Time series

We can track how things change over time.

You might track the daily heat. You might watch the ocean tides. You can even count sunspots in the sky.

We write these numbers in a list. This list shows a pattern. We can use a line chart to see it.

A chart helps us see trends.

Some patterns repeat. These are called seasonal effects. We use these lists to guess the future. This is called forecasting. It helps us predict the weather. It even helps us guess earthquakes.

Imagine you keep a list of the daily heat. Or you track how high the ocean tides go. These lists of numbers are called a time series.

A time series shows data in order. Each number has a time attached to it. We can record data over seconds or years. Some people track sunspots in space. Others track prices in the stock market.

We often use a line chart to see these numbers. A chart helps us find patterns. We might see a trend, which is a general direction. We might see seasonal effects. These are patterns that repeat at certain times.

Scientists use these lists to guess what happens next. This is called forecasting. It helps us predict the weather or even earthquakes. We can also use math to fill in missing numbers. This is called interpolation. If we try to guess numbers far in the future, it is called extrapolation. This can be harder and less certain.

A time series is a special list of information. It keeps data points in a specific order. This order follows the passing of time. You might record how high the ocean tides go. You could also track daily temperatures or sunspots. Each number has a time attached to it. This makes the data discrete-time data. Most of the time, these measurements happen at equal spaces. This could be every second or every year.

We use math to understand these lists. Scientists often use a run chart to see the data. This is a type of line chart. It helps us find patterns in the numbers. We might see a trend, which is a general direction. We might see seasonal effects that repeat. We also look for irregular fluctuations. These are sudden changes that do not follow a pattern.

Many people have helped develop these math tools. During World War II, work moved very quickly. A mathematician named Norbert Wiener helped a lot. He worked on ways to filter signals. He wanted to separate real signals from random noise. Other engineers like Rudolf E. Kálman and Dennis Gabor also helped. They worked on predicting signal values. These ideas are used in many sciences today.

There are many ways to study these patterns. One way is called time-domain analysis. This looks at the data directly. Another way is frequency-domain analysis. This uses tools like the Fourier transform. We can also use curve fitting to help us. This means making a mathematical line that fits the points. We can use interpolation to fill in missing gaps. This is like reading between the lines.

Time series help us see into the future. This process is called forecasting. We use a model to guess future values. We look at what happened in the past to do this. This is very helpful for weather forecasting. It is also used for earthquake prediction. Even the stock market uses these ideas. We can even track things like tuberculosis deaths. A chart shows these deaths in the United States from 1954 to 2021.

A time series is a sequence of data points recorded in chronological order. This means the information is listed according to when it happened. Most often, these observations occur at equally spaced intervals. This type of information is called discrete-time data. It can represent measurements taken over seconds, days, or even centuries. Common examples include the daily temperature or the closing values of the Dow Jones Industrial Average.

To understand these sequences, scientists often use a run chart. This is a specific type of temporal line chart. It helps researchers identify important patterns within the data. One pattern is a trend, which shows a general direction over time. Another is a seasonal effect, which repeats at regular intervals. Finally, researchers look for irregular fluctuations. These are sudden, unpredictable changes in the data.

There are several ways to categorize this data. A time series is a type of one-dimensional panel data. This is different from cross-sectional data. In cross-sectional studies, there is no natural temporal ordering. For example, comparing wages by education level is cross-sectional. In that case, the order of the people does not matter. However, in a time series, the order is essential. If you change the order, you lose the meaning of the data.

Researchers use different mathematical methods to analyze these series. They can use time-domain methods or frequency-domain methods. Time-domain methods include tools like autocorrelation and cross-correlation analysis. Frequency-domain methods include spectral analysis and wavelet analysis. These methods can also be parametric or non-parametric. Parametric approaches assume the data follows a specific structure. Non-parametric approaches do not assume a particular structure.

Much of this math was developed to solve complex problems. During World War II, progress in this field accelerated significantly. The mathematician Norbert Wiener worked on filtering signals from noise. Other engineers, such as Rudolf E. Kálmán and Dennis Gabor, also made major contributions. They focused on predicting signal values at specific points in time. Their work helped create tools like the Kalman filter. These tools are now used in many modern engineering fields.

One major goal of this analysis is forecasting. Forecasting uses a mathematical model to predict future values. This is done by looking at previously observed data. In many cases, the data is modeled as a stochastic process. This accounts for randomness and uncertainty in the measurements. Forecasting is vital for weather forecasting and earthquake prediction. It is also used in econometrics to study financial markets.

Another technique is called curve fitting. This is the process of constructing a mathematical function that fits the data points. This can involve interpolation or smoothing. Interpolation is the process of estimating an unknown quantity between two known values. It is often described as "reading between the lines." On the other hand, extrapolation estimates values beyond the original range of data. Extrapolation is much riskier because it involves higher uncertainty.

Time series analysis is used across many different scientific fields. In signal processing, it is used for signal detection. In meteorology and geophysics, it helps with forecasting. Data mining and machine learning use it for tasks like anomaly detection. It is even used in astronomy to study celestial phenomena. Even medical science uses it, such as tracking tuberculosis deaths in the United States from 1954 to 2021.