# 21 Ways to Visualize a Timeseries

We explore a variety of distinct ways to visualize the same simple dataset. The post is an excursion into the fundamentals of visualization - a partial deconstruction of the process that highlights some common techniques and associated issues.

## What this blog post is about (and what it isn’t)

With the ever more widespread adoption of Data Science tools (defined loosely as the intensive use of data in decision-making), there is a renewed interest in Visualization as an effective channel for humans to understand information at various stages of the data *lifecycle*.

There is a large variety of data visualization tools which can produce an ever more bewildering variety of visualization types:

So far, so good! Yet powerful tools used in wrong ways can have unintended consequences. Visualizations *are* powerful tools and thus come with their own challenges. The message conveyed to the viewer through a visual may be intentionally or unintentionally skewed or biased. We will try to demonstrate in this post that, in a sense, no amount of caution can avoid these challenges.

Any visualization - without exception - must make opinionated choices about how to *transform* the underlying data to produce a visually perceivable artefact

Thus, the best we can hope for, in this respect, is to have *increased awareness* of the risks and opportunities that are intrinsic to the use of visualization to conway information. In turn, this task is best achieved by developing a *deeper understanding* of what we might call the visualization “production” process.

There are two broad classes of *transformations* involved in producing a visualization:

- The first class concerns
**Mathematical Transformations**that are applied in an abstract*data space*. This class of transforms is optional (but very common). It concerns and involves various mathematical*maps*, filters and related operations. These start with the original data set and produce*intermediate data*. These intermediate results will typically have different “numerical appearance”, but still represent the same underlying measurement. - The second classs of transformations involves an or more
**Visual Maps**. These are also mathematical transformations, but are of a very special type: They link the aforementioned data space with the*visual space*we can perceive, and thus with the representation we eventually see.

This post aims to shed some light into this fascinating transformation process that is involved in any visualization no matter how simple. We aim to illustrate the relevant concepts using various distinct visualizations where:

- we use a rather simple dataset
- we keep the dataset intentionally
*the same*throughout the exercise and - we only vary the applied transforms (of both types)

This post is *not* an enumeration or classification of visualization types, nor a cookbook of how to visualize data! It is rather an excursion into the fundamentals of visualization - a partial deconstruction of the process to highlight some common techniques and associated issues.

After the reader completes this journey they will hopefully have a better intuition about how visualizations are put together, and thus be better able to use this fantastic tool in support of their data science objectives. In order to keep the task (and the reading time) finite we will exclude from the discussion several important facets:

- the display of categorical or ordinal data. Our sample will be a
*numerical timeseries* - working with complex timeseries where the observations at each time point are not a single scalar but rather
*objects*. For example, even adding*error bars*to measurements produces a more structured and complicated timeseries - we will not consider visualizations that involve multiple
*distinct*timeseries (so-called mashups of separately measured phenomena) - we will discuss
*some*of the mathematical transformations usually performed in connection with visualization but there are many, many more such examples! - we will ignore
*pictograms*, as they concern a completely different visualization paradigm - we will stick to the two-dimensional plane as this is already a rich enough canvas
- we will ignore
*animations and/or dynamic visualizations*(e.g., graphical elements such as tooltips that appear on hover of a mouse pointer). Such tools inject even more complex structures and transformations into the visualization process.

**NB: The size of the above “ignore list” shows how vast the space of all possible visualizations!**

## Some Preliminaries

We start with some background discussion. We will not define all terms as this would require a book, but will try to be precise about some terms that are relevant in visualization context.

### We will work with a single numerical timeseries. But what is a timeseries?

An (elementary) numerical timeseries is a collection of ordered, timestamped observations (measurements) where each observed value is a numerical scalar obtained from the same measurement process.

### What is a Timestamp?

We will not go deeply into the many details that require precise definition before one can capture “time” as a data point (considerations such as accuracy, formats, calendar conventions, timezones to name but a few). For our purposes it suffices to define a timestamp as *the representation of a point in time with the required accuracy*. This could be, for example, a day convention according to some calendar (e.g., “08-06-2020”), or the number of milliseconds since 1970 expressed as an integer number (No kidding - this is actually the UNIX epoch convention!)

### The Mathematical Graph

To discuss the construction of visual graphics we must discuss first *mathematical graphs*. In mathematics, the graph of a function $f$ is the set of ordered pairs of numbers $(x, y),$ where $f(x) = y$. For concreteness, the following is a random mathematical graph from Wikipedia:

Our core mathematical graph that represent a timeseries will be denoted as the orderedd pair $(t, v)$, where $t$ is a timestamp and $v$ is a scalar value. With this definition under our belt we now move to discuss transformations.

### Mathematical Transformations

Timeseries data can be transformed in an infinite number of ways before they are actually visualized. We will see some of those transformations in the sequel. For example, values can be *binned* to produce a *histogram*. One can also *smooth* or apply *differences* to bring out some particular aspect of the data. One can even apply a *Fourier Transform* to create an entirely different, so-called, *frequency domain representation*.

Each such transformation creates (for visualization purposes) essentially a *new* mathematical graph, even though from a content perspective it is still representing the same underlying phenomenon. Mathematical transformations can be *stacked* or chained: the output of the first transform becomes the input of the next one etc.

### The Visual Map

Once the required mathematical transformations have been applied, a final set of mappings performs the visualization magic: It maps whatever values we have obtained up to this point to the final *visual space* (canvas) that we will be working with. For example, one type of visualization might be obtained by mapping the temporal value $t$ of an observation into a “horizontal” spatial dimension $x$ and anchoring a visual element (e.g. a triangle) to that coordinate.

Those visual maps define the “aesthetics” of a visualization and can be extremely rich in structure, reflecting the many cognitive tools and tricks people have developed to convey information. The landing page of D3, the popular open source visualization library offers testimony for this reality:

As aptly stated by Leland Wilkinson in *The Grammar of Graphics*,

Aesthetics is what turns **graphs** (the mathematical structure of the data) into **graphics**, the visually perceivable object

Aesthetics has various *attributes*. Some of them are more important for our purposes than others. For example one can use different *shapes* to represent data points (as circles, squares etc.) but the conceptual difference between them might not be essential.

Visualizations involve many secondary aesthetic elements that aim to aid comprehension: This includes additional marks and visuals such as *axes, ticks, legends*. While these elements can be very important or even essential for comprehension of the data content, they fall outside the main thrust of visualization structure that we will be discussing here.

## The timeseries data we will use

The single timeseries dataset we will use in all examples is an actual mobility dataset from Google’s open data community report. We will not dig too deeply into the meaning of this data set. Some of it we will discover step-by-step through different visualizations, but the discussion aims to be somewhat generic and not be distracted by the specific example. For those interested in the mobility dataset, a more in-depth exploration of such datasets is available at the Open Risk Dashboard demo

Some stylized facts about this timeseries:

- As of posting it includes about 130 measurements (timepoints).
- The reported measurements start mid-February 2020 and are (normally) daily.
- The values are actually percentage points (e.g. 7.0 means 7.0%). They are versus an (unknown) baseline measurement that follows some complicated algorithm based on prior data.

## Viz 1. The first example of a visualization is actually… the data table itself

The title of this section is a bit facetious, but not completely so! The tabular representation of the timeseries looks like this (we display just the first few observations):

Time | Value |
---|---|

2020-02-15 | -1.0 |

2020-02-16 | 1.0 |

2020-02-17 | 3.0 |

2020-02-18 | 7.0 |

2020-02-19 | 5.0 |

2020-02-20 | 7.0 |

2020-02-21 | 10.0 |

… | … |

Take a closer look: The *tabular visualization* of a timeseries actually uses some very familiar pictograms! The elementary visual building blocks are namely:

- the alphabet and
- the representation of numerals
- (optionally), lines to separate numerals

In the table, two sets of pictograms (dates and values) are displayed next to each other. The spatial association between the two columns indicates that the measured value depicted on the right side of the table corresponds to the date on left side. We can summarize the structure of this “tabular visualization” in a **visualization infobox** (which we will repeat in the sequel for all other visualizations as well):

**Purpose**: Represent the actual numerical data to the desired accuracy.

**Mathematical Transform**: None. The data are shown as-is.

**Visual Map**: $(t, v) \rightarrow (T(t), T(v))$, where the temporal value $t$ is converted into the textual encoding $T(t)$ for the timestamp (following a date format convention) and displayed along the vertical dimension. Similarly, the measurement value $v$ is converted into a string, following the corresponding floating point representation conventions (notice e.g. the use of a dot).

Most people would agree that the tabular representation of data is a very *faithful* visual representation of the timeseries! In fact, tabular representation is the golden standard against which other graphical representations are judged. This is because it does not suffer from the various possible *representation pitfalls* we alluded to in the introduction. As mentioned, the use of pictograms is actually outside the scope of this post.

The excellent usability of the tabular representation does not mean that it does not have its own issues. For example:

- large datasets (exceeding a few dozen rows) may be very difficult to be understood to the average human
- simple stylized facts about the timeseries (such as trends, periodicity etc.) may be very difficult to spot
- in the absence of additional data via e.g. error estimates, the number of significant digits used in the representation can subtly change the message (the
*false sense of accuracy*effect.)

Indeed, if tabular representation could usefully express all the interesting content of data we would obviously not much care about the art and science of data visualization! To paraphrase: If a picture is worth a thousand words, a graph is worth a thousand data rows! Exploratory Data Analysis is indeed the process of systematically analysing data for the purpose of identifying and summarizing their main characteristics in text based (tabular) or visual reports.

The relationship of visualization with Exploratory Data Analysis is covered in an openly accessible Open Risk Academy course: Exploratory Risk Data Analysis using Pandas, Seaborn and Statsmodels as part of the Data Science collection.

### An elementary timeseries is already a complex object

Before we embark on our graphical adventures it is useful to further *deconstruct* the humble timeseries. As we already saw in the mathematical definition of a graph, within each typical timeseries lurk two *simpler* timeseries that we can consider as its two main building blocks:

- A sequence of timestamps $t$.
- A sequence of values $v$.

The two are associated by assigning to each timestamp one and only one value. In the first group of visualizations we explore next we will decouple these two strands (a bit like unfolding the two strands of DNA!). We will focus on truly one dimensional (“1D”) visualizations!

## Viz 2. The 1D Plot of Measurement Times

The first 1D plot focuses on depicting the measurement (observation) times.

The visualization infobox in this instance reads:

**Purpose**: Understand the absolute and relative distribution of observation times

**Mathematical Map**: Projection $(t, v) \rightarrow (t)$

**Visual Map**: The visual map that applies here is $(t) \rightarrow (x)$, where x is a spatial dimension (here horizontal) representing time.

The regular pattern of observation times we have in this particular timeseries does not convey much information. This likely reflects a choice in the preparation of the dataset. In many actual situations (e.g when measuring point processes or other irregular random events) a list of event times might convey a lot of important information encoded in the dataset. Incidentally such a plot can also help us with identifying *missing data*!

## Viz 3. The 1D Plot of Measurement Values

This 1D plot shows us the *distribution* of measured values along a single dimension. Here we ignore all information about the “when” something was measured and focus on the “how much” was measured.

**Purpose**: Understand the distribution of observation values

**Mathematical Map**: Projection $(t, v) \rightarrow (v)$

**Visual Map**: The visual map that applies here is $(v) \rightarrow (x)$, where x is a spatial dimension representing value

This plot starts unveiling some interesting information:

- We grasp immediately the
*range*of measured mobility changes (from +20% to -100%). But we do need the support of ticks and labels to actually grasp this range quantitatively, so there is a bit of cheating involved. - We also perceive a
*gap*(relative data paucity) in the range (-10%, -30%)

Notice further, that even though this example is a one-dimensional visualization, for visibility the marks of the measurements must have *some* extent along the second dimension (of height). The choice of this particular geometry does not carry any other intrinsic meaning beyond making the marks legible.

## Viz 4. The 1D Color Plot of Measurement Values

In this vizualization we take a first look at the complete timeseries (both timestamps and values). We still stick with using one only dimension. The example illustrates that in principle we can use the *color dimension* to capture value variations (and thus be very economic with the use of visual space).

**Purpose**: Understand the distribution of observation values over time

**Mathematical Map**: None. We use both data dimensions as-is

**Visual Map**: Using a map $(t, v) \rightarrow (x, c)$, where $x$ is the spatial dimension representing time and $c$ will be the *color* for the mark representing the measurement value (we use a filled square and the color space spans the blue)

Observations:

- We get immediately a rough overview of when values were relatively high (early on), when they dropped (near the middle of the observation window) and the slow recovery (near the end)
- We also get a glimpse at what seems to be a periodically occurring low measurement
- Notice that the use of a legend is absolutely essential for getting a sense of magnitude associated with color
- The depiction of recognizable dates in the axis occupies quite a bit of space, hence we can only place indicative placemarks

## Viz 5. A 1.5 Dimensional Bubble Plot of Measurement Values

We will now expand a little bit in the second dimension (thats why we call this a 1.5 dimensional plot!). We do this by depicting measurement values using filled circles of varying radii.

**Purpose**: Enhance understanding of the distribution of observation values over time

**Mathematical Map**: None. We use both data dimensions as-is

**Visual Map**: Using a map $(t, v) \rightarrow (x, r)$, where $x$ is the spatial dimension representing time and $r$ will be the *size* dimension for the mark representing the measurement value (we use a filled circle)

We notice that with the effective use of the second dimension we can get a better feeling for the distribution of mobility in time. The decline and recovery of measurements is more visible, and the same applies to the periodic dip. But we still have some perception problems:

- There is an overlap of neighboring points (resolving this might require sampling fewer values - and thus losing information)
- The sense of magnitude coupled to the size of a visual element (here the filled circle) is quite imprecise. Notoriously, this can also be abused when size is coupled to a linear attribute (such as radious), versus the surface attribute
- Here too, using a legend is imperative, as otherwise we have little sense of what of the absolute value of the measurement

The limitations of color and area in representing a dimension are well studied and are linked to the psychophysical function of various stimuli. See e.g., *Shiffman, Sensation and Perception: An Integrated Approach*

## Viz 6. The Purist Scatter Plot

In some sense the scatterplot visualization to which we turn next is the *purest* 2D representation of a timeseries.

**Purpose**: Display an accurate distribution of observation values over time

**Mathematical Map**: None. We use both data dimensions as-is.

**Visual Map**: Using a map $(t, v) \rightarrow (x, y)$, where $x$ is the first spatial dimension representing time and $y$ is the second spatial dimension representing value. Place small circles centered at the corresponding coordinates.

We immediately see the superior ability of using length along a second dimension (the vertical y-axis) to better resolve the content of our data timeseries:

- The ups and downs are captured more vividly (and quantitatively more accurately)
- We can pinpoint turning points
- We get a sense for the local structure (small changes from step to step)

The scatterplot does have a key disadvantage, though: The distribution of points on the graph surface may create confusion as to the actual *temporal order* of observations! This happens because the x and y graph dimensions are intrinsically both spatial in nature, but we are using them in an *overloaded* way. They represent two qualitatively different dimensions. NB: The confusion may be worse if the size of the marks is larger leading to greater visual overlap.

A scatter plot may not be an appropriate visualization type for a time series if it confuses the ordering of observations. Selecting smaller symbols for the representation of points may help.

## Viz 6. The Universal Line Plot

Using a line plots addresses the problem we identified with the scatter plot. The idea is to visually *link* observations.

**Purpose**: Display an accurate distribution of observation values over time with a clear sense of continuity and temporal ordering

**Mathematical Map**: None

**Visual Map**: Using a map $(t, v) \rightarrow (x, y)$, where $x$ is the spatial dimension representing time and $y$ is the spatial dimension representing value. Interpolate the unique line between two observation points.

Finally, the true content of the data seems to be revealed! Besides the patterns we have already identified in previous plots, the recurring “dip” pattern we noticed before is now very visible. The line plot is maybe the most widely used graphical display of a timeseries:

- It capitalizes on
*length perception*being more responsive to stimulus than brightness, volume or area - By creating the impression of
*continuity*, it helps clear the fog associated with nearby values in a scatterplot

The linear (or line) plot is so widely used that when we think about a concrete timeseries dataset we instinctively may use it as a proxy for the dataset itself. Yet we should never forget that a line plot is only one of the 21 **representations** of the timeseries.

In particular, we need to guard against the following pitfal:

The use of a connecting line in a line plot is an *assumption* that is translated into a visual operation. The underlying process that generated the data may or may not be continuous. Representing a discontinuous process with a connected line may be misleading.

Even if the underlying process is continuous, its degree of smoothness may not be best represented by the piecewise linear assumption.

## Viz 7. The Step Plot

The information conveyed by a step plot is largely similar to the line plot we have already seen. But there is a subtle nuance: instead of the spiky and abrupt changes of the line plot that suggest *sampling randomness*, the blocky appearance of the step plot suggests an underlying *discreteness* of the data generation process.

**Purpose**: Display an accurate distribution of observation values over time with minimal assumptions about inter-observation behavior

**Mathematical Map**: None

**Visual Map**: Using a map $(t, v) \rightarrow (x, y)$, where $x$ is the spatial dimension representing time and $y$ is the spatial dimension representing value. Interpolate step-wise between two observation points (with left, right or centered options available).

In reality, we have no information about what (if anything) to insert *between* the known measurement points. Hence, 2D line plots always involve some type of *interpolation* that may accentuate the data set in different ways. Many options exist:

- Linear interpolation (the line plot we have seen already)
- Step-wise interpolation (Assuming values are constant and equal to the previous or next observation). This is the step plot we discuss here
- Some non-linear interpolation that
*smooths*the appearance of the data (we will see next)

Which interpolation is appropriate to use depends on the process producing the data. This knowledge is, at best, a type of *metadata* that is not coming packaged together with the data table! Do we have a strong view that the underlying process is continuous and/or smooth? Then maybe a smooth interpolation like a cubic spline brings this out. Do we know that the process jumps to states while being dormant in-between, then the step approach is more appropriate. The line plot is a generic compromise that implies continuity but no smoothness.

## Viz 8. The Smooth Interpolation Plot

As a last example of a line plot, lets introduce a smooth interpolation (based on cubic splines). This has a very pleasing effect, suggesting a process that gently explores the measurement space.

**Purpose**: Display an accurate distribution of observation values over time with minimal assumptions about inter-observation behavior

**Mathematical Map**: None

**Visual Map**: Using a map $(t, v) \rightarrow (x, y)$, where $x$ is the spatial dimension representing time and $y$ is the spatial dimension representing value. Interpolate between points using, e.g., a cubic spline.

Is a smooth interpolation appropriate? Our data set is about measuring human mobility in some abstract units. We qualitatively know that “true mobility” will be a relatively smoothly varying function of time. It is thus very unlikely that it is linear or step process in appearance. Yet smoothing timeseries is not without its own pitfalls:

- The actual profile may have aspects that we are hidding when smoothing the timeseries. The fact that an entire day lapses between reported measurements suggests that there could be a lot of “microstructure” that is missing
- Potential measurement issues (such as a periodic pattern that might be simply noise from pre-processing artefacts) may get “promoted” into real effects.

The choice of interpolation method in linear plots determines the perceived "fine structure" of the underlying process, which - depending on the context - might be the right or wrong thing to do!

## Viz 10. The Area Chart

The area chart to which we turn next is closely related to the linear and scatter plots in the sense that the location of observations is identical. What changes in this approach is that the visualization colors the area outlined by the observations to create the illusion of a surface. The increased contrast of the areas above and below the line may help with the comprehension of the data.

Obviously the area chart can be combined with any of the linear, step or smooth plots (but not the scatter plot!) to produce slightly different aesthetics.

**Purpose**: Display an accurate distribution of observation values over time, emphasizing fractional value as a positive measure

**Mathematical Map**: None

**Visual Map**: Using a map $(t, v) \rightarrow (x, y)$, where $x$ is the spatial dimension representing time and $y$ is the spatial dimension representing value. Depict the area between measurement points and a horizontal reference line as a solid surface

Whether an area chart is appropriate depends on the nature of the measured value. In our case the measurement is the percentage mobility in a certain region versus the baseline at zero. Hence, the area chart has a meaningful interpretation as a level gauge, where the colored part (starting at -100% change, no mobility) indicates what fraction of the prior state (0% change normal mobility) has been achieved.

One can always color one side of a continuous line plot, but whether the result adds something meaningful to the visualization depends on the nature of the measured variable!

Now let us do a small excursion to the cutting edge of this type of graph, the so-called *horizon chart*.

## Viz 11. New Horizons with The Horizon Chart

The idea behind the horizon chart (1) needs some explaining (which suggest it is not an entirely obvious visual setup). Essentially a horizon chart is a way to compress a visualization in less space, by letting the portion of the plot that *exceeds* a certain threshold (and thus would require extending the spatial domain) fold back. It is a form of *periodic boundary condition*.

**Purpose**: Display an accurate distribution of observation values over time, emphasizing fractional value as a positive measure. Use as little vertical space as possible

**Mathematical Map**: None

**Visual Map**: Using a map $(t, v) \rightarrow (x, y)$, where $x$ is the spatial dimension representing time and $y$ is the spatial dimension representing value. Depict the area between measurement points and a reference line as a solid surface. Apply a periodic boundary condition to effectively fold the graph on a number of overlapping layers.

This process is broken down in the following chart (using other data):

The *folding* exercise of the horizon chart can be repeated several times if needed. We notice some interesting features of the horizon chart:

- The vertical size of the graph has been drastically reduced (as promised!)
- The low values observed in the middle period are showing up as a “valley” in-between the “mountains” to the left and right.
- The opacity level is used to good effect to indicate the value layers (most intense being the highest value)
- The y-axis labels must be omitted, given the folding would make them quite complicated to read. (For a two layered version one could attempt to use both left and right axes). Hence, the insight offered is now strictly qualitative
- There is some difficulty in associating the intermediate (middle layer) values between the left and right parts of the graph (due to the lack of connecting data in between)

Innovative visualizations can add significant value in the right context. User training and careful customization might be required!

## Viz 12. The Bar Chart re-incarnated

The bar chart is of course an extremely common chart, but not necessarily for the depiction of timeseries data! An essential aspect of the usual bar chart is that one of the axes ranges over a *categorical* or ordinal variable. Categorical variables are usually qualitative in nature (a set of species, a list of products). The conceptual trick we perform when adopting the bar chart paradigm is to treat each one of the observation dates as a category (which it obviously *is*).

**Purpose**: Display an accurate distribution of observation values over time using a vertical bar as a familiar visual artifact

**Mathematical Map**: None

**Visual Map**: Using a map $(t, v) \rightarrow (x, y)$, where $x$ is the spatial dimension representing time and $y$ is the spatial dimension representing value. Depict the area between measurement points and a reference line as a rectangular bar.

The result is visually not unlike an area graph or step plot but that is because we have not yet utilized the assumption that measurements are categorical. Overall the information gleaned here is not different from an area chart with step-wise interpolation and the number of categories is too large to effectively annotate as such. What would happen, though, if we treat the observation times as typical categories and ignore the temporal ordering in the representation? We could then e.g. sort the values and obtain the following “sorted bar chart”!

## Viz 13. The Sorted Bar Chart (Also called Top-K Plot)

Treating measured numerical values as categorical means that a range of operations that are more typical for categorical variables is available, for example *sorting* the values (which is a mathematical transformation).

**Purpose**: Display the prevalence of observation values using a familiar visual artifact

**Mathematical Map**: Sort Values Descending $(t, v) \rightarrow (S(t), S(v))$. Re-index the temporal dimension: $(S(t), S(v)) \rightarrow (I, S(v))$

**Visual Map**: Using a map $(I, S(v)) \rightarrow (x, y)$, where $x$ is the spatial dimension representing index and $y$ is the spatial dimension representing (sorted) value. Depict the area between measurement points and a reference line as a rectangular bar.

This sorted bar chart brings out very clearly a pattern we have noticed before: The lack of measured values within a certain range. This suggests that mobility has behaved *discontinuously* after the onset of various lockdown and social distancing measures. Visually this is the first time we obtained a drastically different depiction of the data. This is a general pattern:

Any non-trivial mathematical transformation of the measured data likely also creates new visual patterns.

The sorted bar chart is only one conceptual step away from obtaining the *empirical distribution* of the data generation processes, also known as the *histogram*. For this next step we need to *bin* the values along the y-axis into the desired intervals and then simply count the number of observations falling within each.

## Viz 14. The Histogram

The histogram is an extremely powerful mathematical transformation and visualization of numerical data. It makes tangible and widely accessible the concept of a *statistical distribution*. The end result does not look anything like what we started with, yet it is based on exactly the same information! In fact, in the process of constructing the histogram we lost some temporal information because we used the second spatial dimension to capture not time but *realization frequency* (how frequently the measurement produced a certain range of values).

**Purpose**: Display accurately the prevalence of observation values within predefined ranges

**Mathematical Map**: Project $(t, v) \rightarrow (v)$. Sort $ (v) \rightarrow (S(v))$. Bin $(S(v)) \rightarrow (I, F)$, where $I$ is bin interval and $F$ is frequency
of occurence with interval

**Visual Map**: Using a map $(I, F) \rightarrow (x, y)$, where $x$ is the spatial dimension representing bin index and $y$ is the spatial dimension representing occurence frequency. Depict the area between measurement points and a reference line as a rectangular bar.

Inspecting the distribution we can now definitely characterize it as *bi-modal*. There is a certain clustering of the observations. As with all visualizations there are caveats here as well: The process of binning introduces a new (and subjective structure). This subjectivity is encoded in the selection of the bin intervals (their number and offsets).

One can hide a lot of information using the right (or wrong) bins!

A further caveat associated with a histogram type visualization in connection with timeseries is more subtle: it may suggest that the process is *stationary* as the trend or changing volatility of observations is no longer evident. Our dataset is a clear case where intuitively we know it is not!

## Viz 15. The Density Plot

The concept of visualizing the distribution of values as a histogram leads us naturally to more mathematical pathways and the concept of *kernel density estimation*. We see that the mathematical interventions can become increasingly more heavy-handed. While the kernel density estimation is non-parametric, we are slowly on our way to actually *model* the data, that is, try to describe mathematically rather than just represent the underlying data generation process.

**Purpose**: Display a smoothed distribution of observation values

**Mathematical Map**: Project $(t, v) \rightarrow (v)$. Apply a kernel density
estimator $f_h(v)$ using a kernel $K$ and a smoothing parameter $h$.

**Visual Map**: Using a map $(v, F) \rightarrow (x, y)$, where $x$ is the spatial dimension representing value range and $F$ is the smoothed density estimate.

Visualization is an essential tool in assisting with preliminary analysis on selecting mathematical models for the data. As mentioned in the context of the histogram, attempting to fit a model on this particular data set might be a bit premature.

We next explore some visualizations that are closely tied to modelling timeseries without being actual models. These visualizations explore the temporal relationships of random values that are ordered in time.

## Viz 16. The Lag Plot or Recurrence Plot

Visually the lag plot is yet another complete break with prior patterns. This is because here both dimensions of the graph are used to depict *value* rather than time.

**Purpose**: Explore the persistence of values over time

**Mathematical Map**: Map observations $(t, v) \rightarrow (v_{t-1}, v_{t})$ into consecutive pairs.

**Visual Map**: Using a map $(v, v) \rightarrow (x, y)$, where $x$ is the spatial dimension representing current value and $y$ is the previous (lagged value).

The lag plot brings to the surface the relationship (on a rolling window basis) of an observed value with *prior* values. We see a lot of concentration of points along the diagonal, which indicates that there is persistence across time. This evidently makes sense because mobility today is likely to be heavily related to mobility yesterday.

Lag plots can be constructed for any number of lags, but the first lag is usually already very informative.

## Viz 17. The AutoCorrelation Plot

The autocorrelation plot packs an enormous amount of statistical information in the same two-dimensional space that used to host our timeseries data. What happens here in the background mathematical transformation is that data are grouped repeatedly, according to their temporal distance, and then the distribution within those groups is being estimated.

**Purpose**: Obtain a rigorous estimate of persistence of values over time

**Mathematical Map**: Map observations $(t, v) \rightarrow (\Delta t, K(\Delta t))$ to the autocorrelation function.

**Visual Map**: Using a map $(\Delta t, K) \rightarrow (x, y)$, where $x$ is the spatial dimension representing temporal lag and $y$ is strength of correlation.

## Viz 18. The Phase Diagram

A phase diagram of a dynamical system is the depiction of the system’s *position* (in the generalized sense as the state of the system) alongside its velocity (rate of change of position). For discretely measured (or intrinsically discrete systems, the velocity is proxied by the first differences in values).

**Purpose**: Obtain a rigorous estimate of persistence of values over time

**Mathematical Map**: Map observations $(t, v) \rightarrow (v, \frac{dv}{dt})$ to phase space.

**Visual Map**: Using a map $(v, \frac{dv}{dt}) \rightarrow (x, y)$, where $x$ is the spatial dimension representing value and $y$ is the rate of change of value.

The phase diagram is somewhat related to the lag plot, but it digs deeper into the dynamics of the timeseries. By comparing the two we see that after having essentially differentiated away the persistence pattern, we might be staring at some fundamental aspects of the process (or simply noise!)

## Viz 19. The Periodogram or Frequency Domain Representation

The Fourier transform is another fundamentally modified view of the data. While we have seen several visualizations that (one way or another) mess with the temporal representation, the fourier (or frequency domain representation) is different because it is a complete map from time to inverse time.

**Purpose**: Obtain a rigorous visualization of any periodic patterns

**Mathematical Map**: Map observations $(t, v) \rightarrow (\omega, P)$ to the frequency domain.

**Visual Map**: Using a map $(\omega, P) \rightarrow (x, y)$, where $x$ is the spatial dimension representing frequency (inverse of period) and $y$ is the amount of power in that frequency.

What does it bring out in our case? The number of observations is likely too small for definitive assertions but we do seem to observe excess power in the 7-day interval we have identified before.

The power of sophisticated visualizations like the autocorrelation or the periodogram typically requires substantial number of observations to manifest

We have come a long way. Let’s wrap up by navigating back into more familiar land!

## Viz 20. The Calendar Plot (Monthly Version)

The Calendar Plot is a reversion back to using the less precise color code as value representation, but what we do get in exchange is higher resolution along the temporal dimension: Namely we are now using two dimensions to represent time.

**Purpose**: Obtain a visualization of the evolution of values overlayed on a
familiar calendar theme

**Mathematical Map**: None

**Visual Map**: Using a map $(t, v) \rightarrow (x, y, c)$, where $x$ is the spatial dimension representing time, $y$ is another spatial dimension representing time modulo a monthly interval and $c$ is a color value encoding $v$.

Ofcourse it is possible to use any number of days to wrap the horizontal dimension. But the fact that certain calendars are widely adopted means that reflecting, say, monthly or weekly intervals might reveal patterns associated with behaviors anchored on such labeling of time.

In the first version of the calendar plot we arrange things on a monthly grid. Such an arrangement would highlight any monthly periodicity. It also makes it easy to spot patterns around specific dates (as a lookup table). In this instance there does not appear to be any evident monthly pattern.

## Viz 21. The Calendar Plot (Weekly Version)

Using the Weekly version of the Calendar plot highlights maybe better than any other plot that there is a periodic weekly pattern occuring between early in the weekend

## How was it made?

The visualizations presented here were made using a variety of open source tools. While visualization tools exist more or less for any major programming language or numerical/statistical system, we have focused on two very popular options: javasript and Python.

- A list of Javascript Based visualization tools
- A list of Python Based visualization tools

## Further Reading

The inspiration for this post comes from *Leland Wilkinson, The Grammar of Graphics*.

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