PART 2

Computational & Statistical Thinking

Mathematical Foundation

KU PSYC 500

Jiin Jung

Course Objectives

Specifically, at the conclusion of the course,

1.Students will be able to:

• Curate data from open sources and development relevant questions
• Understand data and organize information
• Produce visualizations of data and communicate information
• Identify and apply statistical methods and analyze data
• Evaluate results to recommend policy or organizational changes
• Demonstrate understanding of Python for use in data science and statistical modeling

Open source data

Datasets are given by organizations

Subsetting

Slicing

Vectorized operation

Reshaping

Combining

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Data visualization

Descriptive statistics

Suggesting hypotheses

Accessing assumptions for inferential statistics

Simulation-based statistics

1. Observe empirical data (EDA)
2. Develop a research question/Choose a statistic of interest
1. Mean, median, mean difference, correlation, etc
3. State H0 (Null Hypothesis)
4. Simulate data assuming that H0 is true
• Bootstrapping
• Permutation
5. Calculate a replicate from a simulated dataset
• Define a function
6. Repeat step 4 & 5 many many times - Iteration
7. Decision
• Compute p-value
• A p-value is a fraction of replicates that is at least as extreme as the observed value.

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Why Simulation?

“Resampling techniques are rapidly entering mainstream data analysis; some statisticians believe that resampling procedures will soon supplant common nonparametric procedures and may displace most parametric procedures as well. “ (Berger, 2012)

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• Requiring fewer assumptions: when assumptions of traditional parametric tests are not met, as with small samples from non-normal distributions
• Simpler and more flexible: Methods for testing means, medians, ratios, or other parameters are the same, so we do not need new methods for these different applications.
• Easier to interpret: e.g. p-value
• Practical: The general availability of cheap rapid computing and new software

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A Replicate

Calculate a replicate from the simulated dataset

• Bootstrap replicates
• Permutation replicates

Iteration

Process

(10k~)

Part 2

Computational & Statistical Thinking

Mathematical Foundation

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Questions?

Intro to NumPy

Lec 11

Objectives

• Intro to NumPy
• Iteration. For loop
• Define Functions

NumPy

Numpy is a library for the Python programming language.

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• It contains a large collection of mathematical functions.
• It operates on large, multi-dimensional arrays and matrices.

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https://numpy.org/

Essential libraries and projects that depend on NumPy’s API gain access to new array implementations that support NumPy’s array protocols (Fig. 3).

Import NumPy

import numpy as np

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Array: a basic data structure of Numpy

np.array()

ndarray

ndarray: a n-dimensional array

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Sorting & adding elements

np.sort()

np.append()

np.concatenate()

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Indexing & slicing elements - 1d array

data = np.array([1,2,3])

Indexing & slicing elements - 2d array

data = np.array([[1,2],[3,4],[5,6]])

Selecting a subset

a = np.array([1,2,3,4,5,6,7,8,9,10])

b = a[3:8]#you can use index positions.

c = a[a > 5] #you can use booleans.

Array operations

a = np.array([1,2,3])

b = np.array([4,5,6])

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a + b

a - b

a * b

a / b

Math formulas & summary statistics

np.square()

np.sum()

np.mean()

np.var()

np.std()

np.percentile()

np.corrcoef()

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https://numpy.org/doc/stable/reference/routines.statistics.html

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Questions?

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