I am a Data Scientist at Fannie Mae and a student in Georgia Tech’s MS in Analytics program. I graduated with honors from Texas A&M University with a BS in Urban and Regional Planning with a minor in Economics. I am excited about housing, risk modeling, and data science.

dataclass is an R package I created to easily define templates for lists and data frames that validate each element. This package is useful for validating data within R processes which pull from dynamic data sources such as databases and web APIs to provide an extra layer of validation around input and output data.

To use dataclass you specify the expected type, length, range, allowable values, and more for each element in your data. Decide whether violations of these expectations should throw an error or a warning.

For example, suppose you wanted to create a data frame in R which contains three columns: date, low_flag, and metric. These columns represent the output of some analytic process in R. Traditionally, you would simply write these columns as a data frame. How can we be sure that the data is correct? Simply describe your data in a declarative fashion:

Now we have a template for our data called my_dataclass. Because we want to validate a data frame (as opposed to a list) we called data_validator() to let dataclass know we are validating a data frame. How do we use it? Simply pass your data to validate as a function. If we pass in valid inputs, dataclass returns the input data. However, invalid inputs throw an error.

tibble::tibble(date = Sys.Date(),low_flag = TRUE,metric = 1) |>my_dataclass()#> # A tibble: 1 × 3#> date low_flag metric#> <date> <lgl> <dbl>#> 1 2023-03-21 TRUE 1tibble::tibble(date = Sys.Date(),low_flag = TRUE,metric = "A string!") |>my_dataclass()#> Error:#> ! The following elements have error-level violations:#> ✖ metric: is not numeric#> Run `rlang::last_error()` to see where the error occurred.

We can also use dataclass to validate lists. Suppose we want to validate that a list contains date, my_data, and note where these elements correspond to the run date, a data frame, and a string respectively:

new_dataclass(date = Sys.Date(),my_data = head(mtcars, 2),note = "A note!")#> $date#> [1] "2023-03-21"#> #> $my_data#> mpg cyl disp hp drat wt qsec vs am gear carb#> Mazda RX4 21 6 160 110 3.9 2.620 16.46 0 1 4 4#> Mazda RX4 Wag 21 6 160 110 3.9 2.875 17.02 0 1 4 4#> #> $note#> [1] "A note!"new_dataclass(date = Sys.Date(),my_data = mtcars,# note is not a single string!note = c(1, 2, 3))#> Error:#> ! The following elements have error-level violations:#> ✖ note: is not a character#> Run `rlang::last_error()` to see where the error occurred.

And that’s it! It’s pretty easy and minimal to get started. The learning curve is very minimal while the benefits of data validation cannot be overstated in a data science workflow!

You can install dataclass from CRAN by running the command below in your R console. Finally, if you want to contribute or submit bugs you can visit the GitHub repository here.

Unlike linear regression, logistic regression does not have a closed-form solution. Instead, we use the generalized linear model approach using gradient descent and maximum likelihood.

First, lets discuss logistic regression. Unlike linear regression, values in logistic regression generally take two forms, log-odds and probability. Log-odds is the value returned when we multiply each term by its coefficient and sum the results. This value can span from -Inf to Inf.

Probability form takes the log-odds form and squishes it to values between 0 and 1. This is important because logistic regression is a binary classification method which returns the probability of an event occurring.

To transform log-odds to a probability we perform the following operation: exp(log-odds) / 1 + exp(log-odds). And to transform probability back to log odds we perform the following operation: log(probability / 1 – probability).

Next, we need to consider our cost function. All generalized linear models have a cost function. For logistic regression, we maximize likelihood. To compute the likelihood of a set of coefficients we perform the following operations: sum(log(probability)) for data points with a true classification of 1 and sum(log(1 – probability)) for data points with a true classification of 0.

Even though we can compute the given cost of a set of parameters, how can we determine which direction will improve our outcome? It turns out we can take the partial derivative for each parameter (b0, b1, … bn) and nudge our parameters into the right direction.

Suppose we have a simple logistic regression model with only two parameters, b0 (the intercept) and b1 (the relationship between x and y). We would compute the gradient of our parameters using the following operations: b0 – rate * sum(probability – class) for the intercept and b1 – rate * sum((probability – class) * x)) for the relationship between x and y.

Note that rate above is the learning rate. A larger learning rate will nudge the coefficients more quickly where a smaller learning rate will approach the coefficients more slowly, but may achieve better estimates.

Now lets put all of this together! The Python function to perform gradient descent for logistic regression is surprisingly simple and requires the use of only Numpy. We can see gradient descent in action in the visual below which shows the predicted probabilities for each iteration.

A Hertzsprung–Russell diagram (HR diagram) is a visualization of star data which shows the relationship between magnitude and spectral characteristics. The diagram was created by Ejnar Hertzsprung and Henry Norris Russell independently in the early 20th century. You can read more about these diagrams here.

While interesting, I am no astronomer and am primarily inspired by how interesting the diagrams appear. I originally saw this diagram on a post my Mike Bostock (creator of D3.js) when learning more about creating data visualizations in JavaScript. You can see his implementation here.

My visual uses the same underlying CSV as Mike Bostock’s visual, but simplifies the output and makes it smaller. It also detects user scrolls to turn individual star data points on and off to create a star-twinkle effect. The effect is most pronounced on smooth scrolls (such as a touchscreen device or trackpad).

In all, this is more of an exercise in art than data analysis. Enjoy!

K-nearest neighbors (KNN) is an algorithm which identifies the k nearest data points in a training sample to a new observation. Typically, nearest is defined by the Euclidian (or straight line) distance, however, other distance norms can be used.

Python is already home to several KNN implementations the most famous of which is the scikit-learn implementation. I still believe there is value in writing your own model implementations to learn more about how they work.

First lets break down what KNN is doing visually and then code up our own implementation. The visual below (built using D3.js) shows several points which are classified into the red and blue groups.

You can hover your mouse over this visual to develop an understanding of how the nearest three points impact the classification of the point.

We can identify the three (k = 3) closest points and determine of those, which classification is the most common. The most common classification becomes our predicted value.

A few notes before we jump into our own implementation. First, it is common to use an odd number for k when performing classification to avoid ties. Second, one downside of KNN when compared to other models is that KNN must be packaged with the training data to make predictions. This is different than linear regression which only requires the coefficients to be known at the time of prediction, for example.

Now let’s look at my implementation of KNN in Python. Only 8 lines of code (excluding function imports)! A safer version of this code may also include several assertion checks to ensure inputs are of the expected type and shape.

Lets look at this function line by line. First, I define a function called knn which accepts a singular new observation called new, the training data called train with its associated labels (the correct prediction), and the mode which is either c for classification or r for regression.

defknn(new, train, labels, k=3, mode="c")

From there I compute how far each of the training points is from the new observation. To accurately compute the distances you would need to take the square root of this value. However, because we are only interested in the rank ordering of points, we can skip that step.

distances = np.sum((new - train) ** 2, axis=1)

Next I use argsort and take from numpy to rank order the indices by how close they are to the new observation. I use index slicing to grab the k nearest points. From there I use take to grab the values of the k closest indices from the label data.

Finally, I take the mode of the values for classification or the mean for regression. To predict over multiple observations I could pass the function into a list comprehension:

[knn(i, train, labels) for i in test]

This was a simple overview of KNN regression using basic numpy and scipy functions!

At first glance, you may think this title is referring to northwestern US states like Oregon or Idaho. While there certainly are wealthy areas in the northwestern US, I am actually referring to which parts of a given city are wealthy.

After traveling across and living in multiple parts of the United States, I have noticed that cities tend to be wealthier on their northern halves. Until now, this was just conjecture but I took the opportunity to utilize publicly available census tract data to investigate my suspicions.

Building the Visual

First, I obtained data from various public data sources. This includes census tract shapefiles, income data, and census tract to county MSA conversions.

I then selected a range of MSAs to analyze. In all I looked at Atlanta, Austin, Boston, Chicago, Dallas, Denver, Houston, Indianapolis, Kansas City, Las Vegas, Los Angeles, Miami, Milwaukee, Minneapolis, Nashville, New Orleans, New York, Oklahoma City, Orlando, Philadelphia, Phoenix, Portland, Salt Lake City, San Antonio, San Francisco, Seattle, Tampa, and Washington DC.

From there, I standardized the latitude and longitude of each MSA such that the most southwestern point in an MSA would have a coordinate of (0,0) while the most northeastern point would have a coordinate of (1,1). This controls for physical size differences between MSAs.

Lastly, I scaled the income of each census tract such that the tract with the highest income in an MSA has an income value of 1 and the lowest income tract has a value of 0. This also controls for wealth differences between MSAs.

I used this dataset to layer all of the MSA data to create a supercity that represents all of the individual MSAs collectively.

And here is the result! The closer to gold a given tract is the higher its income. Conversely, the closer to dark blue a tract is the lower its income. The black dot represents the city center. I observe a fairly clear distinction between the northwest and southeast of US cities.

There are, of course, exceptions to the rule. We can see gold census tracts in the south of some MSAs though wealth generally appears to be concentrated in the northwest.

A Simple Explanatory Model

To add some validity to these findings I estimated a very simple linear model which estimates a census tract’s income using its relative position to the city center. Here are the results:

Term

Coefficient (Converted to USD)

Intercept

$84,288

Longitude (West/East)

-$6,963

Latitude (North/South)

$7,674

Results of income prediction model

The way to read these coefficients is as follows. At the city center census tracts have, on average, a median household income of $84,288. As you move east median household income falls (hence the negative coefficient for Longitude) and as you north income rises (hence the positive coefficient for Latitude).

In other words, northwestern tracts have median household incomes approximately $14,000 wealthier than the city center or $28,000 wealthier than their southeastern counterparts.

Obviously, this model is oversimplified and would not be a good predictor of tract income given the huge variety of incomes across MSAs in the US, but it does illustrate an interesting point about income vs. tract position in an MSA.

Closing Thoughts

Before closing out, I wanted to draw attention to a few specific MSAs where this effect is particularly pronounced. I would argue that this northwest vs southeast impact is pronounced in the following six cities, especially Washington DC.

I hope this high level summary provides some interesting food for thought about the differences in income across US cities.

Suppose you have the following list of numbers in Python and you would like to compute the sum. You use the sum() function and expect it to return 0.3. Yet, when you run the code the console returns a value very slightly above 0.3:

You can round this number of course, but it begs the question as to why the correct sum was not returned in the first place. Enter the IEEE 754 floating point standard.

Floating Point Storage

The double type is a 64 binary digit (bit) numerical storage standard that includes 1 sign bit (determines if number is positive or negative), a 53 bit significand (only 52 are stored for non-zero values), and an 11 bit exponent.

An 11 bit exponent means the smallest positive number that can be stored is 2^{-1022}. Additionally, the largest rounding error possible in this standard is 2^{-52} called machine epsilon. Because this is a binary representation that means numbers that can be represented exactly in base 10 must be approximated when converting to binary.

Going back to our example above, 0.1 is a value that must be rounded for storage in this format. This is because 0.1 in binary is infinite:

0.000110011001100110011...

There are methods to store values exactly but this comes at the speed of computation. What if we want to keep the speed of 64 bit computation but reduce our error, specifically for large number series?

The Algorithm

Enter Kahan’s Summation Algorithm. Developed by William Kahan, this summation methodology allows for more accurate summation using the double storage format. Here is a simple Python implementation:

defkahan_sum(x):sum = 0.0 c = 0.0for i in x: y = i - c t = sum + y c = t - sum - ysum = treturnsum

Okay, so this looks pretty simple. But what do each of the pieces mean? The first two lines establish a function in Python while setting the starting sum and starting error to 0.0:

defkahan_sum(x):sum = 0.0 c = 0.0

The next few lines are the for loop that iterates over each number in the list. First, any error is subtracted from the previous iteration.

y = i - c

Second, the new number is added to the running total minus any error.

t = sum + y

Third, error from this new addition is determined and the new total is assigned. This repeats until there are no more numbers.

c = t - sum - ysum = t

A Practical Example

Okay, so the code is pretty simple but how does this work in practice? Suppose we have a list of two numbers:

[1.0, 1.0]

Step 1

The sum and error terms are set to 0.0 when the algorithm is started. The first step of each iteration is to take the current value and subtract any error from the previous iteration. Because the starting error is 0.0, we subtract 0.0 from the first value.

1.0 - 0.0 = 1.0

Step 2

Next we add the result of the previous operation to the total. Again, the initial total is 0.0 so we just add 0.0 to the value from Step 1 (1.0). Oh no! The computer had to make a rounding error. In this case, the computer was off by 0.1. We can handle this error in the next steps.

0.0 + 1.0 ~ 1.1

Step 3

In this step we determine the error from Step 2. We take the sum from Step 2 (1.1), subtract the total (0.0), and subtract the total from Step 1 (1.0). This leaves us with the approximate error.

1.1 - 0.0 - 1.0 ~ 0.1

Step 4

Finally, we record the current total for the next iteration!

1.1

And Repeat!

Now we repeat Steps 1, 2, 3, and 4 for each additional number. The difference this time is that we have non-zero values for the error and total terms. First, we subtract the error term from the last iteration to the new value:

1.0 - 0.1 = 0.9

Next, add the new value to the previous total:

1.1 + 0.9 = 2.0

Next, take the sum from the last step and subtract the previous iteration’s total and the value from the first step to estimate any error. In this case there is no error so we record a value of 0.0 for the error going into the next iteration:

2.0 - 1.1 - 0.9 = 0.0

Finally, return the sum. We can see that even though the computer made an error of 0.1, the algorithm corrected itself and returned the correct value:

2.0

Final Thoughts

Kahan’s method of summation strikes a balance between the speed of floating point arithmetic and accuracy. Hopefully this walkthrough makes the algorithm more approachable.

Association rule mining is the process of determining conditional probabilities within events that contain items or characteristics. Events can range from tweets, to grocery store receipts, to credit card applications.

Items within these events should also not be unique to each event. For example, words are repeated across tweets, multiple customers will buy the same items at the grocery store, and credit card applicants will share specific characterisitcs.

For all of these applications our goal is to estimate the probability that an event will possess item B given that it has item A. This probability is also called the confidence.

In the example above we might say that we are 23% confident that a customer will purchase rice (item B) given they are purchasing chicken (item A). We can use historical transactions (events) to estimate confidence.

Now for a practical implementation using the tidyverse in R! I am using a groceries dataset from Georgia Tech. This dataset contains rows with items separated by commas.

receipt

citrus fruit, semi-finished bread

ready soups, margarine

One transaction per row with items comma separated.

Because each event contains different items I read it using readLines() and reshape into a longer format. The groceries column contains the item name while transaction contains the transaction ID.

link <- "https://cse6040.gatech.edu/datasets/groceries.csv"groceries <- readLines(link)# Create long form version of datagroceries_long <- tibble::tibble(groceries) |> dplyr::mutate(transaction = dplyr::row_number() ) |> tidyr::separate_rows(groceries,sep = "," )

groceries

transaction

citrus fruit

1

semi-finished bread

1

tropical fruit

2

Long form data with one item per row with a transaction ID.

With our data in the proper format we can develop two functions. The first function takes a vector of items and returns a vector of comma separated combinations as (A,B) and (B,A).

For example, giving this function c("A", "B", "C") would return c("A,B" "A,C" "B,C" "B,A" "C,A" "C,B"). This is because we want to determine the probabilities of A given B and B given A.

Our final function performs the data mining. The first argument called data takes in the data frame of events and items. The last two arguments item_col and event_id tell the function which columns refer to the items and the event identifier respectively.

pair_assoc <- function(data, item_col, event_id, item_min = 1L) {# Derives association pairs for all elements in data# Count all itemsitem_count <-data |> dplyr::count(A = {{ item_col }},name = "A Count" )# Get pairs as probabilitiesdata |> dplyr::group_by({{ event_id }}) |> dplyr::filter(length({{ item_col }}) > 1) |> dplyr::reframe(comb = comb_vec({{ item_col }})) |> dplyr::ungroup() |> dplyr::count(comb,name = "A B Count" ) |> tidyr::separate(col = comb,into = c("A", "B"),sep = "," ) |> dplyr::left_join(y = item_count,by = "A" ) |> dplyr::mutate(Confidence = `A B Count` / `A Count` ) |> dplyr::arrange(desc(Confidence))}

This function works in two stages. First, it determines the count of all individual items in the data set. In the example with groceries, this might be the counts of transactions with rice, beans, etc.

groceries

A Count

baking powder

174

berries

327

Counts of individual items serve as the denominator in the confidence computation.

The second stage uses the comb_vec() function to determine all valid item combinations within each group. This stage only returns valid combinations where the confidence is > 0%.

Finally, the function left joins the item counts to the combination counts and computes the confidence values. I called the function and return the result. I am also filtering to only combinations with a confidence of 50% or more with items purchased more than 10 times.

Here we can see the head of the results table ordered by confidence from highest to lowest. We observe that the confidence of honey and whole milk is 73%! In other words, 73% of the transactions that contain honey also contain whole milk.

A

B

A B Count

A Count

Confidence

honey

whole milk

11

14

0.733

frozen fruits

other vegetables

8

12

0.667

cereals

whole milk

36

56

0.643

rice

whole milk

46

75

0.613

Head of results table.

Association rule mining is a fairly simple and easy to interpret technique to help draw relationships between items and events in a data set.

In the 1970s, professor Robert May became interested in the relationship between complexity and stability in animal populations. He noted that even simple equations used to model populations over time can lead to chaotic outcomes. The most famous of these equations is as follows:

x_{n+1} = rx_{n}(1 – x_{n})

x_{n} is a number between 0 and 1 that refers to the ratio of the existing population to the maximum possible population. Additionally, r refers to a value between 0 and 4 which indicates the growth rate over time. x_{n} is multiplied by the r value to simulate growth where (1 – x_{n}) represents death in the population.

Lets assume a population of animals is at 50% of the maximum population for a given area. We would allow x_{n} to be .5. Lets also assume a growth rate of 75% allowing r to be .75. After the value x_{n+1} is computed, we use that new value as the x_{n} in the next iteration and continue to use an r value of .75. We can visualize how x_{n+1} changes over time.

Within 20 iterations, the population dies off. Lets rerun the simulation with an r value greater than 1.

Notice how the population stabilizes at 20% of the area capacity. When the r value is higher than 3, the population with begin oscillating between multiple values.

Expanding beyond an r value of 3.54409 yields rapid changes in oscillation and reveals chaotic behavior.

Extremely minor changes in the r value yield vastly different distributions of population oscillations. Rather than experiment with different r values, we can visualize the distribution of x_{n+1} values for a range of r values using the R programming language.

Lets start by building a function in R that returns the first 1000 iterations of x_{n+1} for a given r value.

This function returns a dataframe with three columns: the iteration number, the r used for each iteration, and the x_{n+1} value computed for that iteration.

Now we need to iterate this function over a range of r values. Using purrr::map_dfr we can row bind each iteration of r together into a final dataframe.

Min refers to the lower limit of r while the max refers to the upper limit. The function will return a dataframe of approximately 400,000 values referring to each of the 1000 iterations for the 400 r values between the lower and upper bound. The function returns all 400,000 values in less than a quarter of a second.

With the dataframe of values assembled, we can visualize the distribution of values using ggplot.

Notice how r values of less than 1 indicate the population dies out. Between 1 and just under three, the population remains relatively stable. At around 3, the populations being oscillating between two points. Beyond an r of 3.54409, chaos ensues. It becomes extremely difficult to predict the value of x_{n+1} for a given iteration with an r value above 3.54409. So difficult, in fact, that this simple deterministic equation was used as an early random number generator.

So what are the practical applications for this? Representations of chaos (or systems that yield unpredictable results and are sensitive to starting conditions) can be seen across many industries and fields of study. In finance, for example, intra-day security prices have been described as a random walk – extremely difficult to predict. While long term outlooks may show seasonality, chaos theory can help model the extremely chaotic and unpredictable nature of stock prices.

Text is all around us; essays, articles, legal documents, text messages, and news headlines are consistently present in our daily lives. This abundance of text provides ample opportunities to analyze unstructured data.

Imagine you are playing a game where someone hands you a index card with the misspelled name of a popular musician. In addition, you have a book containing correctly spelled names of popular musicians. The goal is for you to return the correct spelling of the misspelled name.

In this example, suppose someone hands you a card with “Billie Jole” written on it. You quickly open the book of musicians, find names beginning with the letter B, and find the name “Billy Joel.”

As a human, this was easy for you to complete, but what if you wanted to automate this task? This can be done using Fuzzy Logic, or more specifically, the Levenshtein distance.

The Levenshtein distance considers two pieces of text and determines the minimum number of changes required to convert one string into another. You can utilize this logic to find the closest match to any given piece of text.

I am going to focus on implementing the mechanics of finding a Levenshtein distance in Python rather than the math that makes it possible. There are many resources on YouTube which explain how the Levenshtein distance is calculated.

First, import numpy and define a function. I called the function lv as shorthand for Levenshtein distance. Our function requires two input strings which are used to create a 2D matrix that is one greater than the length of each string.

If you were to use the strings “pear” and “peach” in this instance, the function should create a 5 by 6 matrix filled with zeros.

Next, the first row and column need to count up from zero. Using for loops, we can iterate over the selected values. Our Python function now creates the following matrix.

defld(s1, s2): rows = len(s1)+1 cols = len(s2)+1 dist = np.zeros([rows,cols])for i inrange(1, rows): dist[i][0] = ifor i inrange(1, cols): dist[0][i] = i

Finally, we need to iterate over every column and row combination. By doing this, we can find the minimum value of the cells directly above, to the left, and above to the left of each cell. After the minimum is found, our Python script adds one to this value to the location in question.

Our matrix should now look like the following with the far bottom right cell representing the number of changes required to convert one string into another. In this instance, it requires 2 changes to convert “peach” into “pear”; deleting the letter “c” in “peach” and replacing the letter “h” with the letter “r”.

What is so great about this function is that it is adaptable and will accept a string of any length to compute the number of changes required. While the mechanics behind this function are relatively simple, its use cases are vast.