questions stringlengths 80 1.7k | answers stringlengths 74 1.45k |
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True or False: For a fixed number of observations in a data set, introducing more vari ables normally generates a model that has a better fit to the data. What may be the drawback of such a model fitting strategy? | True. However, when an excessive and unnecessary number of variables is used in a lo gistic regression model, peculiarities (e.g., specific attributes) of the underlying data set dis proportionately affect the coefficients in the model, a phenomena commonly referred to as “overfitting”. Therefore, it is important that a l... |
Define the term “odds of success” both qualitatively and formally. Give a numerical example that stresses the relation between probability and odds of an event occurring. | and the probability of failure 1 − p. Formally: Odds(p) ≡ ! p 1 − p . For instance, assuming the probability of success of an event is p = 0.7. Then, in our example, the odds of success are 7/3, or 2.333 to 1. Naturally, in the case of equal probabilities where p = 0.5, the odds of success is 1 to 1. |
1. Define what is meant by the term "interaction", in the context of a logistic regression predictor variable. LOGISTIC REGRESSION 2. What is the simplest form of an interaction? Write its formulae. 3. What statistical tests can be used to attest the significance of an interaction term? | 1. An interaction is the product of two single predictor variables implying a nonadditive effect. 2. The simplest interaction model includes a predictor variable formed by multiplying two ordinary predictors. Let us assume two variables X and Z. Then, the logistic regression model that employs the simplest form of inte... |
True or False: In machine learning terminology, unsupervised learning refers to the mapping of input covariates to a target response variable that is attempted at being predicted when the labels are known. | False. This is exactly the definition of supervised learning; when labels are known then supervision guides the learning process. LOGISTIC REGRESSION |
Complete the following sentence: In the case of logistic regression, the response vari able is the log of the odds of being classified in [...]. | In the case of logistic regression, the response variable is the log of the odds of being clas sified in a group of binary or multiclass responses. This definition essentially demonstrates that odds can take the form of a vector. |
Describe how in a logistic regression model, a transformation to the response variable is applied to yield a probability distribution. Why is it considered a more informative repres entation of the response? | When a transformation to the response variable is applied, it yields a probability distribu tion over the output classes, which is bounded between 0 and 1; this transformation can be employed in several ways, e.g., a softmax layer, the sigmoid function or classic normalization. This representation facilitates a softdec... |
Complete the following sentence: Minimizing the negative log likelihood also means maximizing the [...] of selecting the [...] class. 2.2.2 Odds, Logodds | Minimizing the negative log likelihood also means maximizing the likelihood of selecting the correct class. 2.3.2 Odds, Logodds |
Assume the probability of an event occurring is p = 0.1. 1. What are the odds of the event occurring?. 2. What are the logodds of the event occurring?. 3. Construct the probability of the event as a ratio that equals 0.1. 2.2. PROBLEMS | 1. The odds of the event occurring are, by definition: odds = (0.1 0.9) = 0.11. 2. The logodds of the event occurring are simply taken as the log of the odds: logodds = ln(0.1/0.9) = −2.19685. 3. The probability may be constructed by the following representation: probability = odds odds + 1 = 0.11 1.11 = 0.1, or, altern... |
ility of success is 0.8. True or False: If the odds of success in a binary response is 4, the corresponding probab | True. By definition of odds, it is easy to notice that p = 0.8 satisfies the following relation: odds = (0.8 0.2) = 4 |
their respective odds. Draw a graph of odds to probabilities, mapping the entire range of probabilities to | The graph of odds to probabilities is depicted in Figure 2.12. Odds odds(p) = p 1−p 10,0 8,0 6,0 4,0 2,0 0,1 0,2 0,3 0,4 0,5 0,6 Probability 0,7 0,8 0,9 FIGURE 2.12: Odds vs. probability values. LOGISTIC REGRESSION |
The logistic regression model is a subset of a broader range of machine learning models known as generalized linear models (GLMs), which also include analysis of variance (AN OVA), vanilla linear regression, etc. There are three components to a GLM; identify these three components for binary logistic regression. | A binary logistic regression GLM consists of there components: 1. Random component: refers to the probability distribution of the response variable (Y ), e.g., binomial distribution for Y in the binary logistic regression, which takes on the values Y = 0 or Y = 1. 2. Systematic component: describes the explanatory vari... |
logit forms the linear decision boundary: Let us consider the logit transformation, i.e., logodds. Assume a scenario in which the log Pr(Y = 1|X) Pr(Y = 0|X) ! = θ0 + θT X, for a given vector of systematic components X and predictor variables θ. Write the mathem atical expression for the hyperplane that describes the... | The hyperplane is simply defined by: θ0 + θT X = 0. Note: Recall the use of the logit function and derive this decision boundary rigorously. |
of each other. True or False: The logit function and the natural logistic (sigmoid) function are inverses LOGISTIC REGRESSION 2.2.3 The Sigmoid The sigmoid (Fig. 2.1) also known as the logistic function, is widely used in binary classification and as a neuron activation function in artificial neural networks. σ(x) = 1 σ(... | True. The logit function is defined as: z(p) = logit(p) = log ! , p 1 − p 2.3. SOLUTIONS for any p ∈ [0, 1]. A simple set of algebraic equations yields the inverse relation: p(z) = exp z 1 + exp z , which exactly describes the relation between the output and input of the logistic function, also known as the sigmoid. 2... |
Compute the derivative of the natural sigmoid function: σ(x) = 1 + e−x ∈ (0, 1). | or derivation via the softmax function. There are various approaches to solve this problem, here we provide two; direct derivation d 1. Direct derivation: dx σ(x) = d dx((1 + e 2. Softmax derivation: −x)−1) = −((1 + e −x)(−2)) d dx(1 + e −x) = e −x (1+e−x)2 . In a classification problem with mutually exclusive classes, ... |
Remember that in logistic regression, the hypothesis function for some parameter vector β and measurement vector x is defined as: hβ(x) = g(βT x) = = P(y = 1|x; β), 1 + e−βT x 2.2. PROBLEMS Suppose the coefficients of a logistic regression model with independent variables are as where y holds the hypothesis value. follow... | 1. The logit value is simply obtained by substituting the values of the dependent variables and model coefficients into the linear logistic regression model, as follows: logit = β0 + β1x1 + β2x2 = −1.5 + 3 · 1 + −0.5 · 5 = −1. 2. According to the natural relation between the logit and the odds, the following holds: odds... |
Proton therapy (PT) [2] is a widely adopted form of treatment for many types of cancer including breast and lung cancer (Fig. 2.2). FIGURE 2.2: Pulmonary nodules (left) and breast cancer (right). A PT device which was not properly calibrated is used to simulate the treatment of cancer. As a result, the PT beam does not... | 1. Tumour eradication (Y ) is the response variable and cancer type (X) is the explanatory variable. 2. Relative risk (RR) is the ratio of risk of an event in one group (e.g., exposed group) versus the risk of the event in the other group (e.g., nonexposed group). The odds ratio (OR) is the ratio of odds of an event in... |
Consider a system for radiation therapy planning (Fig. 2.3). Given a patient with a ma lignant tumour, the problem is to select the optimal radiation exposure time for that patient. A key element in this problem is estimating the probability that a given tumour will be erad icated given certain covariates. A data scien... | 1. By using the defined values for X1 and X2, and the known logistic regression model, substitution yields: ˆp(X) = −6+0.05X1+X2 e (1 + e−6+0.05X1+X2) = 0.3775. 2. The equation for the predicted probability tells us that: −6+0.05X1+3.5 e (1 + e−6+0.05X1+3.5) = 0.5, which is equivalent to constraining: −6+0.05X1+3.5 = 1.... |
Recent research [3] suggests that heating mercury containing dental amalgams may cause the release of toxic mercury fumes into the human airways. It is also presumed that drinking hot coffee, stimulates the release of mercury vapour from amalgam fillings (Fig. 2.4). FIGURE 2.4: A dental amalgam. To study factors that af... | 2.3. SOLUTIONS For the purpose of this exercise, it is instructive to predefine z as: z (X1, X2) = −6.36 − 1.02 × X1 + 0.12 × X2. 1. By employing the classic logistic regression model: odds = exp(z (X1, X2)). 2. By substituting the given values of X1, X2 into z (X1, X2), the probability holds: p = exp(z (1, 100))/(1 + e... |
To study factors that affect Alzheimer’s disease using logistic regression, a researcher considers the link between gum (periodontal) disease and Alzheimer as a plausible risk factor [1]. The predictor variable is a count of gum bacteria (Fig. 2.5) in the mouth. FIGURE 2.5: A chain of spherical bacteria. The response v... | 1. The estimated probability of improvement is: ˆπ(gum bacteria) = exp(−4.8792 + 0.0258 × gum bacteria) 1 + exp(−4.8792 + 0.0258 × gum bacteria). Hence, ˆπ(33) = 0.01748. LOGISTIC REGRESSION 2. For ˆπ(gum bacteria) = 0.5 we know that: ˆπ(gum) = exp(ˆα + ˆβx) 1 + exp(ˆα + ˆβx) = 0.5 gum bacteria = −ˆα/ ˆβ = 4.8792/0.025... |
Recent research [4] suggests that cannabis (Fig. 2.6) and cannabinoids administration in particular, may reduce the size of malignant tumours in rats. FIGURE 2.6: Cannabis. To study factors affecting tumour shrinkage, a deep learning researcher collects data from two groups; one group is administered with placebo (a su... | 1. The sample odds ratio is: ˆθ = 130 × 6833 60 × 6778 = 2.1842. 2. The estimated standard error for log cid:16 cid:17 log ˆθ ˆσ = 3. According to previous sections, the 95% CI for the true log odds ratio is: 0.7812 ± 1.96 × 0.1570 = (0.4734, 1.0889). Correspondingly, the 95% CI for the true odds ratio is: (e0.4734, e1... |
defined as: The entropy (see Chapter 4) of a single binary outcome with probability p to receive 1 is H(p) ≡ −p log p − (1 − p) log(1 − p). 1. At what p does H(p) attain its maximum value? 2. What is the relationship between the entropy H(p) and the logit function, given p? LOGISTIC REGRESSION 2.2.6 Python/PyTorch/CPP | 1. The entropy (Fig. 2.13) has a maximum value of log2(2) for probability p = 1/2, which is the most chaotic distribution. A lower entropy is a more predictable outcome, with zero providing full certainty. 2. The derivative of the entropy with respect to p yields the negative of the logit func tion: dH(p) dp = −logit(p... |
The following C++ code (Fig. 2.7) is part of a (very basic) logistic regression implement ation module. For a theoretical discussion underlying this question, refer to problem 2.17. 1 #include ... 2 std::vector<double> theta {6,0.05,1.0}; 3 double sigmoid(double x) { 4 double tmp =1.0 / (1.0 + exp(x)); 5 std::cout << "... | LOGISTIC REGRESSION FIGURE 2.13: Binary entropy. 1. During inference, the purpose of inner_product is to multiply the vector of logistic re gression coefficients with the vector of the input which we like to evaluate, e.g., calculate the probability and binary class. 2. The line hypo(x) > 0.5f is commonly used for the e... |
matrix. The following Python code (Fig. 2.8) runs a very simple linear model on a twodimensional 1 import torch 2 import torch.nn as nn 4 lin = nn.Linear(5, 7) 5 data = (torch.randn(3, 5)) 7 print(lin(data).shape) 8 >? FIGURE 2.8: A linear model in PyTorch Without actually running the code, determine what is the size o... | Because the second dimension of lin is 7, and the first dimension of data is 3, the result ing matrix has a shape of torch.Size([3, 7]) . 2.3. SOLUTIONS |
tion module in Python. The following Python code snippet (Fig. 2.9) is part of a logistic regression implementa LOGISTIC REGRESSION 1 from scipy.special import expit 2 import numpy as np 3 import math 5 def Func001(x): e_x = np.exp(x np.max(x)) return e_x / e_x.sum() 9 def Func002(x): return 1 / (1 + math.exp(x)) 12 d... | the interviewer. Ideally, you should be able to recognize these functions immediately upon a request from 1. A softmax function. 2. A sigmoid function. 3. A derivative of a sigmoid function. |
Python. The following Python code snippet (Fig. 2.10) is part of a machine learning module in 2.2. PROBLEMS 1 ^^I^^I 2 from scipy.special import expit 3 import numpy as np 4 import math 5 ^^I^^I 6 def Func006(y_hat, y): if y == 1: return np.log(y_hat) else: return np.log(1 y_hat)^^I FIGURE 2.10: Logistic regression me... | The function implemented in Fig. 2.10 is the binary crossentropy function. |
same function. The following Python code snippet (Fig. 2.11) presents several different variations of the LOGISTIC REGRESSION 1 ^^I^^I 2 from scipy.special import expit 3 import numpy as np 4 import math 6 def Ver001(x): return 1 / (1 + math.exp(x)) 9 def Ver002(x): return 1 / (1 + (np.exp(x))) 12 WHO_AM_I = 709 14 def... | 1. All the methods are variations of the sigmoid function. 2. In Python, approximately 1.797e + 308 holds the largest possible valve for a floating point variable. The logarithm of which is evaluated at 709.78. If you try to execute the following expression in Python, it will result in inf : np.log(1.8e + 308). 3. I wou... |
DataScience-Interview-Preparation-Bot Dataset
This dataset contains questions and answers related to data science interview preparation. It is designed to help individuals prepare for data science interviews by providing a variety of questions and detailed answers.
Dataset Structure
The dataset is provided in CSV format with the following columns:
question: The question or problem statement.answer: The solution or answer to the problem.
Example
Here is an example of the dataset:
questions,answers
"True or False: For a fixed number of observations in a data set, introducing more variables normally generates a model that has a better fit to the data. What may be the drawback of such a model fitting strategy?","True. However, when an excessive and unnecessary number of variables is used in a logistic regression model, peculiarities (e.g., specific attributes) of the underlying data set disproportionately affect the coefficients in the model, a phenomena commonly referred to as “overfitting”. Therefore, it is important that a logistic regression model does not start training with more variables than is justified for the given number of observations."
"Define the term “odds of success” both qualitatively and formally. Give a numerical example that stresses the relation between probability and odds of an event occurring.","The odds of success are defined as the ratio between the probability of success p ∈ [0, 1] and the probability of failure 1 − p. Formally: Odds(p) ≡ p / (1 − p). For instance, assuming the probability of success of an event is p = 0.7. Then, in our example, the odds of success are 7/3, or 2.333 to 1. Naturally, in the case of equal probabilities where p = 0.5, the odds of success is 1 to 1."
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