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The data will stand out—and you will too.

Art is science in love.

— E.F. Weisslitz

The final art pieces are the outcome of a long process of exploration, experimentation and more than a few dead-ends. Here I'll take you through the process of parsing, exploring and drawing the data and finding a story.

The first step was to identify how to present the theme of "differences" in the exhibit. We wanted to draw attention to the fact that the extent to which we can answer questions about biological states depends on how accurate and precise the measurements are. Initially, I thought that this might be a useful theme—highlighting both biological and technical variation.

In parallel, I thought about different functional sources of variation and the kinds of questions that the data might be used to answer. I narrowed them down to differences that cause disease, differences that cause disease progression and differences that allow for a variety of normal function.

We also considered the idea of showing differences due to purely experimental error, which are fluctuations due to the technology not necessarily the biology.

For each of the difference scenarios, the input data set were single-cell gene expression counts.

cell1 sample cell_type gene1 count cell1 sample cell_type gene2 count ... cell2 sample cell_type gene1 count cell2 sample cell_type gene2 count ...

Each cell was identified by its sample (e.g. blood normal vs tumor) and type (e.g. B cell, T cell, etc). Typically, we had counts for about 500 genes of 100's or 1000's of cells of a given type and sample.

The transcriptome of each cell is a point in high-dimensional space—one dimension for each gene for which we have a count. To find a projection of the data onto the page, we dimensionally reduced the data using tSNE (t-distributed stochastic neighbour embedding) into either 2 or 4 dimensions.

# 2 dimensions (x,y) cell1 sample cell_type tsne_x tsne_y cell2 sample cell_type tsne_x tsne_y ... # 4 dimensions (x,y,u,v) cell1 sample cell_type tsne_x tsne_y tsne_u tsne_v cell2 sample cell_type tsne_x tsne_y tsne_u tsne_v ...

When the cells are drawn as points based on their tSNE coordinates, typically they will cluster both by type and disease status. They cluster by type because the gene expression profile for each cell type is different—this is what causes cells to have different function.

What we're looking for here is cells that cluster by disease state within a given cell type cluster, like the classical monocytes highlighted in the figure above. What this happens, we can say that there's something fundamentally different for this cell type between the normal and diseased states. For cell types that don't have a differential expression in disease, we see more of a random mixing of the normal and disease cell populations within their clusters.

I like to start by exploring ways to map the data onto the page. Typically, at this stage I try a lot of different approaches and many take the shortest route to the trash.

The focus of our story is "differences". So, I'll be looking for visuals that capture the Gestalt of a difference. Because we're aiming for an equal mix of art and data visualization, it's not critical that we can judge the differences quantitatively—ability to make qualitative assessments will suffice—but it is important that something obviously appears to be different, both within one panel and across panels.

In the search for an encoding, two basic questions have to be addressed: how cells are to be (a) placed and (b) represented on the page. For example, placement could be systematic, such as on a grid, with order based on some property such as total gene count. With this approach, we can achieve a tiling that covers the full canavs.

Or, placement could be based on properties such as tSNE coordinates. In this case, we settle in not populating the canvas evenly and hope that the cells fall into groups in a meaningful way.

Below is one attempt at a tiling encoding. The cells are represented by a grid of squares and each square shows a comparison of gene counts in disease vs normal for four genes. This layout can be adapted into triangles (3 genes) or hexagons (6 genes).

The genes can be chosen based on ones that are known to be of clinical significance or, as below, identified from the data set as ones that have the largest change in expression between normal and disease. Cells can be ordered based on similarity in counts between normal and disease.

I then tried using the 4-dimensional `(x,y,u,v)` tSNE coordinates to represent each cell, still sticking to a tiling of cells. Cells are drawn in a row-dominant order and those with more similar tSNE coordinates are drawn closer together. For example, for the circular encoding, each concentric circle radius is `(x,x+y,x+y+u,x+y+u+v)`.

Playing with colors and shapes makes for interesting tilings.

As much as all these attempts have pretty shapes and colors, it's hard to point your finger at any part of these encodings and say, "Here's a difference worth investigating."

As well, by placing cells on a grid makes for a rigid represntation. We agreed that we needed something that looked a little more organic. Could the way the cells were drawn look like actual cells?

We started looking at drawing the cells based on their 2-dimensional tSNE coordinates. In this representation, cells that are closer together on the plane have more similar transcription profiles.

With this approach, it was really easy to draw attention to differences—either by cell type (which we wanted to do in the normal function case) or disease status. We also felt that this would be a familiar approach and one that showed populations of cells at the resolution of single cells.

One of the data sets that we identified early was the internal validation set, which captured the variability between operators and technical replication. Below are the results of six experiments, each done by a different operator on two different days. Here, we don't expect (and we don't see) any clustering based on days or operators. In the end, we chose not make use of this data set in the final exhibit and focus instead on clinically relevant differences.

We then explored ways in which the tSNE coordinates could be used to derive different representations, such as a network or tesselation.

By hiding the cells and filling the polygons based on cell type and mixing the neighbouring polygon colors, we get a

A more organic feel can be achieved by perturbing the edges of both the cells and polygons. We were happy enough with this representation to use it on the introductory panel in the exhibit.

The distance between a given tumor cell and its nearest normal cell of the same type can be emphasized by encoding the distance with color, such as below.

At this point we decided to explore mapping this difference by the 3rd dimension, literally.

The stereoscopic image shown above is a quick prototype generated in Illustrator. The final images were rendered in a full 3D environment using Cinema4D.

We discuss the many ways in which analysis can be confounded when data has a large number of dimensions (variables). Collectively, these are called the "curses of dimensionality".

Some of these are unintuitive, such as the fact that the volume of the hypersphere increases and then shrinks beyond about 7 dimensions, while the volume of the hypercube always increases. This means that high-dimensional space is "mostly corners" and the distance between points increases greatly with dimension. This has consequences on correlation and classification.

Altman, N. & Krzywinski, M. (2018) Points of significance: Curse(s) of dimensionality *Nature Methods* **15**:399–400.

Inference creates a mathematical model of the datageneration process to formalize understanding or test a hypothesis about how the system behaves. Prediction aims at forecasting unobserved outcomes or future behavior. Typically we want to do both and know how biological processes work and what will happen next. Inference and ML are complementary in pointing us to biologically meaningful conclusions.

Statistics asks us to choose a model that incorporates our knowledge of the system, and ML requires us to choose a predictive algorithm by relying on its empirical capabilities. Justification for an inference model typically rests on whether we feel it adequately captures the essence of the system. The choice of pattern-learning algorithms often depends on measures of past performance in similar scenarios.

Bzdok, D., Krzywinski, M. & Altman, N. (2018) Points of Significance: Statistics vs machine learning. Nature Methods 15:233–234.

Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: a primer. Nature Methods 14:1119–1120.

Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: supervised methods. Nature Methods 15:5–6.

Celebrate `\pi` Day (March 14th) and go to brand new places. Together with Jake Lever, this year we shrink the world and play with road maps.

Streets are seamlessly streets from across the world. Finally, a halva shop on the same block!

Intriguing and personal patterns of urban development for each city appear in the Boonies, Burbs and Boutiques series.

No color—just lines. Lines from Marrakesh, Prague, Istanbul, Nice and other destinations for the mind and the heart.

The art is featured in the Pi City on the Scientific American SA Visual blog.

Check out art from previous years: 2013 `\pi` Day and 2014 `\pi` Day, 2015 `\pi` Day, 2016 `\pi` Day and 2017 `\pi` Day.

We examine two very common supervised machine learning methods: linear support vector machines (SVM) and k-nearest neighbors (kNN).

SVM is often less computationally demanding than kNN and is easier to interpret, but it can identify only a limited set of patterns. On the other hand, kNN can find very complex patterns, but its output is more challenging to interpret.

We illustrate SVM using a data set in which points fall into two categories, which are separated in SVM by a straight line "margin". SVM can be tuned using a parameter that influences the width and location of the margin, permitting points to fall within the margin or on the wrong side of the margin. We then show how kNN relaxes explicit boundary definitions, such as the straight line in SVM, and how kNN too can be tuned to create more robust classification.

Bzdok, D., Krzywinski, M. & Altman, N. (2018) Points of Significance: Machine learning: a primer. Nature Methods 15:5–6.

Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: a primer. Nature Methods 14:1119–1120.

In a Nature graphics blog article, I present my process behind designing the stark black-and-white Nature 10 cover.

Nature 10, 18 December 2017

In this primer, we focus on essential ML principles— a modeling strategy to let the data speak for themselves, to the extent possible.

The benefits of ML arise from its use of a large number of tuning parameters or weights, which control the algorithm’s complexity and are estimated from the data using numerical optimization. Often ML algorithms are motivated by heuristics such as models of interacting neurons or natural evolution—even if the underlying mechanism of the biological system being studied is substantially different. The utility of ML algorithms is typically assessed empirically by how well extracted patterns generalize to new observations.

We present a data scenario in which we fit to a model with 5 predictors using polynomials and show what to expect from ML when noise and sample size vary. We also demonstrate the consequences of excluding an important predictor or including a spurious one.

Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: a primer. Nature Methods 14:1119–1120.