This document discusses quantiles and quantile functions in Matlab. It defines quantiles as points taken at regular intervals from a cumulative distribution function that divide ordered data into essentially equal subsets. It lists some common quantile names like median, tertiles, quartiles, quintiles, deciles, and percentiles. It provides an example of using the quantile function in Matlab to return quantiles from a data set at specified probability values.

1. Matlab:Linear Methods

2. QuantileQuantiles are points taken at regular intervals from the cumulative distribution function (CDF) of a random variable. Dividing ordered data into n essentially equal-sized data subsets is the motivation for n-quantiles; the quantiles are the data values marking the boundaries between consecutive subsets.

3. QuantileSome quantiles have special names:The 2-quantile is called the medianThe 3-quantiles are called tertiles or terciles -> TThe 4-quantiles are called quartiles -> QThe 5-quantiles are called quintiles -> QUThe 9-quantiles are called noniles (common in educational testing)-> NOThe 10-quantiles are called deciles -> DThe 12-quantiles are called duo-deciles -> DdThe 20-quantiles are called vigintiles -> VThe 100-quantiles are called percentiles -> PThe 1000-quantiles are called permillages -> Pr

4. QuantileY = quantile(X,p) returns quantiles of the values in X. p is a scalar or a vector of cumulative probability values. When X is a vector, Y is the same size as p, and Y(i) contains the p(i)thquantile. When X is a matrix, the ith row of Y contains the p(i)thquantiles of each column of X. For N-dimensional arrays, quantile operates along the first nonsingleton dimension of X.

5. QuantileExamples:y = quantile(x,.50); % the median of xy = quantile(x,[.025 .25 .50 .75 .975]); % Summary of x

6. Least Squares FittingLeast squares fitting is a mathematical procedure for finding the best-fitting curve to a given set of points by minimizing the sum of the squares of the offsets ("the residuals") of the points from the curve.

7. Least Squares Fitting

8. Least Squares FittingIn practice, the vertical offsets from a line (polynomial, surface, hyper-plane, etc.) are almost always minimized instead of the perpendicular offsets.

9. mldivide, mrdividemldivide(A,B) and the equivalent A\B perform matrix left division (back slash). A and B must be matrices that have the same number of rows, unless A is a scalar, in which case A\B performs element-wise division — that is, A\B = A.\B.

10. mldivide, mrdividemrdivide(B,A) and the equivalent B/A perform matrix right division (forward slash). B and A must have the same number of columns.

11. Generalized Linear ModelsLinear regression models describe a linear relationship between a response and one or more predictive terms. Many times, however, a nonlinear relationship exists. Nonlinear Regression describes general nonlinear models. A special class of nonlinear models, known as generalized linear models, makes use of linear methods.

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