Week 6: Regression and Classification Flashcards
1
Q
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Advantage: have the potential to accurately fit a wider range of possible shapes for f
Disadvantage: a very large number of observations is required to obtain an accurate estimate for f
A
What is the advantage and disadvantage of non-parametric methods
2
Q
RevERSED
predict(model)
A
How do you predict the outcome of the model in R?
3
Q
RevERSED
Presence of a funnel shape in the residual plot. Transform Y to log(Y) or sqrt(Y)
A
How can you detect non-constant variance of error terms? What is a solution?
4
Q
RevERSED
Use least squared approach to minimise RSS (residual sum of squares)
A
How do you find the coefficients of a simple linear regression model?
5
Q
RevERSED
LOOCV has higher levels of variance than k-fold because the model outputs are highly correlated with each other and therefore the mean has higher variance
A
Why does k-fold CV give more accurate estimates of MSE than LOOCV?
6
Q
RevERSED
One less than the number of levels, because there is a baseline level with no dummy variable
A
How many dummy variables will there be when there is a predictor with more than 2 levels?
7
Q
RevERSED
- Forward selection: begin with null model. Then fit p simple linear regressions and add to the null model the variable that results in the lowest RSS. Continue adding variables until some stopping rule is satisfied
- Backward selection: start with all variables, remove the variable with the largest p-value, continue removing variables until a stopping rule is reached
- Mixed selection: combination of forward and backward. Start with no variables in the model. Add the variable that provides the best fit. Continue to add variables one by one. If at any point the p-value for one of the variables in the model rises above a certain threshold, then remove that variable from the model. Continue until all the 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
A
What are the three approaches for deciding which variables to include in a model? How do they work?
8
Q
RevERSED
Do no make explicit assumptions about the functional form of f
A
What are non-parametric methods?
9
Q
RevERSED
the percentage of Falses that are identified correctly = TN/(TN+FP)
A
What is specificity?
10
Q
RevERSED
Randomly divide the set of observation into k groups (or folds) of approximately equal size. The first fold is treaded as a validation set and the method is fit on the remaining k-1 folds. Repeat k times and get k estimates of the MSE. Find the average
A
What is k-fold cross validation?
11
Q
RevERSED
irreducible and reducible error
A
What does the accuracy of Y* as a prediction for Y depend on?
12
Q
RevERSED
Automatically outputs the log odds. To change it:
predict(lr_mod, type=“response”)
A
What is the default when using predict with logistic? How do you change it?
13
Q
RevERSED
predicted probability versus observed proportion, should be a straight line with slope 1
A
What is a calibration plot?
14
Q
RevERSED
splits
A
What is a code for splitting data into test and train in R?
15
Q
RevERSED
recursive partitioning . Find the split that makes observations as similar as possible on the outcome within that split. Do that again with each resulting group. Stop at stopping parameter
A
What are classification trees?
16
Q
RevERSED
predict(model, newdata = data)
A
How do you predict the outcome of the model in R based on new data?
17
Q
RevERSED
Tend to overfit
Use them as a basic building block for ensembles
A
What is the problem with regression trees? What can they be used for?
18
Q
RevERSED
Compute the standard error of B0 and B1
A
How do you assess the accuracy of the coefficient estimates?
19
Q
RevERSED
- Additivity assumption: that the association between a predictor X and the response Y does not depend on the values of the other predictors
- the error terms e1, e2, … are uncorrelated
- the error terms have a constant variance, Var(ei) = sigma squared
A
What are the assumptions of the linear model? (3)
20
Q
RevERSED
Use dummy variables. 0 for one, 1 for other. Or -1 and 1
A
How do you put a categorical variable in a linear regression model?
21
Q
RevERSED
as p increases (more dimensions), a given observation has no nearby neighbours
A
What is the curse of dimensionality?
22
Q
RevERSED
bias initially decreases faster than variance increases, so the MSE declines. But at some point increasing flexibility has more impact on the variance, so the MSE increases.
A
What happens to the MSE as you increase flexibility?
23
Q
RevERSED
Given a value for K and a prediction point x0, KNN regression first identifies the K training observations that are closest to x0 represented by N0. Then it estimates f(x0) using the average of all the training responses in N0
A
How does k nearest neighbours regression work?
24
Q
RevERSED
if we include an interaction in a model we should also include the main effects, even if the p-values associated with their coefficients are not significant
A
What is the hierarchical principle?
25
# RevERSED
Advantage: reduce the problem of estimating f down to one of estimating a set of parameters
Disadvantage: will usually not match the true unknown form of f
What is the advantage and disadvantage of parametric methods?
26
# RevERSED
Y = B0 + B1X1 + B2X2 + B3X1X2 + e
Combination of predictors
What are interaction terms?
27
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p(X) = [e^ B0+B1X] / [1 + e^ B0+B1X]
What is the logistic function?
28
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the percentage of Trues that are identified correctly = TP/(TP+FN)
What is sensitivity/recall?
29
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small training MSE but large test MSE
What happens to MSE when model is overfitted?
30
# RevERSED
predicting class one if Pr (Y = 1 | X = x0) \> 0.5
What does Bayes classifier correspond to in a two-response value setting?
31
# RevERSED
lr\_mod
How do you fit a logistic regression model using R?
32
# RevERSED
increasing X by one unit changes the log odds by B1
How do you interpret B1 in a logistic regression model?
33
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mean squared error
MSE = 1/n SUM(y - predicted y)^2
What is the most commonly used measure for measuring the quality of fit? what is the formula?
34
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A model is perfectly calibrated if for any probability value p, a prediction of a class with confidence p is correct 100\*p percent of the time
What is callibration?
35
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Z-statistic
What is used for hypothesis testing in logistic regression?
36
# RevERSED
pred\_prob 0.5, labels = c(“No”, “yes”)
table(true=, predicted = pred\_lr)
How do you produce table of observed vs predicted results when classified as probability?
37
# RevERSED
e: cannot be predicted using X, therefore the error introduced by e cannot be reduced
What is irreducible error?
38
# RevERSED
TP/(TP+FP)
What is the positive predictive value (PPV) / precision?
39
# RevERSED
When performing regressions with a single predictor shows a very different outcome to performing regressions with multiple predictors that are also relevant
What is confounding?
40
# RevERSED
ifelse(student==“Yes”, 1, 0)
How would you turn a vector of "yes" and "nos" into a vector of 1s and 0s
41
# RevERSED
(TP+TN)/(TP+FP+FN+FN)
What is accuracy?
42
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training MSE will decrease, but test MSE may not
What happens to MSE as model flexibility increases?
43
# RevERSED
There is a dataset, set of people trying to find prediction rule and a referee.
The referee runs the prediction rule against a testing dataset, which is sequestered behind a Chinese wall
The referee objectively and automatically reports the score achieved by the submitted rule
Results in declining error rate
What is the common task framework /benchmarking?
44
# RevERSED
B0 can be interpreted as the average Y among non-students. B0 + B1 as the average Y among students. B1 as the average difference in Y between students and non students
How do you interpret B0 and B1 when there is a dummy variable 1, when someone is a student, 0 when they are not a student
45
# RevERSED
Assume that X= (X1, …Xp) is drawn from multivariate normal distribution with a class-specific mean vector and common covariance matrix
What are the assumptions for linear discriminant analysis when p\>1?
46
# RevERSED
For each observation xi, there is an associated response measurement yi
What is supervised learning?
47
# RevERSED
P(A|B) = P(B|A) . P(A) / P(B)
What is Bayes theorem in general terms?
48
# RevERSED
e is a random error term which is independent of X and has mean 0
In the formula Y = f(X) + e, what are the properties of e?
49
# RevERSED
Model the distribution of the predictors X separately in each of the response classes. Then use Bayes theorem to flip these around into estimates for Pr(Y=k|X=x)
What is a generative model for classification?
50
# RevERSED
```
assigns an observation X=x to the class for which pk(x) is the largest (assigns each observation to the most likely class given predictor values)
-produces the lowest possible test error rate, called the Bayes error rate
```
What does Bayes classifier do?
51
# RevERSED
TSS measures the total variance in the response Y before the regression is performed
RSS measures the variability that is left unexplained after performing the regression
TSS - RSS measures the amount of variability in the response that is explained by performing the regression
What do TSS and RSS each measure? and TSS-RSS
52
# RevERSED
Measures the proportion of variability in Y that can be explained using X
R^2 = (TSS-RSS)/TSS
What is the R squared statistic?
53
# RevERSED
less true positives and less false positives (ie less positives overall)
What will happen if you increase the cutoff value?
54
# RevERSED
Low-levels of smoothness can lead to overfitting
When selecting a level of smoothness for non-parametric methods, what is the trade-off?
55
# RevERSED
A single observation (x1, y1) is used for the validation set and the remaining observations make up the training set.
Find the MSE and repeat this approach n times and get the average of the n MSE estimates
What is the leave one out cross validation (LOOCV) approach?
56
# RevERSED
false positive rate vs true positive rate. The area under the curve gives the overall performance of a classifier
What is the ROC curve?
57
# RevERSED
- validation estimate of the test-error rate can be highly variable depending on which observations are included in the training set and which are included in the validation set
- not as many observations in the training set
What are 2 drawbacks of the validation set approach?
58
# RevERSED
Find estimate for fk(x) to estimate pk(x) by approximating bayes classifier
What is the aim of linear discriminant analysis?
59
# RevERSED
the error that is introduced by approximating a real-life problem which may be very complicated, by a simpler model. In general, more flexible methods result in less bias
What is bias?
60
# RevERSED
log(p(X)/(1-p(X))) = B0 + B1X
What are the log odds from a logistic function?
61
# RevERSED
seq(0, 40, length.out = 1000)
Code to generate sequence of 1000 equally spaced values from 0 to 40
62
# RevERSED
B0\* +- 2.SE(B0\*)
where 2 is actually the 97.5% quantile of a t-distribution with n-2 degrees freedom
How do you find the confidence intervals of the coefficient estimates?
63
# RevERSED
models the probability that Y belongs to a particular category
What is a logistic regression model?
64
# RevERSED
knn(train, test, cl, k)
train is the training set with only the predictor variables, test is the test set with only the predictor variables, cl is the outcomes from the training set, k is the desired k
How do you make a prediction vector using knn in R?
65
# RevERSED
1 - Accuracy
What is the error rate?
66
# RevERSED
Y = B0 + B1X
What is the simple linear regression model?
67
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An unusual value for xi
for multiple regression: point that is unusual in terms of the full set of predictors
What is a high leverage point?
68
# RevERSED
- only has to be fit k times compared to n times in LOOCV
- variability in the test error estimate is lower than when using the validation set approach
What are the 2 advantages of k-fold cross validation?
69
# RevERSED
```
F1 score = sqrt(precision x recall)
it is not affected by uneven class distributions
```
What is the F1 score and what is the point of it?
70
# RevERSED
E(MSE) = bias-squared + variance + e
How can you write the expected test MSE
71
# RevERSED
ensemble of weak learners. Make trees that are too simple and make more of them for observations with big residuals, then average them
What is boosting?
72
# RevERSED
the estimated standard errors will be too low -\> unwanted sense of confidence in model
What happens to the linear model if the error terms are correlated?
73
# RevERSED
- fk(x) is normal
- there is a common variance across all K classes
What are the 2 assumptions for linear discriminant analysis when p=1?
74
# RevERSED
yi = yi\*. difference between observed and predicted response
What is the residual ei for each data point?
75
# RevERSED
It is still a linear model
What type of model is polynomial regression?
76
# RevERSED
model
How do you create a linear model in R? and get its coefficients and r^2 etc
77
# RevERSED
sum of all the residuals squared:
e1^2 + e2^2 ...
What is the residual sum of squares (RSS)?
78
# RevERSED
Classifying a response variable with more than 2 classes
What is multinomial logistic regression?
79
# RevERSED
Cor(Y, Y\*)
What is R-squared always equal to in multiple linear regression?
80
# RevERSED
F-statistic, large F-statistic to reject null hypothesis
What do you need to check the significance of multiple coefficients together e.g. B0 = B1 = B2 = 0?
81
# RevERSED
Pr(X|Y=k)
What does fk(x) represent in linear discriminant analysis?
82
# RevERSED
randomly divide the available set of observations into two parts, a training set and a validation set. The model is fit on the training set and the fitted model is used to predict the responses for the observations in the validation set
What is the validation set approach?
83
# RevERSED
Given a positive integer K and a test observation x0, the KNN classifier first indentifies the K points in the training data that are closest to x0, represented by N0. It then estimates the conditional probability for class j as the fraction of points in N0 whose response values equal j. KNN classifies the test observation x0 to the class with the largest probability.
How does the k-nearest-neighbours classifier work?
84
# RevERSED
f\* will not be a perfect estimate for f and will cause some error, this error is reducible because we can potentially improve the accuracy of f
What is reducible error?
85
# RevERSED
AutoML performs all the steps of comparing different models using cross-validation, choosing parameters automatically
fit\_aml
What is autoML and its code in R?
86
# RevERSED
Plot the standardised residuals. Those with absolute values greater than 3 may be outliers
How can you detect outliers?
87
# RevERSED
use the number of mis-classified observations rather than the MSE
How do you quantify the test error on classification problems?
88
# RevERSED
lda\_mod
How do you fit a linear discriminant analysis model in R?
89
# RevERSED
A very low K value means the decision boundary is overly flexible and finds patterns in the data that don’t correspond to the Bayes decision boundary. This classifier has low bias but very high variance
A very high K value means the classifier becomes less flexible and produces a decision boundary that is close to linear. This is a low-variance but high-bias classifier
What is the trade-off for a low or high K value in KNN classifier?
90
# RevERSED
predict(model, newdata = data, interval = “confidence”)
How do you predict the outcome of the model in R based on new data and get the upper and lower confidence intervals?
91
# RevERSED
- Prediction intervals are used to answer the question how much will Y vary from Y\*
- Prediction intervals are always wider than confidence intervals because they incorporate both the reducible error and irreducible error
What is a prediction interval?
92
# RevERSED
residual standard error:
RSE = sqrt[(1 / n-2)\*RSS]
It is the average amount that the response will deviate from the true regression line.
It is an absolute measure of the lack of fit of the model
How do you assess the accuracy of the linear regression model?
93
# RevERSED
For each observation we observe a vector of measurements xi but no response yi. Seek to understand relationship between observations
What is unsupervised learning?
94
# RevERSED
If epilepsy is set as the baseline, the B(stroke)0 is interpreted as the log odds of stroke versus epilepsy given that x1 = … = x1 = 0. A one unit increase in Xj is associated with a B(stroke)j increase in the log odds of stroke over epilepsy
How do you interpret the coefficients of multinomial logistic regression? with stroke, overdose and epilepsy as the 3 classifications
95
# RevERSED
TN/(TN+FN)
What is the negative predictive value (NPV)?
96
# RevERSED
Hypothesis test. Can compute t-statistic using standard errors. p-value is the probability of observing a value equal or larger than t. small p-value indicates unlikely to observe such a substantial association between the predictor and the response due to chance.
If p-value is small, reject hypothesis that co-efficient is 0
How do you test for significance of the coefficients?
97
# RevERSED
Change the threshold at which an observation is assigned to a class - default is 0.5
How do you alter the sensitivity and specificity of a classifier?
98
# RevERSED
the amount by which f\* would change if we estimated it using a different training data set. In general, more flexible methods have higher variance
What is variance?
99
# RevERSED
table(true=, predicted=)
How do you produce table of observed vs predicted results when classified discretely?
100
# RevERSED
Plot the residuals ei versus the predictor xi
How can you detect non-linearity of data with a linear model?
101
# RevERSED
Plot the residuals as a function of time. Adjacent residuals may have similar values if they are correlated
How can you detect correlation in the error terms?
102
# RevERSED
following the errors or noise too closely
What is overfitting?
103
# RevERSED
First make an assumption about the functional form or shape of f, then use a procedure that uses the training data to fit or train the model
How do you find f with a parametric method?
104
# RevERSED
Develop an agent that improves its performance based on interactions with the environment
What is reinforcement learning?
105
# RevERSED
Collinearity reduces the accuracy of the estimates of the regression coefficients, and causes the standard error to grow
What happens when there is collinearity between the predictor variables?
106
# RevERSED
Advantage: It has less bias because the training set is bigger
Disadvantage: time consuming to implement
What is an advantage and a disadvantage of the LOOCV approach?
107
# RevERSED
Y = B0 + B1X1 + B2X2 + … + BpXp + e
What is the standard multiple linear regression formula?
108
# RevERSED
When there is a small number of observations per predictor
When will parametric methods outperform non-parametric methods?
109
# RevERSED
Instead of selecting a baseline classes, treat all K classes symmetrically. Estimate coefficients for all K classes
What is softmax coding for multinomial logistic regression?
110
# RevERSED
bagged trees with feature sampling. Make trees that are too complex and average over bootstrapped samples to cancel out the overfitting parts
What are random forests?
111
# RevERSED
2 variables: correlation matrix
multiple variables: variance inflation factor (VIF). value exceeding 5 or 10 is problematic
How can you assess collinearity between 2 variables and between multiple variables?
112
# RevERSED
Prediction: predict Y using Y\* = f\*(X),
Inference: understanding the association between Y and X
What is the difference between prediction and inference?
113
# RevERSED
p(X)/(1-p(X)) = e^ B0+B1X
What are the odds from a logistic function?
