Of textbook B - weblectures quiz
12 important questions on Of textbook B - weblectures quiz
What is an input vector?
- Y is the dependent variable, referred to as response or target.
- independent variables are denotes as feature, input, or predictors.
- you usually have a set of predictors, x1, x2, etc.
- this set can be collectively referred to as the input vector. Which is a vector with all features in it.
- Large X's are variables, while small x's are datapoints
How can you write the regression formula for three regressors?
- F(x) = f(x1, x2, x3) = E(Y | X1 = x1, X2 = x2, X3 = x3)
- this just means that f(x) returns the average of all Y's for which holds: X1 = x1, X2 = x2, X3 = x3
What is a problem with estimating f, and how is it solved?
- Particulary in small datasets, it might be the case that we have no datapoints with X = x.
- we thus cannot compute E(Y | X = x), the average of Y for X = x.
- we can still compute f^(x) through the following:
- f^(x) = Ave(Y | X ∊
N(x))
- which means estimate f^(x) on the basis of neighbouring points.
- this is called nearest neighbor averaging
it works best with small amount of variables and largish amount of datapoints.
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Why does nearest neighbor averaging not work for datasets with more variables?
- If you want to include a specific proportion of the sample to find f^(x), let's say 10%;
- you need a bigger radius to achieve this goal is you have a second variable compared to when you have one variable
- the more dimensions, the further you have to look to find ten procent of the sample.
- the further you have to look for neighbors, the less similar these neighbors are going to be which means less accurate predictions
- this concept is called the curse of dimensionality
How can we solve the curse of dimensionality?
- By adding more structure to our model; using a pre-specified formula, like the linear regression formula, we can predict Y for X = x without using nearest neighbor averaging.
What is meant by flexibility of a model/method?
- Flexibility is the degree to which the model has freedom to fit to the data.
- making our model the mean of y (so for every x value the predicted y value is it's mean) is not very flexible
- a linear regression is slightly more flexible
- a polynomial is even more flexible.
- as models become more flexible they have the opportunity to fit to the data better, but they also have a larger tendency to overfit to trainingdata.
What is smoothing and how is K nearest Neighbor averaging an example of it?
- Smoothing is the collective of methods of forming a model that work not by fitting a prespecified model, like a regression, to the data but specify prediction (points) based on what the data in that location indicates.
- k nearest neighbor averaging forms a line through the data cloud by taking the average of the values of the k nearest neighbors around that point. Y^i is thus based on what the data around xi is like.
How does K nearest naighbor classifying work for qualitative y?
- KNN classifying assigns individual cases to a class by looking at which classes the
K nearest neighbors belong to. - Then it estimates the conditional probability for each class j as the proportion of class j members in the neighborhood.
- it assigns the point to the class to which the highest amount of neighbors belong
dimension curse is also present here, similarly to in regression problems.
How is performance of a classification system assessed?
- The performance of C^(x) is assessed by looking at the misclassification error rate:
- Err te = Ave i ∊ te I [yi =! C^(xi)]
- which states that the misclassification error rate is the average of instances in a test set where the
predicted class did not match the actual class.
How can the dimension curse be solved for knn classifiers?
- Again use structured models like support-vector machines.
- also structured models for obtaining conditional class probabilities like logistic regression can be used.
- the dimension curse is a bigger problem for modelling the probabilities compared to just classifying
What is the decision boundary?
in this case the black, or purple dotted line
How do the size of k (amount of neighbors evaluated) and the error rate relate?
- In this figure, K becomes smaller towards the right side. The model thus becomes more flexible.
- again we see that as the model becomes more flexible, training errors become smaller.
- test errors however, benefit from a certain degree of flexibility but becoming more flexible after this optimum causes more test error since the model is overfitting to the training data.
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