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Published byBrenda Cropsey Modified over 3 years ago

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Nonmetric Multidimensional Scaling input data are ranks: most similar pair AB

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Some basics We will be attempting to estimate coordinates X ik 's that result in model-fitted distances to match our dissimilarity data as best as possible. for metric mdsd ij = linear function (δ ij ) for nonmetric d ij = monotonic function (δ ij ) a.monotonically increasing function (always increases or stays same, never goes down): actually -- 2 definitions of monotonicity i.strong monotonicity: whenever δ ij < δ kl then d ij < d kl ii.weak monoticity: whenever δ ij < δ kl then d ij d kl We'll go with "b" (less restrictive, less demanding assumptions on data that could be errorful). Nonmetric Multidimensional Scaling ^^ ^^

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b.input dissimilarities data {δ ij 's} are immediately translated to ranks (thus input data can certainly be ranks). example: ---ranked like:---.20---1 ---.53.41---3 2 --- c.How to handle "ties" in the data (δ ij = δ kl ranks?) example: ---ranked like:---.20---1 ---.41.41--- 2?3? 2?3? --- i.primary approach to ties: if δ ij = δ kl then d ij may or may not equal d kl ii.secondary approach to ties: if δ ij = δ kl then d ij d kl We'll go with i because again it's more flexible and less restrictive for data we know are likely errorful. Nonmetric Multidimensional Scaling ^^ ^^

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3.Monotonic (or "isotonic") regression 3 values to watch for each pair of points (i & j): *data point = dissimilarity = δ ij, immediately translated to ranks. *distances = d ij computed at each iteration from model (estimate coordinates, compute d ij 's, etc.) *disparities = d ij (in k & w notation) values needed in the monotonic regression These are close to d ij values, but d ij 's are not distances (i.e., they will not neccessarily satisfy axioms like triangle inequality). Nonmetric Multidimensional Scaling ^ ^

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4.An example of what's done in monotonic regression a.Begin with data b. Translate to ranks: c.We'll have d ij 's at each iteration (those are computed using formula for Euclidean distance based on estimated coordinates X ik 's at that iteration). (The X ik 's are estimated via algorithm of "steepest descent"). Nonmetric Multidimensional Scaling

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d.Estimate disparities d ij 's These are distances (so they satisfy our distance axioms), but we want a model to obtain values that are monotonically increasing for our dissimilarities. Note there are deviations (i.e., decreases) at points denoted by "*". d ij 's = 0 Nonmetric Multidimensional Scaling stimulus pair ordered from max to min δ ij (i,j) δ ij rank of δ ij d ij estimated at "current" iteration 5,4.2313 5,1.2526 2,1.3433*3* 3,2.3945 4,2.4558 3,1.47610 4,3.507 13 5,2.568 11 * 5,3.7299*9* 4,1.921015 ^ Rank of δ ij

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d ij 's do not form monotonically increasing function, but disparities d ij 's do: 3 3 6 4.5 3 4.5 5 5 8 8 10 10 13 11 11 11 9 11 15 15 Nonmetric Multidimensional Scaling Shepard Diagram Model Assessment: Kruskal’s “Stress” badness-of-fit:

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Figure S1. Ambient temperature and disinfectant types of sampling periods and water temperature. In panel A, red lines indicate periods of stable disinfectant.

Figure S1. Ambient temperature and disinfectant types of sampling periods and water temperature. In panel A, red lines indicate periods of stable disinfectant.

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