2011-05-09/jpv, 2012-07-06/jpv

This note is an edited summary of the ideas underlying the design of the Time2 Library.

With time playing a rather important role in this world, sequences of things ordered by time are present in many kinds of systems. Because the idea of time is intuitively familiar, it is tempting to choose a simplistic design when modeling a system or even not to think about it at all. Ironically, simplistic designs can lead to needlessly complex implementations, as annoying issues are addressed one after the other.

Stated informally, a time series is a set of elements uniquely identified by a discrete point in time or by a time interval. The Time Series Framework requires a more precise definition:

Note that the definition does not explicitly allow for time ranges identifying values. This is generally not a problem because of the nature of time domains.A time series is characterized by a value type and a time domain. All elements of a time series have a value of the same type, the value type, or can be recognized as missing. Any element can be uniquely identified by a point in the time domain of the series.

The terms used in the definition will be explained shortly, but before that a few typical problems will be discussed. These problems should illustrate why a framework for time series is useful.

**Tel Aviv, Cairo, New York**

Beyond this contrived example, there are many situations with no data on weekends. A familiar case is provided by stock markets. These are also a good illustration of the annoying details hiding in seemingly simple problems: weekends are not the same in Tel Aviv, Cairo, and New York. When designing a database for global stock market data, or when drawing charts to compare the prices of some American, Egyptian, or Israeli stocks, such issues must be dealt with.

Getting a good grip on the time domain is the **first important design
goal** of the Time Series Framework.

**Missing values**

The following table lists the first *eleven* Olympic Games.

1896 | Athens |

1900 | Paris |

1904 | Saint-Louis |

1908 | London |

1912 | Stockholm |

1916 | |

1920 | Antwerp |

1924 | Paris |

1928 | Amsterdam |

1932 | Los Angeles |

1936 | Berlin |

Eleven? It is true that there are ten Games, but it is also true that
the list has eleven elements. Is something wrong here? No. Games had
been scheduled in Berlin in 1916 but were canceled because of the
war. Conceptually, the Games of 1916 are a *missing value*, and
there are many real world phenomena modeled with time series where it
is common to have some values missing. A system dealing with time
series must be capable of dealing with such cases gracefully and in a
useful way. You don't want your software to return the nine first
Games when you asked for ten, or to crash on the Games of 1916. Or you
don't want your portfolio evaluation software to give up when a quote
is missing. And as a software developer, you don't want to invent ad-hoc
solutions all the time.

Detecting missing values and handling them intelligently is the
**second important design goal** of the Time Series Framework.

**Time domain**

**Time index**

A time index is in a time domain and carries a discrete offset which defines a point in time. Two time indexes in the same time domain can be compared, with a larger index corresponding to a later point in time. Adjacent points in time are represented by adjacent offsets.

Because time indexes are expected to be used as keys it is important to implement them as immutable objects.

**Time range**

A time range is a pair of time indexes in the same time domain, called the lower and upper bound. If the lower bound is larger than the upper bound the range is said to be empty.

**Observation, value type, missing value**

An observation has a time index and a value of some type. The value type must allow the definition of a special value representing missing values, without restricting the set of useful values; null (nil) can only be used as this special value if it has no other meaning in the relevant context.

**Time series**

A time series maps a set of time indexes to values. All time indexes are in the same time domain and all values are of the same type. For this reason we talk of the time domain and the value type of a time series.

A time series can be defined alternatively as a set of observations, with each element of the set uniquely identified by its time index.

A time series has a range defined by the smallest and largest time indexes of the included observations.

A time series maintains the abstraction of missing values, which correspond to time indexes in the range for which no observation exist. This definition implies that a time series can never have missing values at its boundaries. This is also true for a subset of a time series, because it is also a time series.

Two cases must be considered, depending on the frequency of missing
values. In a **regular** time series, missing values are
exceptional. In a **sparse** time series, they are the rule.

Sparse series are useful for managing irregular events. They are typically implemented as dictionaries. Missing values are not stored. Instead of time indexes, only offsets are used as keys. The time domain is stored only once. Time indexes are reconstructed if and when needed.

The storage of regular series is straightforward and efficient. All values, missing or not, are stored into an array. With missing values the exception, the overhead is reasonable, and because they are self-signaling, missing values are always detectable. In addition to the array, the series stores also the time domain and the offset of the time index of the value in the first array element. A time index can be reconstructed from the time domain and the sum of the stored offset and the array index of the value.

Time Series Framework Design
by Jean-Paul Vetterli is licensed under a Creative Commons Attribution 3.0 Unported License.