time.

5.1.2.3 hybrid approaches

After adding each new variable, the method may also remove any variables that no longer provide an improvement.

5.2 Shrinkage

5.2.1 Ridge Regression

The ordinary least squares minimize:

\begin{eqnarray} RSS = \sum_{i=1}^n (y_i - \beta_0 - \sum_{j=1}^p \beta_j x_{ij})^2 \end{eqnarray}

and ridge regression introduce a shrinkage penalty:

\begin{eqnarray} RSS + \lambda \sum_{j=1}^p \beta^2_j \end{eqnarray}

When \(\beta_j\) is small toward 0, the above added penalty will be small. So it will make the \(\beta_j\) smaller, i.e. shrinkage. The parameter \(\lambda\) is critical for the influence of the penalty.

Actually it uses the l2 norm.

\begin{eqnarray} ||\beta||_2 = \sqrt{\sum_{j=1}^p} \beta^2_j \end{eqnarray}

5.2.2 Lasso

Ridge regression cannot remove any features, unless \(\lambda = \infty\). This may not be a problem for prediction accuracy but it can create a challenge in model interpretation.

The Lasso uses l1 norm penalty.

\begin{eqnarray} ||\beta||_1 = \sum |\beta_j| \end{eqnarray}

\(l_1\) penalty has the effect of forcing some of the coefficient estimates to be exactly equal to 0 when the tuning parameter \(\lambda\) is sufficiently large.

It is much easier to interpret, it yields sparse model, i.e. models that involve only a subset of the variables.

5.3 Dimension Reduction

linear combination of the predictors into M new predictors.

\begin{eqnarray} Z_m = \sum_{j=1}^p \phi_{jm} X_j \end{eqnarray}

5.3.1 Principal Components Regression

5.3.1.1 Principal Component Analysis (PCA)

The following are some criteria for the direction selection, they all talk about the same thing:

  • The first principal component direction is that along which the observation vary the most.
  • This also yields the highest variance.
  • It also defines the line that is as close as possible to the data.
  • projected observations are as close as possible to the original observations.

The second principal component \(Z_2\) is a linear combination of the variables that is uncorrelated with \(Z_1\), and has largest variance subject to this constraint. Actually \(Z_1\) and \(Z_2\) are always orthogonal.

5.3.1.2 Principal Component Regression (PCR)

Construct the first M principal components, and do linear regression on the new predictors.

5.3.2 Partial Least Squares (PLS)

The directions identified by PCA is in an unsupervised way, i.e. it does not use response Y.

Set each \(\phi_{j1}\) equal to the coefficient from teh simple linear regression of Y onto Xj. Intuitively PLS places the highest weight on the variables that are most strongly related to the response.

Second PLS direction is by

  1. adjust each of the variables for Z1, by regressing each variable on Z1 and taking residuals This captures the remaining information that has not been explained by the first PLS direction
  2. use this orthogonalized data in exactly the same fashion as Z1.
  3. Repeat M times.

6 Non-linear

6.1 polynomial regression

It is just replace standard linear model to higher dimension ones (typically less than 4). The one with \(X,X^2,X^3\) is called cubic regression.

6.2 step function

Also called piecewise constant regression. It actually piecewise the data, and do linear regression. The linear model is

\begin{eqnarray} y_i = \beta_0 + \beta_1 C_1(x_i) + \beta_2 C_2(x_i) + ... + \beta_K C_K(x_i) + \epsilon_i \end{eqnarray}

Given a value X, there's at most one of \(C_i\) can be non-zero.

6.3 basis function

a set of basis function \(b_i(X)\):

\begin{eqnarray} y_i = \beta_0 + \beta_1 b_1(x_i) + \beta_2 b_2(x_i) + ... + \beta_K b_K(x_i) + \epsilon_i \end{eqnarray}

6.4 regression spline

It is piecewise polynomial. But it ensures the smooth at the knots.

We have K knots, and fit a cubic regression. At the knots, we need to ensure the 0,1,2 deviation is the same.

6.5 smoothing spline

This is a different approach, but also produces a spline. Instead of making RSS minimal, we make the following minimal

\begin{eqnarray} RSS = \sum_{i=1}^n (y_i - g(x_i))^2 \end{eqnarray}

We need to find a \(g\). If we do not put any constraints, we can simply let \(g\) equal to \(y_i\). But this is overfitting. We need some constraints on \(g\). We want to find the \(g\) that minimizes:

\begin{eqnarray} \sum_{i=1}^n (y_i - g(x_i))^2 + \lambda \int g''(t)^2dt \end{eqnarray}

The function \(g\) that minimizes it is a smoothing spline.

The first term is a loss function, nd second is a penalty term.

6.6 local regression

  1. gather the fraction \(s = k/n\) of training points whose \(x_i\) are closest to \(x_0\).
  2. assign a weight \(K_{i0} = K(x_i, x_0)\) to each point, so that the point furthest from x0 has weight 0, and closest has highest weight. All but these k nearest neighbors get 0.
  3. fit a weighted least squares regression, by finding \(\beta_0\) and \(\beta_1\) that minimize
\begin{eqnarray} \sum_{i=1}^n K_{i0} (y_i - \beta_0 - \beta_1 x_i)^2 \end{eqnarray}

local regression can perform poorly if p is much larger than 3 or 4, because there will be very few training observations close to x0.

6.7 Generalized Additive Models (GAMs)

It is called additive model because we calculate a separate \(f_j\) for each \(X_j\), and add together all of their contributions.

7 Tree-based Methods

7.1 Decision tree

7.1.1 regression tree

  1. divide the predictor space into J distinct and non-overlapping regions \(R_1,...,R_J\).
  2. for each observation fail into Rj, make the prediction using the mean in Rj.

To get the regions, use recursive binary splitting, a top-down, greedy approach.

  • From the root
  • every split choose the best split that leads to the greatest possible reduction of RSS

It is likely to overfit the data. So we can grow a very large tree, and then prune it back in order to obtain a subtree.

The whole algorithm goes here:

  1. recursive binary splitting to grow a large tree
  2. apply cost complexity pruning
  3. use K-fold cross-validation

7.1.2 classification tree

intuitive

7.2 bagging

The decision tree suffers from high variance. If we split the training data into two parts at random, the result two trees can be very different.

Bagging can reduce the variance. It is related to bootstrap.

Bagging involves

  1. creating multiple copies of the original training data set using the bootstrap,
  2. fitting a separate decision tree to each copy,
  3. and then combining all of the trees in order to create a single predictive model.

Each tree is built on a bootstrap data set, independent of the other trees.

The key idea is averaging a set of observations reduces variance.

7.3 boosting

This is another approach for improving the prediction results from a decision tree. The different from bagging is,

  • the trees are grown sequentially: each tree is grown using information from previously grown trees.
  • Boosting does not involve bootstrap sampling. Each tree is fit on a modified version of the original data set.

8 Support Vector Machine

the best “out of the box” classifiers.

A hyperplane is something like this:

\begin{eqnarray} \beta_0 + \beta_1 X_1 + \beta_2 X_2 = 0 \end{eqnarray}

A hyperplane can separate two clusters.

8.1 Maximal Margin Classifier

Also known as optimal separating hyperplane.

In fact exist an infinite number of such hyperplanes. This is the motivation.

The separating hyperplane that is farthest from the training observations. That is, we can compute the (perpendicular) distance from each training observation to a given separating hyperplane; the smallest such distance is the minimal distance from the observations to the hyperplane, and is known as the margin

The maximal margin hyperplane is the separating hyperplane for which the margin is largest—that is, it is the hyperplane that has the farthest minimum dis- tance to the training observations.

The closest observations are support vectors. they “support” the maximal margin hyperplane in the sense that if these points were moved slightly then the maximal margin hyperplane would move as well.

guarantees that each observation will be on the correct side of the hyper- plane, provided that M is positive.

8.2 Support Vector Classifier

The above method is

not stable
An additional blue observation has been added, leading to a dramatic shift in the maximal margin hyperplane.
not allow mistake

Also known as soft margin classifier.

allow some observations to be on the incorrect side of the margin, or even the incorrect side of the hyperplane.

The model formula is TODO.

The parameters in the model:

  1. the slack variable εi tells us where the ith observation is located, relative to the hyperplane and relative to the margin
  2. C: determines the number and severity of the vio- lations to the margin (and to the hyperplane) that we will tolerate C is treated as a tuning parameter that is generally chosen via cross-validation

Some observations:

  1. only observations that either lie on the margin or that violate the margin will affect the hyperplane
  2. an observation that lies strictly on the correct side of the margin does not affect the support vector classifier
  3. Observations that lie directly on the margin, or on the wrong side of the margin for their class, are known as support vectors.
  4. When the tuning parameter C is large, then the margin is wide

8.3 Support Vector Machine

enlarging the feature space in a specific way, using kernels. Can deal with non-linear, just use a non-linear kernel.

linear kernel:

\begin{eqnarray} K(x_i, x_{i'}) = \sum_{j=1}^p x_{ij} x_{i'j} \end{eqnarray}

polynomial kernel:

\begin{eqnarray} K(x_i, x_{i'}) = (1 + sum_{j=1}^p x_{ij} x_{i'j})^d \end{eqnarray}

Radial kernel:

\begin{eqnarray} K(x_i, x_{i'}) = exp(-\gamma \sum_{j=1}^p (x_{ij} - x{i'j})^2) \end{eqnarray}

8.4 SVMs with More than two classes

8.4.1 one-versus-one classification

compute all pairs SVMs.

8.4.2 one-versus-all classification

compute all one versus all other SVMs.

9 Unsupervised Learning (Clustering)

Euclidean distance

9.1 K-means

W(Ck)
the amount by which the observations within a cluster differ from each other

It says the total within-cluster variation, summed over all K clusters, is as small as possible. It defines the within-cluster variation. The formula for it is: the sum of all of the pairwise squared Euclidean distances between the observations in the kth cluster.

\begin{eqnarray} W(C_k) = \frac{1}{|C_k|} \sum_{i,i' \in C_k} \sum_{j=1}^p (x_{ij} - x_{i'j})^2 \end{eqnarray}

where \(|C_k|\) denotes the number of observations in the kth cluster.

The algorithm:

  1. select a number K, randomly assign a clustering from 1 to K for each observation
  2. iterate until cluster assignments stop changing 2.1 for each cluster, compute centroid: the vector of the p features means for the observations in the kth cluster. 2.2 assign each observation to the cluster whose centroid is closest.

This algorithm guarantee to decrease the objective formula above.

9.2 Hierarchical Clustering

Does not predefine the number of clusters. The result is called a dendrogram, a tree-based representation of the observations.

It is constructed bottom-up. The tree node means a fusion. The height of the fusion indicates how different the two observations are. Never compare the horizontal distance.

Construction algorithm: examine all pairwise inter-cluster dissimilarities among all clusters. Fuse the most similar ones.

The four most commonly used types of linkage:

  • complete: maximal intercluster dissimilarity
  • single: minimal intercluster dissimilarity
  • average: mean intercluster dissimilarity
  • centroid: dissimilarity between the centroid of cluster A and B

Author: Hebi Li

Created: 2017-03-27 Mon 14:36

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