The ideas of the Second Sophistic were reflected in Asia. A new method of production was introduced. Small denominations were discontinued. The cities struggled to recognise Hadrian’s lover Antinous.

Forecasting is an important problem that spans many fields, including business and industry, government, economics, environmental sciences, medicine, social science, politics, and finance. Forecasting problems are often classified as short term, medium term, and long term. Short-term forecasting problems involve predicting events only a few time periods (days, weeks, and months) into the future. Medium-term forecasts extend from 1 to 2 years into the future, and long-term forecasting problems can extend beyond that by many years.

The conquest of Italy was followed by an increase of Roman coinage. All non-Roman coinages were obliterated in the Second Punic War.

Often we look at the relationships between categorical variables, such as which hospital a patient is admitted to, or whether a person has diabetes, pre-diabetes, or no diabetes at all. These variables can be nominal (like the hospital) or ordinal (like diabetes, pre-diabetes, or no diabetes). In many cases we want to know something about how these variables are related.

In the 1990s and before, most of the world’s information was stored on paper and other analog media, such as film. However, with the proliferation of personal computers and the internet, by 2000 one-quarter of the world’s information was stored digitally. Since that time, the amount of digital data has exploded, roughly doubling every couple of years, so that now more than 98% of all stored information is digital.

The city coinages reflected the debasements of the central empire in different ways. The monetary system became fragmented, and started to collapse in the 250s, before finally ending in c. 275.

Graphical plots are the means by which data are most easily visualized and understood. Indeed, there is no better tool for finding patterns in data than the human eye applied to appropriate displays of relevant data, particularly patterns that are ill-specified or unknown.

Overfitting refers to the use of a model with more parameters than can be justified by the data. Models that are overfit are often poor at predicting the outcome of new observations, that is, observations that were not used in the construction of the model. The next example illustrates this concept.

We began the last chapter by reviewing the terms population, parameter, sample, and statistic. Parameters are numerical characteristics of a population that we would like to know, but since the population is nearly always too large to make a measurement on every unit, we often rely on a sample from the population.

In Chapter 9 we discussed point estimation for a parameter or a vector of parameters. In Chapters 10 and 11, on confidence intervals and hypothesis testing, we needed the idea of the standard error of an estimator.

A common problem in statistics is to compare groups. Does a new drug work better at reducing the time of hospitalization from COVID? Which pop-up ad generates a higher click-rate? Which type of metal – aluminum, brass, or stainless steel – will produce the most reliable product? Usually, the question involves either the mean response or the proportion of responses.

The problem of statistical inference can be described as follows. There is a population and we would like to know certain aspects of the units that make up the population. For example, we might want to know what proportion have a certain property, or what the mean value (of some measure) of all units in the population is. The population is too large to sample in its entirety, so we rely on information from a sample taken from the population.

The Roman conquests in the western Mediterranean saw the arrival of Roman coins, but in the east the local coinages at first remained and were manipulated.