代写Chemistry 125-225: Machine Learning in Chemistry Homework Assignment #2代写Python编程

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Chemistry 125-225: Machine Learning in Chemistry

Fall Quarter, 2024

Homework Assignment #2 - Due: November 11, 2024. Turn in a writeup with your responses to a ll questions below, codes, outputs (e.g. graphs, etc.). Attach all your Python files as well, so we can run them.

Problem 1: Linear Regression and Least Squares Fitting

In this problem, you will derive and apply the least squares method to fit a straight line to data from a real-world chemistry dataset. This exercise will help you understand the derivation of least squares fitting, its implementation, and how it can be applied in a chemical context.

Background and Derivations

The least squares method is widely used to fit models to data, one of the simplest applications being to fit a straight line y = mx + b, where m is the slope and b is the intercept.

(a) Derivation of the Normal Equations

Suppose we have a dataset consisting of N observations. Each observation includes an independent variable xi and a dependent variable yi , which are related (approximately linearly) by the equation yi ≈ mxi + b. Our objective is to find the slope m and intercept b that best fit the data in a least- squares sense, minimizing the sum of squared residuals between the actual y-values and the predicted values ˆ(y) = mx + b.

Define the following:

• Vector y: The N-dimensional vector of observed y-values.

• Matrix A: An N × 2 matrix containing the x-values of the observations in the second column and a column of ones in the first column. This setup allows us to solve for both m and b simultaneously.

• Vector x: The vector containing our unknown parameters, m and b.

The goal is to find the vector x such that Ax is as close as possible to y in a least-squares sense. This leads to minimizing the squared error ||Ax − y||2.

i. Show that the optimal solution for x that minimizes the squared error satisfies the normal equa- tion:

ATAx = AT y.

*Hint:* Start by expanding ||Ax − y||2 and then set the gradient with respect to x to zero.

ii. Using the normal equation ATAx = AT y, derive explicit formulas for m and b that apply when fitting a line. Show that these are equivalent to:

iii. Suppose in a chemical experiment, x represents the concentration of a reactant (e.g., molarity, M) and y represents the rate of reaction observed at each concentration. Explain how finding m and b could help you interpret the relationship between concentration and reaction rate, possibly leading to insights about reaction kinetics.

(b) Download and Explore the Data

We will use the ESOL (Delaney) dataset from MoleculeNet, which provides information on water solubility (log solubility) of organic molecules. Access this dataset using the DeepChem Python library as shown below.

i. Install DeepChem (if not already installed) by running: pip install deepchem

ii. Use the following Python code to load the dataset and extract features and labels:

import deepchem as dc # Load the ESOL dataset tasks , datasets , transformers = dc . molnet . load_delaney () train_dataset , valid_dataset , test_dataset = datasets # Extract features (X) and labels (y) from the training dataset X_train = train_dataset .X y_train = train_dataset .y

iii. Display the structure of X train and y train to understand what these variables represent in this dataset.

iv. Answer the following questions:

• What are the features in this dataset?

• Which feature(s) would you expect to correlate with solubility?

i. Using the LinearRegression model from scikit-learn, train a linear regression model on X train and y train.

ii. Use the following code to train the model and calculate relevant parameters (slope, intercept, and RMSE):

from sklearn . linear_model import LinearRegression from sklearn . metrics import mean_squared_error import numpy as np # Train a simple linear regression model on the dataset linear_regressor = LinearRegression () linear_regressor . fit ( X_train , y_train ) # Obtain slope ( coefficients ) and intercept slope = linear_regressor . coef_ . flatten () # Flatten if it ’s a multidimensional array intercept = float ( linear_regressor . intercept_ ) # Convert intercept to float # Predict target variable for training data y_train_pred = linear_regressor . predict ( X_train ) # Calculate RMSE rmse = np . sqrt ( mean_squared_error ( y_train , y_train_pred )) # Print results in the terminal print (f" Slope (s): { slope }") print (f" Intercept : { intercept }") print (f" RMSE : { rmse }")

iii. Answer the following questions:

• Explain the meaning of each parameter (slope, intercept) in the context of solubility.

• What does the RMSE value indicate about the model’s accuracy?

(d) Plot the Fitted Line

i. Plot the actual versus predicted values of solubility to assess the fit visually.

import matplotlib . pyplot as plt # Plot actual vs predicted solubility values plt . scatter ( y_train , y_train_pred , label =’Predicted vs Actual ’) plt . plot ([ min ( y_train ) , max( y_train )] , [ min ( y_train ) , max( y_train )] , color =’red ’, linestyle. =’--’, label =’Ideal Fit ’) plt . xlabel (" Actual Solubility ") plt . ylabel (" Predicted Solubility ") plt . legend () plt . title (" Linear Regression on ESOL Dataset ") plt . show ()

ii. Answer the following questions:

• How well does the fitted line match the data visually?

• Are there any potential outliers or deviations?

(e) Assumptions in Least Squares Fitting

i. Discuss the assumptions behind least squares fitting.

ii. Answer the following questions:

• What assumptions are made about the distribution of noise in least squares fitting?

• How might non-Gaussian noise affect the accuracy of your linear model?

Submission

Submit a report containing:

• Code for each part.

• Plots and answers to the questions.

• Interpretation of results.

Problem 2: Nonlinear Least Squares Fitting on Experimental Chem- istry Data

In this problem, you will perform. a nonlinear least squares fit on experimental chemistry data from the Free- Solv dataset, a curated database of experimental and calculated hydration free energies for small molecules in water. This exercise will introduce you to concepts in nonlinear regression, data filtering, and model evaluation.

The FreeSolv dataset can be accessed at https://github.com/MobleyLab/FreeSolv. You will download the dataset and perform. the following steps to fit a model that describes the relationship between calculated and experimental hydration free energies.

1. Data Download and Preparation

(a) Download the database .txt file from the FreeSolv GitHub repository, or use Python code to download it programmatically.

(b) Load the dataset using pandas in Python, skipping comment lines (lines starting with #) and specifying the delimiter as a semicolon. Name the columns as follows:

• Compound ID, SMILES, IUPAC Name, Expt Free Energy, Uncertainty, Calc Free Energy, DOI, Notes, Additional Column 1, Additional Column 2

2. Outlier Detection and Filtering

(a) Remove any rows in X or y that contain NaN or infinite values.

(b) Filter out outliers in the dataset by removing data points where the Expt Free Energy value is more than 3 standard deviations away from the mean. This step ensures that extreme values do not unduly influence the fit.

3. Model Definition and Fitting

(a) Define a quadratic model of the form.

f(x) = ax2 + bx + c

where a, b, and c are parameters to be determined.

(b) Use scipy .optimize .curve fit to fit the model to the filtered data and extract the best-fit parameters a, b, and c.

4. Plotting the Results

(a) Plot the filtered data points (as a scatter plot) and the fitted quadratic model (as a smooth curve).

(b) Label the axes appropriately as ”Calculated Free Energy” (x-axis) and ”Experimental Free En- ergy” (y-axis).

5. Model Evaluation

(a) Calculate the Root Mean Square Error (RMSE) to evaluate the model’s accuracy. The RMSE is defined as:

where yi are the actual experimental values and f(xi ) are the values predicted by your quadratic model.

(b) Print the RMSE to assess the quality of your model fit.

Questions

(a) Why is it important to remove outliers when performing regression? How could outliers affect the quality of your model?

(b) Explain why a quadratic model was chosen here instead of a simple linear or exponential model. Under what circumstances would each model type be appropriate?

(c) The RMSE provides a measure of fit quality. Would a lower RMSE always indicate a better model in a physical or chemical context? Why or why not?

(d) What assumptions are implicit in least squares fitting regarding the distribution of errors in the data? Discuss how violations of these assumptions might influence your fit.

Hints:

• Use the pandas, numpy, and scipy libraries in Python to handle data processing, model fitting, and numerical calculations.

• For plotting, use matplotlib.pyplot. The plot function can be used to draw the quadratic fit, and the scatter function is suitable for data points.

• Ensure that any non-numeric values (e.g., missing data or text) in X or yare handled before performing the fit.

Problem 3: Nonlinear Least Squares Fit Using Gradient Descent in numpy

In this problem, you will perform a nonlinear least squares fit by implementing a gradient descent algorithm from scratch using only numpy. Unlike previous problems where you used scipy to handle optimization, here you will manually compute the gradient vector of the loss function and use it in a gradient descent loop. This exercise will deepen your understanding of the principles behind nonlinear least squares fitting and gradient-based optimization.

Objective: Fit a quadratic model to a dataset using gradient descent. Specifically, given a set of data points (xi , yi ), find parameters a, b, and c that minimize the sum of squared errors:

where

f(x) = ax2 + bx + c

is the quadratic model function.

1. Data Normalization (Scaling)

(a) Normalize the predictor variable X and the response variable y to have zero mean and unit variance:

where µX and σX are the mean and standard deviation of X, and similarly for y. This step helps stabilize gradient descent by preventing large gradients.

2. Define the Loss Function and Gradient

(a) Derive the loss function L for the nonlinear least squares fit:

where xi and yi are the data points.

(b) Compute the partial derivatives of L with respect to each parameter a, b, and c. This will form. the gradient vector you will use in gradient descent. You are given the partial derivative with respect to a:

Using a similar approach, compute the partial derivatives with respect to b and c, ∂b/∂L and ∂c/∂L , respectively.

(c) Write a Python function compute gradient(X, y, a, b, c) that takes in the data X andy along with the parameters a, b, and c, and returns the gradient vector as anumpy array [∂L/∂a,∂L/∂b,∂L/∂c].

3. Implement Gradient Descent for Optimization

(a) Initialize the parameters a, b, and c with some starting values (e.g., all zeros).

(b) Set a small learning rate η (e.g., 0.00001) to prevent parameter values from growing too quickly during updates.

(d) Write a loop that performs the following steps:

i. Compute the current loss L based on the current values of a, b, and c.

ii. Use compute gradient to calculate the gradient vector for the current values of a, b, and c. iii. Update the parameters using the gradient descent update rule:

iv. Break out of the loop if the change in loss between iterations is smaller than the convergence threshold.

(e) After convergence, print the optimized values of a, b, and c.

4. Convert Parameters Back to Original Scale

(a) Since you optimized the parameters on the normalized data, convert the parameters back to the original scale for accurate interpretation. Use the following transformations:

5. Plotting and Evaluation

(a) Plot the original data points and the fitted quadratic model using the de-normalized parameters. (b) Plot the loss over iterations to observe the convergence of gradient descent.

Questions

(a) Why is it important to remove outliers when performing regression? How could outliers affect the quality of your model?

(b) Explain why a quadratic model was chosen here instead of a simple linear or exponential model. Under what circumstances would each model type be appropriate?

(c) The RMSE provides a measure of fit quality. Would a lower RMSE always indicate a better model in a physical or chemical context? Why or why not?

(d) What impact does the learning rate η have on the convergence of gradient descent? What happens if η is too large or too small?

(e) Explain the purpose of data normalization in gradient descent. How does normalization affect conver- gence and stability?

What to Turn In:

1. Your Python code implementing gradient descent and data normalization.

2. A plot showing the original data and the fitted quadratic curve.

3. A plot of the loss over iterations, illustrating the convergence behavior.

4. A brief explanation answering each of the questions posed above.

5. The final optimized parameters a, b, and c on the original scale, along with the final loss.

Hints:

• To track convergence, keep a record of the previous loss and compare it with the current loss in each iteration.

• If you observe very large or NaN values in your parameters, reduce the learning rate or check your gradient computation.

• Use numpy operations (e.g., np .sum, np.dot) to implement the gradient calculations efficiently.