scaling parameter

Examples of scaling parameter in the following topics:

• The probability of a dyadic tie: Leinhardt's P1

  • The equation also contains scaling constants for each actor in the pair (ai and aj), as well as a global scaling parameter (lambda).
  • That is, if ties are not mutual or asymmetric, they must be null.Only the scaling constant "lambda," and no causal parameters enter the third equation.
  • Two descriptive parameters are given for each actor (these are estimated across all of the pair-wise relations of each actor):
  • Using the equations, it is possible to predict the probability of each directed tie based on the model's parameters.
  • Which actors are likely to have ties that are not predicted by the parameters of the model can also be shown in a dendogram, as in figure 18.28.

• Absolute Temperature

  • It is one of the principal parameters of thermodynamics and kinetic theory of gases.
  • By using the absolute temperature scale (Kelvin system), which is the most commonly used thermodynamic temperature, we have shown that the average translational kinetic energy (KE) of a particle in a gas has a simple relationship to the temperature:
  • Note that this equation would not look this elegant if the Fahrenheit scale were used instead.
  • By international agreement, the unit kelvin and its scale are defined by two points: absolute zero and the triple point of Vienna Standard Mean Ocean Water (water with a specified blend of hydrogen and oxygen isotopes).
  • Here, the size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure.

• Characteristics of Estimators

  • This section discusses two important characteristics of statistics used as point estimates of parameters: bias and sampling variability.
  • Bias refers to whether an estimator tends to either over or underestimate the parameter.
  • Scale 2 is a cheap scale and gives very different results from weighing to weighing.
  • A statistic is biased if the long-term average value of the statistic is not the parameter it is estimating.
  • More formally, a statistic is biased if the mean of the sampling distribution of the statistic is not equal to the parameter.

• Change of Scale

  • In order to consider a normal distribution or normal approximation, a standard scale or standard units is necessary.
  • In order to consider a normal distribution or normal approximation, a standard scale or standard units is necessary.
  • In the simplest cases, normalization of ratings means adjusting values measured on different scales to a notionally common scale, often prior to averaging.
  • The $z$-score is only defined if one knows the population parameters.
  • It requires knowing the population parameters, not the statistics of a sample drawn from the population of interest.

• Variation and Prediction Intervals

  • A prediction interval bears the same relationship to a future observation that a frequentist confidence interval or Bayesian credible interval bears to an unobservable population parameter.
  • Alternatively, in Bayesian terms, a prediction interval can be described as a credible interval for the variable itself, rather than for a parameter of the distribution thereof.
  • Since prediction intervals are only concerned with past and future observations, rather than unobservable population parameters, they are advocated as a better method than confidence intervals by some statisticians.
  • A general technique of frequentist prediction intervals is to find and compute a pivotal quantity of the observables $X_1, \dots, X_n, X_{n+1}$ – meaning a function of observables and parameters whose probability distribution does not depend on the parameters – that can be inverted to give a probability of the future observation $X_{n+1}$ falling in some interval computed in terms of the observed values so far.
  • The usual method of constructing pivotal quantities is to take the difference of two variables that depend on location, so that location cancels out, and then take the ratio of two variables that depend on scale, so that scale cancels out.

• Distorting the Truth with Descriptive Statistics

  • Descriptive statistics can be manipulated in many ways that can be misleading, including the changing of scale and statistical bias.
  • As an example of changing the scale of a graph, consider the following two figures, and .
  • A statistic is biased if it is calculated in such a way that is systematically different from the population parameter of interest.
  • The bias of an estimator is the difference between an estimator's expectations and the true value of the parameter being estimated.
  • Omitted-variable bias appears in estimates of parameters in a regression analysis when the assumed specification is incorrect, in that it omits an independent variable that should be in the model.

• Lab 1: Chi-Square Goodness-of-Fit

  • Scale the axes.
  • Label and scale the x-axis.
  • Test to see if grocery receipts follow the exponential distribution with decay parameter 1/x .
  • Label and scale the x-axis.

• Normal distribution model

  • Specifically, the normal distribution model can be adjusted using two parameters: mean and standard deviation.
  • Because the mean and standard deviation describe a normal distribution exactly, they are called the distribution's parameters.
  • The normal models shown in Figure 3.2 but plotted together and on the same scale.

• Interpreting confidence intervals

  • Incorrect language might try to describe the confidence interval as capturing the population parameter with a certain probability.
  • This is one of the most common errors: while it might be useful to think of it as a probability, the confidence level only quantifies how plausible it is that the parameter is in the interval.
  • Another especially important consideration of confidence intervals is that they only try to capture the population parameter.
  • Confidence intervals only attempt to capture population parameters.

• Capturing the population parameter

  • A plausible range of values for the population parameter is called a confidence interval.
  • If we report a point estimate, we probably will not hit the exact population parameter.
  • On the other hand, if we report a range of plausible values – a confidence interval – we have a good shot at capturing the parameter.
  • If we want to be very certain we capture the population parameter, should we use a wider interval or a smaller interval?
  • Likewise, we use a wider confidence interval if we want to be more certain that we capture the parameter.