Mean, Deviation, & Standard Deviation

The mean describes the center of the data. The standard deviation describes the spread of the data. But to compare two variables, it is convenient to normalize both. In this exercise, you are provided with two arrays of data, which are highly correlated, and you will compute and visualize the normalized deviations of each array.

Deze oefening maakt deel uit van de cursus

Introduction to Linear Modeling in Python

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Oefeninstructies

Compute the deviations dx and dy.
Compute the normalized deviations zx and zy.
Call plot_cdfs() to see how both the raw and the normalized data compares.

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Probeer deze oefening eens door deze voorbeeldcode in te vullen.

# Compute the deviations by subtracting the mean offset
dx = x - np.____(x)
dy = y - np.____(y)

# Normalize the data by dividing the deviations by the standard deviation
zx = dx / np.____(x)
zy = dy / np.____(y)

# Plot comparisons of the raw data and the normalized data
fig = plot_cdfs(dx, dy, zx, zy)

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Deze oefening maakt deel uit van de cursus

Introduction to Linear Modeling in Python

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We start the course with an initial exploration of linear relationships, including some motivating examples of how linear models are used, and demonstrations of data visualization methods from matplotlib. We then use descriptive statistics to quantify the shape of our data and use correlation to quantify the strength of linear relationships between two variables.

Exercise 1: Introduction to Modeling Data Exercise 2: Reasons for Modeling: Interpolation Exercise 3: Reasons for Modeling: Extrapolation Exercise 4: Reasons for Modeling: Estimating Relationships Exercise 5: Visualizing Linear Relationships Exercise 6: Plotting the Data Exercise 7: Plotting the Model on the Data Exercise 8: Visually Estimating the Slope & Intercept Exercise 9: Quantifying Linear Relationships Exercise 10: Mean, Deviation, & Standard Deviation

Huidige oefening

Exercise 11: Covariance vs Correlation Exercise 12: Correlation Strength

Here we look at the parts that go into building a linear model. Using the concept of a Taylor Series, we focus on the parameters slope and intercept, how they define the model, and how to interpret the them in several applied contexts. We apply a variety of python modules to find the model that best fits the data, by computing the optimal values of slope and intercept, using least-squares, numpy, statsmodels, and scikit-learn.

Exercise 1: What makes a model linear Exercise 2: Terms in a Model Exercise 3: Model Components Exercise 4: Model Parameters Exercise 5: Interpreting Slope and Intercept Exercise 6: Linear Proportionality Exercise 7: Slope and Rates-of-Change Exercise 8: Intercept and Starting Points Exercise 9: Model Optimization Exercise 10: Residual Sum of the Squares Exercise 11: Minimizing the Residuals Exercise 12: Visualizing the RSS Minima Exercise 13: Least-Squares Optimization Exercise 14: Least-Squares with `numpy`Exercise 15: Optimization with Scipy Exercise 16: Least-Squares with `statsmodels`

Next we will apply models to real data and make predictions. We will explore some of the most common pit-falls and limitations of predictions, and we evaluate and compare models by quantifying and contrasting several measures of goodness-of-fit, including RMSE and R-squared.

Exercise 1: Modeling Real Data Exercise 2: Linear Model in Anthropology Exercise 3: Linear Model in Oceanography Exercise 4: Linear Model in Cosmology Exercise 5: The Limits of Prediction Exercise 6: Interpolation: Inbetween Times Exercise 7: Extrapolation: Going Over the Edge Exercise 8: Goodness-of-Fit Exercise 9: RMSE Step-by-step Exercise 10: R-Squared Exercise 11: Standard Error Exercise 12: Variation Around the Trend Exercise 13: Variation in Two Parts

In our final chapter, we introduce concepts from inferential statistics, and use them to explore how maximum likelihood estimation and bootstrap resampling can be used to estimate linear model parameters. We then apply these methods to make probabilistic statements about our confidence in the model parameters.

Exercise 1: Inferential Statistics Concepts Exercise 2: Sample Statistics versus Population Exercise 3: Variation in Sample Statistics Exercise 4: Visualizing Variation of a Statistic Exercise 5: Model Estimation and Likelihood Exercise 6: Estimation of Population Parameters Exercise 7: Maximizing Likelihood, Part 1 Exercise 8: Maximizing Likelihood, Part 2 Exercise 9: Model Uncertainty and Sample Distributions Exercise 10: Bootstrap and Standard Error Exercise 11: Estimating Speed and Confidence Exercise 12: Visualize the Bootstrap Exercise 13: Model Errors and Randomness Exercise 14: Test Statistics and Effect Size Exercise 15: Null Hypothesis Exercise 16: Visualizing Test Statistics Exercise 17: Visualizing the P-Value Exercise 18: Course Conclusion