When fitting a non-linear model, which statement is always true? Select one: O a. Take the log of both variables and fit a straight line to those variables O b. The error is always worse in a non-linear fit O c. The linear coefficients must be transformed to get the non-linear coefficients O d. All of the above
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- Solve in R programming language: 5) Use R to calculate and simulate with the exponential distribution as follows. (a) For an exponential random variable X with λ = 4, simulate 1000 independent exponential random variables by using the R function rexp(n, λ). Calculate the mean and variance of this sample. (b) Compare the empirical results from part (a) with the distribution mean 1/λ and the distribution standard deviation 1/λ.When maximizing a function, the gradient at a given point will always point in Notes: In machine learning, when we are trying to learn parameters to solve a problem the direction of the gradient will be crucial to finding "good" parameters! any direction the direction of steppest ascent the direction away from the origin the direction of steppest descentFor logistic regression, the gradient of the cost function is given by J(0) = (i) E (he (x) – y')x;). Write down mathematical expression(s) for the correct m gradient descent update for logistic regression with a learning rate of a. (In the expression, he(x^) should be replaced by the sigmoid function.)
- Answer the given question with a proper explanation and step-by-step solution. You will be required to run Expectation Maximization for estimating parameters of a Gaussian mixture model in this example. Refer to lecture slides (Lecture E.2) for the exact formulation. You can use python or do the computations by hand. Recall that you need to use the pdf formulation for computing p(x|theta_k) Consider the following one dimensional data set: 2.3 3.2 3.1 1.6 1.9 11.5 10.2 12.3 8.6 10.9 Assume that we are interested in learning a mixture model with two components (k = 2). Let pi_k denote the probability P(z_i = k) for any i. Let (mu_1,sigma_1) be the parameters for the first Gaussian component of the mixture and (mu_2,sigma_2) be the parameters for the second Gaussian component of the mixture. Given the following initialization: pi_1 = pi_2 = 0.5, mu_1 = mu_2 = 0, and sigma_1 = sigma_2 = 1. Answer the following: a) After first M step, mu_1 = mu_2 = 6.56 b) After…1. The impulse response of a causal system is: h(t) = A cos(wt) e¯¹/¹u(t) where u(t) is the Heaviside step function. The response is measured experimentally with a sampling interval of T. a. Write an expression for the sampled impulse response h[n]. b. Calculate the z transform of h[n] and write an expression for H[z]. Use the tables provided below as necessary. c. Does the system have an infinite impulse response (IIR) or finite impulse response (FIR)? Justify your answer. d. What is the DC gain of H[z]? e. Write a difference equation that describes the output y[n] in terms of input x[n].You run a logistic regression model in R using the glm() function. The dependent variable is the factor variable Y and independent variables are X1 and X2 (in other words, the formula is Y~X1+X2). In the model output, the coefficient of the constant term is a0, the coefficient of X1 is a1, and the coefficient of X2 is a2. Assuming a cutoff = 0.5, which of the following defines the equation of a decision boundary? a0 + a1X1 + a2X2 = 0.5 exp(-(a0 + a1X1 + a2X2)) = 0 a0 + a1X1 + a2X2 = 0 O exp(-(a0 + a1X1 + a2X2)) = 0.5
- "When conducting a binary regression with a skewed predictor, it is often easiest to assess the need for x and log(x) by including them both in the model so that their relative contributions can be assessed directly." Show that indeed the log odds are a function of x and log(x) for the gamma distribution.Implement a simple linear regression model using Python without using any machine learning libraries like scikit-learn. Your model should take a dataset of input features X and corresponding target values y, and it should output the coefficients w and b for the linear equation y =wX + bWhich of the following are true about principal components analysis (PCA)? Assume that no two eigenvectorsof the sample covariance matrix have the same eigenvalue. A: Appending a 1 to the end of every sample point doesn’t change the results of performing PCA (except thatthe useful principal component vectors have an extra 0 at the end, and there’s one extra useless component witheigenvalue zero). B: If you use PCA to project d-dimensional points down to j principal coordinates, and then you run PCA againto project those j-dimensional coordinates down to k principal coordinates, with d > j > k, you always get the sameresult as if you had just used PCA to project the d-dimensional points directly down to k principle coordinates. C: If you perform an arbitrary rigid rotation of the sample points as a group in feature space before performingPCA, the principal component directions do not change. D: If you perform an arbitrary rigid rotation of the sample points as a group in…
- We are intrested in predicting the percentage of people commuting to work by walking given some input variables. Each observation corresponds to a different city and each input variable summarizes some characteristic of a given city, such as density, urban sprawl and average income per capita. This is 1. not a machine learning problem. Only social scientists would be interested in such a problem. 2. both a classification and a regression problem as it depends on the way one codes the output variable as either 0, 1 or a a particular number in the [0,1] interval. 3. a regression problem. The output variable is continuous. 4. a classification problem. Walking to work is a discrete variable and can only take two values: to walk to work and not to walk to worYou have built a classification model to predict if a patient will be readmitted within 30 days of discharge from the hospital. When you examine the ROC curve you find that it essentially coincides with the central diagonal of the curve. Based on this, which of the following can you infer: Your model performs about as good as random guessing Your model performs much worse than random guessing Your model performs much better than random guessingWe create a simple regression model and call the fit function as follows: Im=LinearRegression() Im.fit(X,Y) in a multilinear model we proceed in the same way: mlm=LinearRegression() mlm.fit(Z,Y) How does the linear regression model knows if we are doing a simple or multiple linear regression? Answer: