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An example of isotonic regression

In numerical analysis, isotonic regression (IR) involves finding a weighted least-squares fit $x\in {\mathbb {R}}^{n}$ to a vector $a\in {\mathbb {R}}^{n}$ with weights vector $w\in {\mathbb {R}}^{n}$ subject to a set of non-contradictory constraints of kind $x_{i}\geq x_{j}$.

Such constraints define partial order or total order and can be represented as a directed graph $G=(N,E)$, where N is the set of variables involved, and E is the set of pairs (i, j) for each constraint $x_{i}\geq x_{j}$. Thus, the IR problem corresponds to the following quadratic program (QP):

$\min \sum _{i=1}^{n}w_{i}(x_{i}-a_{i})^{2}$ ${\text{subject to }}x_{i}\geq x_{j}~{\text{ for all }}(i,j)\in E.$

In the case when $G=(N,E)$ is a total order, a simple iterative algorithm for solving this QP is called the pool adjacent violators algorithm (PAVA). Best and Chakravarti (1990) have studied the problem as an active set identification problem, and have proposed a primal algorithm in O(n), the same complexity as the PAVA, which can be seen as a dual algorithm.

IR has applications in statistical inference, for example, to fit of an isotonic curve to mean experimental results when an order is expected. A benefit of isotonic regression is that it does not assume any form for the target function, such as linearity assumed by linear regression.

Another application is nonmetric multidimensional scaling, where a low-dimensional embedding for data points is sought such that order of distances between points in the embedding matches order of dissimilarity between points. Isotonic regression is used iteratively to fit ideal distances to preserve relative dissimilarity order.

Isotonic regression is also sometimes referred to as monotonic regression. Correctly speaking, isotonic is used when the direction of the trend is strictly increasing, while monotonic could imply a trend that is either strictly increasing or strictly decreasing.

Isotonic Regression under the $L_{p}$ for $p>0$ is defined as follows:

$\min \sum _{i=1}^{n}w_{i}|x_{i}-a_{i}|^{p}$ $\mathrm {subject~to~} x_{i}\geq x_{j}~{\text{ for all }}(i,j)\in E.$

Simply ordered case

To illustrate the above, let $x_{1}\leq x_{2}\leq \ldots \leq x_{n}$, and $f(x_{1})\leq f(x_{2})\leq \ldots \leq f(x_{n})$, and $w_{i}\geq 0$.

The isotonic estimator, $g^{*}$, minimizes the weighted least squares-like condition:

$\min _{g}\sum _{i=1}^{n}w_{i}(g(x_{i})-f(x_{i}))^{2}$

Where $g$ is the unknown function we are estimating, and $f$ is a known function.

Software has been developed in the R statistical package for computing isotone (monotonic) regression.

References

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  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
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