curve fitting

Examples of curve fitting in the following topics:

• Fitting a Curve

  • Curve fitting with a line attempts to draw a line so that it "best fits" all of the data.
  • Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints.
  • Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data.
  • Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables.
  • Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to a greater degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data.

• Plotting Lines

  • It is highly unlikely that the experimental error in the data is negligible, yet the curve falls exactly through each of the data points.
  • In either case, the best-fit layer can reveal trends in the data.
  • Such curve fitting functionality is often found in graphing software or spreadsheets.
  • Best-fit curves may vary from simple linear equations to more complex quadratic, polynomial, exponential, and periodic curves.
  • The so-called "bell curve", or normal distribution often used in statistics, is a Gaussian function.

• Diagnostics for the email classifier

  • There are two key conditions for fitting a logistic regression model:
  • This may at first seem very discouraging: we have fit a logistic model to create a spam filter, but no emails have a fitted probability of being spam above 0.75.
  • The curve fit using natural splines is shown in Figure 8.18 as a solid black line.
  • If the logistic model fits well, the curve should closely follow the dashed y = x line.
  • We have added shading to represent the confidence bound for the curved line to clarify what fluctuations might plausibly be due to chance.

• Facts About the Chi-Square Distribution

  • When df > 90, the chi-square curve approximates the normal.
  • A left-tailed test could be used to determine if the fit were "too good. " A "too good" fit might occur if data had been manipulated or invented.

• Goodness of Fit

  • The goodness of fit test determines whether the data "fit" a particular distribution or not.
  • Goodness of fit means how well a statistical model fits a set of observations.
  • For example, we may suspect that our unknown data fits a binomial distribution.
  • If the observed values and the corresponding expected values are not close to each other, then the test statistic can get very large and will be way out in the right tail of the chi-square curve.
  • These hypotheses hold for all chi-square goodness of fit tests.

• The Normal Approximation to the Binomial Distribution

  • The process of using the normal curve to estimate the shape of the binomial distribution is known as normal approximation.
  • This is exactly what he did, and the curve he discovered is now called the normal curve.
  • The process of using this curve to estimate the shape of the binomial distribution is known as normal approximation.
  • Independently the mathematicians Adrian (in 1808) and Gauss (in 1809) developed the formula for the normal distribution and showed that errors were fit well by this distribution.
  • The smooth curve is the normal distribution.

• Properties of Continuous Probability Distributions

  • The graph of a continuous probability distribution is a curve.
  • Probability is represented by area under the curve.
  • The curve is called the probability density function (abbreviated: pdf).
  • The entire area under the curve and above the x-axis is equal to 1.
  • When using a continuous probability distribution to model probability, the distribution used is selected to best model and fit the particular situation.

• Introduction

  • You see the bell curve in almost all disciplines.
  • Often real estate prices fit a normal distribution.

• The F-Test

  • It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model that best fits the population from which the data were sampled.
  • The hypothesis that a proposed regression model fits the data well (lack-of-fit sum of squares).
  • The curve is not symmetrical but is skewed to the right.
  • There is a different curve for each set of degrees of freedom.
  • As the degrees of freedom for the numerator and for the denominator get larger, the curve approximates the normal.

• Using simulation for goodness of fit tests

  • Simulation methods may also be used to test goodness of fit.
  • Figure 6.21 shows the simulated null distribution using 100,000 simulated values with an overlaid curve of the chi-square distribution.