From 02a41a169556fbdb93eba169ba6da3916eb76472 Mon Sep 17 00:00:00 2001 From: Anurag Sharma Date: Wed, 24 Jun 2015 13:32:41 +0200 Subject: [PATCH 1/2] Tutorial for Sketchmap and landmark selection added. --- index.html | 1 + pages/plumed_tuts.html | 21 +++ pages/plumed_tuts/ala12_helix.jpg | Bin 0 -> 9644 bytes pages/plumed_tuts/ala_gmdsplot.png | Bin 0 -> 83566 bytes pages/plumed_tuts/ala_histosigma.png | Bin 0 -> 15349 bytes pages/plumed_tuts/ala_histosolo.png | Bin 0 -> 12024 bytes pages/plumed_tuts/ala_landmark.png | Bin 0 -> 84660 bytes pages/plumed_tuts/ala_mdsplot.png | Bin 0 -> 60629 bytes pages/plumed_tuts/ala_phipsi.png | Bin 0 -> 71659 bytes pages/plumed_tuts/ala_smap_angles.gif | Bin 0 -> 464929 bytes pages/plumed_tuts/ala_smap_ani.gif | Bin 0 -> 233959 bytes pages/plumed_tuts/ala_smap_combo.gif | Bin 0 -> 612542 bytes pages/plumed_tuts/ala_smap_proj.png | Bin 0 -> 176810 bytes pages/plumed_tuts/analysis.html | 140 +++++++++++++++++++ pages/plumed_tuts/gnuplot.html | 47 +++++++ pages/plumed_tuts/gp_land.png | Bin 0 -> 160807 bytes pages/plumed_tuts/gp_proj.png | Bin 0 -> 308749 bytes pages/plumed_tuts/landmarks.html | 132 ++++++++++++++++++ pages/plumed_tuts/projecting.html | 50 +++++++ pages/plumed_tuts/results.html | 52 +++++++ pages/plumed_tuts/sketch-map.html | 194 ++++++++++++++++++++++++++ pages/plumed_tuts/summary.html | 41 ++++++ pages/tuts/landmarks.html | 97 ++++++++++++- pages/tuts/sketch-map.html | 45 +++++- sitemap.xml | 5 + style.css | 22 +++ 26 files changed, 840 insertions(+), 7 deletions(-) create mode 100644 pages/plumed_tuts.html create mode 100644 pages/plumed_tuts/ala12_helix.jpg create mode 100644 pages/plumed_tuts/ala_gmdsplot.png create mode 100644 pages/plumed_tuts/ala_histosigma.png create mode 100644 pages/plumed_tuts/ala_histosolo.png create mode 100644 pages/plumed_tuts/ala_landmark.png create mode 100644 pages/plumed_tuts/ala_mdsplot.png create mode 100644 pages/plumed_tuts/ala_phipsi.png create mode 100644 pages/plumed_tuts/ala_smap_angles.gif create mode 100644 pages/plumed_tuts/ala_smap_ani.gif create mode 100644 pages/plumed_tuts/ala_smap_combo.gif create mode 100644 pages/plumed_tuts/ala_smap_proj.png create mode 100644 pages/plumed_tuts/analysis.html create mode 100644 pages/plumed_tuts/gnuplot.html create mode 100644 pages/plumed_tuts/gp_land.png create mode 100644 pages/plumed_tuts/gp_proj.png create mode 100644 pages/plumed_tuts/landmarks.html create mode 100644 pages/plumed_tuts/projecting.html create mode 100644 pages/plumed_tuts/results.html create mode 100644 pages/plumed_tuts/sketch-map.html create mode 100644 pages/plumed_tuts/summary.html diff --git a/index.html b/index.html index 62572e22..5f693c19 100644 --- a/index.html +++ b/index.html @@ -33,6 +33,7 @@ mkMenu("gismo", "GISMO"); mkMenu("manu", "MANUAL"); mkMenu("tuts", "TUTORIAL"); + // mkMenu("plumed_tuts", "PLUMED"); diff --git a/pages/plumed_tuts.html b/pages/plumed_tuts.html new file mode 100644 index 00000000..2d2a5a31 --- /dev/null +++ b/pages/plumed_tuts.html @@ -0,0 +1,21 @@ + + +
+ + +
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+

Analysis and choice of parameters

+ 2d slice of torsional angle data +

+ Sketch-map builds on the notion that in many cases the high-dimensional data + that one is trying to represent with few parameters does actually contain + some genuinely high-dimensional features, which would interfere with the + dimensionality reduction procedure. +

+

The histogram of pairwise distances

+

+ A first step is therefore to analyze the data in order to understand how + the samples are distributed in high-dimensional space. A cursory + examination of two-dimensional slices of the data set can be useful to + achieve a superficial understanding of the structure of the data. + Such a plot for the 12-alanine data shows that samples + are gathered in seemingly Gaussian blobs, connected by a spiders web + of less-populated transition states.
+ To better guide the choice of parameters one should perform a more quantitative + analysis. To this aim, we can use the dimdist + utility to generate a histogram of the pairwise + distances of samples, i.e. the probability distribution for two points picked + from the sample at random to be at a given distance. +

+ +
+
+ Running dimdist [+-] +
+
+

+ The dimdist utility can be used to compute the distribution + of pairwise distances within the high-dimensional data set. For this example, you should type +

+
+ dimdist -D 24 -P highd-dataset -pi 6.283185 -maxd 15 -nbin 300 -wbin 0.05 -lowmem > highd-histo +
+

+ Here the -D options specifies the dimensionality, + -P the location of the data file, -pi + indicates that in computing distances the coordinates have to be taken to have the + specified periodicity, and the other options control the range, bins and smoothening of + the histogram. See --MANUAL-- for a more detailed description of the options which may + apply to a different kind of data.
+ Note that the computational expense of this analysis scales with the square of the number of + data points, so it can be quite time-consuming. It there are more than a few tens of thousands + points, it is often sufficient to use a random subset of the samples. +

+
+
+ + + Histogram of pairwise distances in D=24 +

+ The resulting histogram may be at first surprising. In most cases, as for the 12-ala + data, one would observe a very low probability of finding two points very close together. + This is a consequence of the high dimensionality: + taking two points from a high-D-dimensional Gaussian distribution it will be unlikely for + them to be at distances smaller than several standard deviations! As a rule of thumb, if + on all the d-dimensional slices of the original data the samples seems to have a Gaussian + distribution with standard deviation s, the distribution of distances in the full, + D-dimensional description will be peaked at s*sqrt(D/d). In the example, the 2-d plot + indicates a standard deviation of about 0.5, and the distribution shows a shoulder which + corresponds to these Gaussian correlations at about R=2.
+ A second observation is that the overall distribution is likely to be strongly peaked at + rather large distances. In fact, in this case + it can be shown that the long-range part of the distribution is very similar to a uniform + distribution of samples, even if clearly the samples are not uniform at all! + This is again a consequence of the quirks of working in high dimension: the "intrinsic" volume + element grows so rapidly with distance that far-away points weight a lot in the histogram, + even though they carry very little information on the structure of the data set. +

+

Proximity in high dimension

+

+ The idea behind sketch-map is to transform the distances in such a way that the (not-so-relevant) + information contained in the short-range and long-range point-point distances is not considered, + and the dimensionality reduction algorithm can focus on reproducing the intermediate distances, + which carry information on the relative position of different stable conformations rather + than on Gaussian fluctuations or on far-away relations which are dominated by the topology + of the high-dimensional space rahter than on the specific features of the problem. +

+ n(r) and a superimposed s(R) +

+ This is achieved by means of a sigmoid function which is applied to distances,
+\[ + s_{\sigma,a,b}(r) = 1 - (1+ (2^{a/b}-1)(r/\sigma)^a)^{-b/a} +\] + which is zero for x→0, one for x→∞ and 1/2 for x=σ. The a and be parameters + define the rate at which s goes to zero and to one, respectively. The critical parameter to select + is the switching distance σ: points which are closer than σ will be considered "close" + and points which are farther apart will be considered "far". σ should be placed somewhere + between the range of distances characteristic of Gaussian fluctuations (0-3 in this example) + and the range dominated by the intrinsic high dimensionality (8 and larger in this example). + This leaves a range of σ to be chosen in the 4-7 range. We choose here σ=6, but + values not too far from this yield comparable results.
+ Once σ is chosen, one should select a and b. While the method is not very sensitive to these + values, one should make sure that the step around R=σ is not too sharp, and that at the same + time distances within the Gaussian basins are mapped to zero. b should be chosen in such a way that + s(x) approaches 1 by the time n(R) has become very small. If one observes that the final map projects + some outliers very far away from anything else, it may help to reduce the value of b in high dimension, + so that s(x) does not approach 1 too rapidly. +

+ +
+
+ Further examples of histogram and effective sigmoid functions [+-] +
+
+

+ -- PUT HERE HISTOGRAM/CHOSEN FUNCTION PLOTS FOR THE LIGAND BINDING, THE 3D EXAMPLE, AND FURTHER + DISCUSS THE RATIONALE -- +

+
+
+ + + +

Proximity in low dimension

+

+ One of the key-features of sketch-map is that also in low dimensions the similarity of points + is not evaluated using Euclidean distances. Another sigmoid function is applied, so that the + close→close/far→far mapping can be enforced. The σ parameter in low dimension is + irrelevant, since using a value different from the one chosen in high dimension simply corresponds + to a rigid scaling of the low-dimensional map. The high-dimension σ should always be used.
+ The a and b parameters do not need to be the same. If they are chosen to be equal to their + high-dimension counterparts, sketch-map would ultimately try to match distances, since distances + are transformed in low and high dimension in the same way. This may be desirable when the mapping + is not too complex, e.g. when one knows already that high-dimensional points lie very close to a + hyperplane.
+ When this is not the case, it is useful to choose different values: the advantage is that one can + better accommodate the counter-intuitive features of high-dimensional spaces in the low-dimensional + projection. For instance, if D is the original dimension and d the dimension of the projection, + one can "equalize" the volume elements in the low and high-dimensional cases by choosing + ad/d≈aD/D. For the same reason, it is usually a good idea to choose + bd to be smaller than bD. As a rule of thumb, one or two are good values + to start experimenting for both ad and bd.
+ In this example, we take ad=1 and bd=2. +

+ +
diff --git a/pages/plumed_tuts/gnuplot.html b/pages/plumed_tuts/gnuplot.html new file mode 100644 index 00000000..13169025 --- /dev/null +++ b/pages/plumed_tuts/gnuplot.html @@ -0,0 +1,47 @@ +
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z#xH&5ep0Fr78e%4Ms_ADh#u&!L9;VYkr()#SzAshah1J-gjWImY*Ftd*8U&|n|Zn~ zSQEx;TO#GWca~W4E3;|n9S|>bWFpE+0h3`ul&ejd~js)E=heo7^gnuSpz}Bz48Z4Ik~8=MrZZJBV;G_ zt#(=_YN6qP3Y+b$Z-}esgfyPgK0_1dey) z?g8~n^<8b}CgJSLpx?boJshQUVF}2YWOa=xtg^Cnha#F%M6zODMjY&`JR%)%3l?kZ zFA0$qLk>et+9UX7^t1wBb5bWabevMQ7pqxSuudnr-7#6U`@IxEH?NrUb(&@V^Aw|- z&@eN3iry#qVeUA_et&C7JHlKCafl-ilTqs?a#Xb5xtL(Gz!Bq-V$VBhp(q9mY*lrY z4Sx3wokQS2K{DxJ4~=w{-{aAn{NVsyY;`_9`=d&;45nJzj^CySGmL;3^>LT?v&yA9 z!P!~#&vCIF@Y1`eAR0+_qB2+0 z&hImVnTn!C(O%wh#K>cT^1$t;2BqGsggK4!ZO?s1xPc(t>lJxu9)EECksPKtYLW{- zsXSf~{d5l4Ueowd&qtb0xR{r$TEGgF ziRU=!mgdB~K4#rQBnu>Ha`*VA1$4$3>RGiZ9nRkf35LD*D!j@|>`q5mZi8B1Ogp-#yvkQmkoeMch zBIs%E^7_X-3~_v<#6`XE+ + + 100 minmax landmarks picked from 12ala data. Circles area is proportional to weight. +

Selecting landmark points with PLUMED

+

+ The sketch-map calculation is expensive and its expense increases sharply with the number of points whose projections you are + endeavoring to find simultaneously. Moreover, the iterative minimization has more chances of getting stuck in a poor local minimum + when the number of variables to be optimized grows larger. It is therefore not feasible to fit projections for all the points in the trajectory + at once. Hence, before running the sketch-map algorithm we must select a set of representative, landmark, points from the trajectory. + There are following strategies we use to do this selection: +

+ +
+ + +

+ All of these strategies are implemented in the SelectRandomFrames, FarthestPointSampling, SelectWithStride code and StagedSampling. +

+ The input file for the following routines can be generated using the script provided [here] + + diff --git a/pages/plumed_tuts/projecting.html b/pages/plumed_tuts/projecting.html new file mode 100644 index 00000000..40b9cc8f --- /dev/null +++ b/pages/plumed_tuts/projecting.html @@ -0,0 +1,50 @@ +
+ +

Out-of-sample embedding

+ +

+ After the low-d embedding of the landmark points has been + worked out, one can use it to define a map which allows to + project additional points. In other terms, a set of sketch-map + collective coordinates can be computed for any configuration + of the system. This final step is delegated to the + dimred utility. +

+ +
+
+ running dimproj [+-] +
+
+

+ We need first to prepare two files with the landmarks, one containing + the high-dimensional coordinates (plus weights) and another containing the sketch-mapped + ones, without headers or extra columns. Then we run +

+
+ dimproj -D 24 -d 2 -P highd-landmarks -p lowd-landmarks -pi 6.283185 -w + -grid 15.0,21,201 -fun-hd 6,12,12 -fun-ld 6,1,2 -cgmin 3 < highd-new > lowd-new +
+

+ Here most of the options correspond to those in the sketch-map step, + given the close analogy between the sketch-map stress function and the + one which is used for the out-of-sample embedding. After having performed + a grid search based on the parameters provided, a few conjugate gradient + steps are performed (-cgmin). +

+
+
+ + + +

+ The image below shows the embedding of the circa 50000 configurations of + the trajectory, colored as usual based on just one of the backbone dihedral. + Despite the fact that the landmarks are a very coarse sampling of the + configuration space, the out-of-sample sketch-map embedding separates nicely + between different structures, and shows a remarkably rich picture of the + many metastable states accessible to this simple model polypeptide. +

+ Linear MDS embedding of 12-ala data, colored based on one of the dihedrals + +
diff --git a/pages/plumed_tuts/results.html b/pages/plumed_tuts/results.html new file mode 100644 index 00000000..4d225d4f --- /dev/null +++ b/pages/plumed_tuts/results.html @@ -0,0 +1,52 @@ +
+ + Sketch-map embedding of the 12-ala colored according to the different dihedrals +

Analyzing the results

+

+ Unless you are working for a model system for which you know already which kind of embedding + to expect, it is not straightforward to judge the success of the fitting procedure. There are + many ways to post-process the projected points to get some qualitative -- and quantitative -- + assessment of the outcome of the fit.
+ A first possibility is to color the projected points according to different coordinates of the + corresponding high-dimensional samples: if the fitting was successful one would expect points + of similar color to be clustered together, rather than mixed up with points of different color. +

+

+ Another, more quantitative, analysis can be performed using the dimdist + utility. With the appropriate parameters, dimdist generates a D-d plot, + i.e. an histogram which measures the probability that, for any high-dimensional pairwise distance D, + the distance between the corresponding projections will be d. A completely faithful embedding would + preserve the distances, so the D-d histogram should be a series of delta function along the D=d diagonal. + Having performed sketch-map, instead, one expects would already be content when the majority of the + density is concentrated in the D<σ,d<σ and D>σ,d>σ regions. +

+ +
+
+ computing D-d plots [+-] +
+
+ D-d plots can be generated using the dimdist utility. + Note that both the high-dim and low-dim data set must be provided, in files containing + one point per line, without headers. Optionally, the high-dimension file may contain + weights (-w). +
+ dimdist -D 24 -d 2 -w -P highd_landmarks_noheader -p lowd_projection -pi 6.283185 -lowmem + -gnuplot -maxd 15 50 -nbin 150 250 -wbin 0.2 > ddplot.histo +
+ Dd-plot and the corresponding embedding for different mixing parameters +

+ Mixed metric MDS/sketch-map embedding, with mixing parameter going from pure MDS to pure sketch-map. + The embedding (coloured according to one of the dihedrals) is shown side by side with the corresponding + D-d plot. Note how metric MDS tends to have high density in the D>σ,d<σ. This means that + "far" points are mapped to "close" points in low-dimension: this is detrimental if the mapping is to + be used to classify configurations, to compute free energies or perform enhanced sampling. + Sketch-map, on the other hand, manages to keep most of the density in the "good" regions: even though + the structure of the data set is deformed, most of the topological relations that matter + have been preserved. +

+
+
+ + +
diff --git a/pages/plumed_tuts/sketch-map.html b/pages/plumed_tuts/sketch-map.html new file mode 100644 index 00000000..d397fdaa --- /dev/null +++ b/pages/plumed_tuts/sketch-map.html @@ -0,0 +1,194 @@ +
+ +

Running sketch-map

+ +

+ Having determined some reasonable parameters which define + "proximity" for the data set we want to project, we are now + ready to run sketch-map. The minimization procedure can be quite + time-consuming, in particular if one is using many landmarks + and/or very high dimensionality. +

+ +
+
+ running Sketchmap [+-] +
+
+

+ Following code uses all dataset to run the sketch-map optimization which is decsribed in detail below. +

+
+ d: READ FILE=input.in VALUES=c.*
+ SKETCHMAP ...
+ ARG=d.*
+ NLOW_DIM=2
+ USE_ALL_DATA
+ HIGH_DIM_FUNCTION={SMAP R_0=1 A=3 B=2}
+ LOW_DIM_FUNCTION={SMAP R_0=1 A=1 B=2}
+ OUTPUT_FILE=smap-out
+ EMBEDDING_OFILE=smap-embed
+ ... SKETCHMAP
+ +
+ The parameters required and the functionality is detailed below: +
    +
  • The NLOW_DIM allows user to specify the dimension to which the projection is done
  • +
  • The USE_ALL_DATA allows user to perform sketchmap optimization on the complete dataset. This can be replaced with Landmark Selection to do optimization on only the selected landmark points.
  • +
  • The SMAP is specified to use SMACOF algorithm for sketch-map optimization.
  • +
  • The R_0, A and B specify the sigma value and the parameters of sigmoid function respectively.
  • +
+ +
+
+ + + +
+
+ +

A detailed description of the process

+ + In order to better understand the inner workings of the minimization, we will + discuss in some length how to manually perform the steps which are automated + by the sketch-map script. + + Linear MDS embedding of 12-ala data, colored based on one of the dihedrals + +

Getting a starting map: running linear MDS

+

+ Sketch-map owes its flexibility to the fact of using an iterative minimization + of a strongly non-linear fitness function. The downside is that it is exposed + to the risk of getting stuck in a poor local minimum, which doesn't preserve + as well as possible the proximity relations found in the high-dimensional data. + In order to generate a reasonable starting configuration for the low-dimensional + projections, we perform linear MDS by running dimred + without any iterative minimization parameters. +

+

+ Images show the two-dimensional embedding at different stages of + the optimization (here after the initial linear MDS projection). + Dots have areas proportional to the weight of the corresponding landmark, and + are colored according to the value of one of the dihedrals + (so points of different colors should be projected far apart). +

+
+
+ running linear MDS [+-] +
+
+

+ You can get a linear MDS (PCA if the distance is Euclidean with no periodicity) + by just running dimred without iterative minimization options +

+
+ grep -v \# highd-landmarks | dimred -D 24 -pi 6.283185 -d 2 -w -center > lowd-mds +
+

+ Note that dimred expects on standard input the sequence of the + landmarks given as +

+
+ CLASSICAL_MDS ...
+ ATOMS=1-256
+ METRIC=OPTIMAL-FAST
+ USE_ALL_DATA
+ NLOW_DIM=2
+ OUTPUT_FILE=rmsd-embed
+ ... CLASSICAL_MDS
+ +
+

+ In this case we specify the -w option, so the weights must be present + as the last entry on each line. Note that for MDS weights are ignored, but we keep them in the file + as will be used for later steps. Comments lines which may be present in the landmark file should be + deleted or filtered out as we are doing here.
+ The output will contain a just a list of the low-dimensional projections. If options are specified + which make the output more verbose, a header or some additional columns may be present as well. +

+
+
+ + + Linear MDS embedding of 12-ala data, colored based on one of the dihedrals +

Iterative metric MDS and pointwise-global optimization

+

+ Conventional MDS is not really trying to match pairwise distances in high dimension to low-dimension, + minimizing --write here chi2_id--. It is trying to do so with the constraint of the embedding + being a linear projection. This additional assumption makes the problem solvable with spectral + methods, but means that the projected distances won't match the original ones as well as they could. + On the other hand, iterative minimization of the stress function is prone to getting stuck in local + minima. To partially circumvent this problem, dimred allows one to + perform a pointwise-global optimization. This means that one the program tries to find the + global minimum of the stress for one point at a time, keeping the others fixed, and performing + local conjugate-gradient steps whenever one projection is moved from its initial location. + This global optimization is performed by grid search: the stress is computed on a coarse grid as + a function of the coordinates of the projected point. Then an interpolation is built on a finer + grid, and one sees if there exists a node in the fine grid with lower stress than the current one. +

+
+
+ running metric MDS (with pointwise global optimization) [+-] +
+
+

+ We start from the embedding obtained by linear MDS and perform a metric MDS optimization + with pointwise-global steps by launching +

+
+ grep -v \# highd-landmarks | dimred -D 24 -pi 6.283185 -d 2 -w -center -init lowd-mds -grid 10.0,21,201 -preopt 50 -gopt 3 > lowd-gmds-metric +
+

+ Here -grid gw,g1,g2 specifies the extent of the grid (which runs from -gw to gw in all directions), + the number of points in the coarse grid g1 and the number of points in the fine grid g2. -preopt + indicates the number of conjugate-gradient steps to be performed before starting the pointwise-global procedure + and -gopt the number of CG steps after each successful global move. + -init specify a file with the initial embedding positions, which should just contain the + d-dimensional embedding coordinates, one point per line. +

+
+
+ + +

+ While the resulting embedding is still far from perfect, one can see how points with similar dihedrals start + clustering together, even though there is still considerable overlap and regions where all the values are + mixed up. +

+ + Simulated annealing optimization of sketch-map: varying from metric MDS to sketchmap as imig goes to zero +

Finally - sketch-map

+

+ In simpler cases, it is now possible to just run again dimred, starting from the + metric MDS embedding, and perform sketch-map with the chosen parameters. In complex cases such as this one, where + the sketch-map embedding is very different (and much more informative!) than the metric MDS one, doing + sketch-map optimization directly may lead to a poor optimization -- just because the very bad initial configuration. + In this case, it can prove very effective to perform a simulated annealing procedure in which + the stress function of a metric MDS and that of sketch-map with the chosen parameters are mixed together, + and successive optimizations are performed with a decreasing fraction of the metric MDS stress. +

+ +
+
+ getting an optimized sketch-map embedding (simulated annealing) [+-] +
+
+

+ In order to perform sketch-map, we must specify the sketch-map function parameters + σ, a, b in both high and low dimension, with the options -fun-hd sigma,a,b + and -fun-ld sigma,a,b: +

+
+ grep --color=auto -v \# highd-landmarks | dimred -D 24 -pi 6.283185 -d 2 -w -center -init lowd-gmds-metric -grid 15.0,21,201 -fun-hd 6,12,12 -fun-ld 6,1,2 -imix 0.1 -preopt 50 -gopt 3 > lowd-gmds-smap_0.1 +
+

+ -imix frac specifies that a fraction frac of metric MDS stress will be + mixed with (1-frac) sketch-map stress. Ideally, one should proceed iteratively, running with + say frac=0.1, and using the outcome embedding as the starting configuration from an optimization + with frac=0.01 and so on until the embedding does not change anymore. +

+
+
+ + +
diff --git a/pages/plumed_tuts/summary.html b/pages/plumed_tuts/summary.html new file mode 100644 index 00000000..0e4f699b --- /dev/null +++ b/pages/plumed_tuts/summary.html @@ -0,0 +1,41 @@ +
+ Dodecaalanine in a alpha-helix configuration +

A sketch-map tutorial

+
+ This tutorials aims at being a "walk-through" to help you apply sketch map to your + high-dimensional data. We will use as an example a quite challenging case, namely + the mapping in two dimensions of a set of configurations of the model polypeptide dodeca-alanine + which are represented by the 24 backbone dihedral angles.
+ Before you start, you should download and compile the sketch-map tools. + You can then either experiment with the data file for + dodeca-alanine, or prepare your own data set in a file formatted with one sample per line, + e.g +
+ x1(1) x2(1) .... xD(1)
+ x1(2) x2(2) .... xD(2)
+ .... +
+
+

+ As with most of non-linear dimensionality reduction techniques, sketch-map works + by finding a set of low-dimensional points which reproduce the pair-wise similarity + between the samples of the high-dimensional data set. Whereas other NLDR methods + try to match some sort of distance between the points, sketch-map focuses on + reproducing proximity, i.e. making sure that points that are "close" in high dimension + will be "close" in low dimension, and those which are "far" in high dimension will + be "far" in high dimension. Finding a viable definition of proximity requires + some preliminary data analysis, which allows one to determine optiimal parameters + for running sketch-map. +

+

+ The tutorial covers all the steps necessary to obtain a satisfactory low-dimensional + embedding, including: +

+
    +
  • data analysis and selection of sketch-map parameters
  • +
  • selection of landmark points
  • +
  • the actual sketch-map optimization
  • +
  • out-of-sample embedding of the remaining configurations
  • +
+ +
diff --git a/pages/tuts/landmarks.html b/pages/tuts/landmarks.html index 3ad1772c..bcc17ea6 100644 --- a/pages/tuts/landmarks.html +++ b/pages/tuts/landmarks.html @@ -22,7 +22,6 @@

Selecting landmark points

space, dimlandmark also allows to assign weights proportional to the density of samples surrounding each sample.

-
running dimlandmark [+-] @@ -45,6 +44,100 @@

Selecting landmark points

- + +

+ Both of these strategies can also be run in PLUMED +

+ + +
+
+ using PLUMED for Random Selection and FPS [+-] +
+
+

+ The SelectRandomFrames allows user to select a small, representative subset of the high-dimensional data randomly, so that dimensionality reduction can be performed more effectively. USE_ALL_DATA can be replaced with RUN=ii (a number) to use datapoints only after ii steps while selecting randomly from the data. +

+
+ d: READ FILE=highd-dataset.in VALUES=c.*
+ OUTPUT_LANDMARKS ...
+ ARG=d.*
+ USE_ALL_DATA
+ METRIC=EUCLIDEAN
+ LANDMARKS={RANDOM N=300 SEED=-3241 }
+ LIST_FILE=landmark-random.out
+ ... OUTPUT_LANDMARKS
+
+

+ METRIC options specify the distance function employed, RANDOM keyword is used to specify the Random Sampling method to be employed and N is used for the number of landmarks to be selected. LIST_FILE is used to specify the output file where the landmarks are generated. SEED specifies the seed for the random number generator. It should be given different values for different runs. You can download the example file, the plumed input file and the script to prepare the input file as a tarball. +

+ +

+ The FarthestPointSampling allows user to select a small, representative subset of the high-dimensional data using Farthest Point Sampling technique. All the parameters remain same as defined in Random Sampling method except the key word FPS is used instead of RANDOM +

+
+ d: READ FILE=highd-dataset.in VALUES=c.*
+ OUTPUT_LANDMARKS ...
+ ARG=d.*
+ USE_ALL_DATA
+ METRIC=EUCLIDEAN
+ LANDMARKS={FPS N=300 SEED=1 }
+ LIST_FILE=landmark-fps.out
+ ... OUTPUT_LANDMARKS
+
+ +
+
+ +

+ Apart from the above mentioned strategies, there are some more strategies which can be used to select the landmarks when running the sketchmap algorithm with PLUMED. +

+
    +
  • Select points using stride - the landmarks are selected after skipping uniform number of frames between two selections. The algorithm calculates the uniform stride length with the number of required landmarks and the total data.
  • +
  • Select points using a staged sampling from the trajectory - the landmarks are selected in two stages. First a relatively larger number of intermediate landmarks are selected, using farthest point sampling. After this these landmarks are used to calculate voronoi weights and random sampling is done based on the weights in each voronoi region and final selection is made.
  • +
+ + + +
+
+ using PLUMED for Selection with Stride and Staged Sampling [+-] +
+
+

+ The Stride Sampling allows user to select a representative subset of the high-dimensional data using strides of predefined constant step length. +

+
+ d: READ FILE=highd-dataset.in VALUES=c.*
+ OUTPUT_LANDMARKS ...
+ ARG=d.*
+ USE_ALL_DATA
+ METRIC=EUCLIDEAN
+ LANDMARKS={STRIDE N=300 }
+ LIST_FILE=landmark-stride.out
+ ... OUTPUT_LANDMARKS
+
+

+ +

+

+ The StagedSampling allows user to select a representative subset of the high-dimensional data using Staged Sampling technique. All the parameters remain same as defined in Random Sampling method except the key word STAGED is used and another parameter GAMMA is specified which determines TODO. +

+
+ d: READ FILE=highd-dataset.in VALUES=c.*
+ OUTPUT_LANDMARKS ...
+ ARG=d.*
+ USE_ALL_DATA
+ METRIC=EUCLIDEAN
+ LANDMARKS={STAGED N=300 SEED=1 GAMMA=0.5}
+ LIST_FILE=landmark-staged.out
+ ... OUTPUT_LANDMARKS
+
+

+ +

+
+
+ diff --git a/pages/tuts/sketch-map.html b/pages/tuts/sketch-map.html index 293e81a8..f060d193 100644 --- a/pages/tuts/sketch-map.html +++ b/pages/tuts/sketch-map.html @@ -47,12 +47,46 @@

Running sketch-map

+ +

+ The sketchmap can also be run under PLUMED framework. The following code explains the functionality available in the current implementation of the sketchmap algorithm in PLUMED. +

+
+
+ running Sketchmap in PLUMED [+-] +
+
+

+ Following code uses all dataset to run the sketch-map optimization which is decsribed in detail below. +

+
+ d: READ FILE=input.in VALUES=c.*
+ SKETCHMAP ...
+ ARG=d.*
+ NLOW_DIM=2
+ USE_ALL_DATA
+ HIGH_DIM_FUNCTION={SMAP R_0=1 A=3 B=2}
+ LOW_DIM_FUNCTION={SMAP R_0=1 A=1 B=2}
+ OUTPUT_FILE=smap-out
+ EMBEDDING_OFILE=smap-embed
+ ... SKETCHMAP
+ +
+ The parameters required and the functionality is detailed below: +
    +
  • The NLOW_DIM allows user to specify the dimension to which the projection is done
  • +
  • The USE_ALL_DATA allows user to perform sketchmap optimization on the complete dataset. This can be replaced with Landmark Selection to do optimization on only the selected landmark points.
  • +
  • The SMAP is specified to use SMACOF algorithm for sketch-map optimization.
  • +
  • The R_0, A and B specify the sigma value and the parameters of sigmoid function respectively.
  • +
+ +
+
+
-
-
-
+

A detailed description of the process

@@ -110,8 +144,9 @@

Getting a starting map: running linear MDS

- - + + + Linear MDS embedding of 12-ala data, colored based on one of the dihedrals

Iterative metric MDS and pointwise-global optimization

diff --git a/sitemap.xml b/sitemap.xml index 4b9adacb..2f66fec6 100644 --- a/sitemap.xml +++ b/sitemap.xml @@ -13,6 +13,11 @@ http://epfl-cosmo.github.io/sketchmap/index.htm?page=tuts weekly 0.5000 + + + http://epfl-cosmo.github.io/sketchmap/index.htm?page=plumed_tuts + weekly + 0.5000 http://epfl-cosmo.github.io/sketchmap/index.htm?page=manu diff --git a/style.css b/style.css index d040dbc4..67bca531 100644 --- a/style.css +++ b/style.css @@ -130,5 +130,27 @@ div.code { padding:1em; font-family:"Courier"; font-weight:bold; background: #ffe4be; } +/* for plumed codes */ +div.pcbox { + padding:0; margin:2em; background: #fff5f1; clear:both; display:block +} + +div.pcbox .head { + background: #FFAC8A; font-weight:bold; + padding:5px; margin:0; cursor:pointer; + } + +div.pcbox .body { + padding:0.5em; margin:0; + } + +/* links & pc */ +a img { border:0 } + +.pcode {font-family:"Courier"; font-weight:bold;} +div.pcode { + padding:1em; font-family:"Courier"; font-weight:bold; + background: #ffe4be; +} From f92dbd6b99f8173a517aad7c3b50f6817df8b4ec Mon Sep 17 00:00:00 2001 From: Anurag Sharma Date: Wed, 24 Jun 2015 13:44:42 +0200 Subject: [PATCH 2/2] Removed duplicate files --- pages/plumed_tuts/ala12_helix.jpg | Bin 9644 -> 0 bytes pages/plumed_tuts/ala_gmdsplot.png | Bin 83566 -> 0 bytes pages/plumed_tuts/ala_histosigma.png | Bin 15349 -> 0 bytes pages/plumed_tuts/ala_histosolo.png | Bin 12024 -> 0 bytes pages/plumed_tuts/ala_landmark.png | Bin 84660 -> 0 bytes pages/plumed_tuts/ala_mdsplot.png | Bin 60629 -> 0 bytes pages/plumed_tuts/ala_phipsi.png | Bin 71659 -> 0 bytes pages/plumed_tuts/ala_smap_angles.gif | Bin 464929 -> 0 bytes pages/plumed_tuts/ala_smap_ani.gif | Bin 233959 -> 0 bytes pages/plumed_tuts/ala_smap_combo.gif | Bin 612542 -> 0 bytes pages/plumed_tuts/ala_smap_proj.png | Bin 176810 -> 0 bytes pages/plumed_tuts/analysis.html | 140 ------------------- pages/plumed_tuts/gnuplot.html | 47 ------- pages/plumed_tuts/gp_land.png | Bin 160807 -> 0 bytes pages/plumed_tuts/gp_proj.png | Bin 308749 -> 0 bytes pages/plumed_tuts/landmarks.html | 132 ------------------ pages/plumed_tuts/projecting.html | 50 ------- pages/plumed_tuts/results.html | 52 ------- pages/plumed_tuts/sketch-map.html | 194 -------------------------- pages/plumed_tuts/summary.html | 41 ------ 20 files changed, 656 deletions(-) delete mode 100644 pages/plumed_tuts/ala12_helix.jpg delete mode 100644 pages/plumed_tuts/ala_gmdsplot.png delete mode 100644 pages/plumed_tuts/ala_histosigma.png delete mode 100644 pages/plumed_tuts/ala_histosolo.png delete mode 100644 pages/plumed_tuts/ala_landmark.png delete mode 100644 pages/plumed_tuts/ala_mdsplot.png delete mode 100644 pages/plumed_tuts/ala_phipsi.png delete mode 100644 pages/plumed_tuts/ala_smap_angles.gif delete mode 100644 pages/plumed_tuts/ala_smap_ani.gif delete mode 100644 pages/plumed_tuts/ala_smap_combo.gif delete mode 100644 pages/plumed_tuts/ala_smap_proj.png delete mode 100644 pages/plumed_tuts/analysis.html 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-

Analysis and choice of parameters

- 2d slice of torsional angle data -

- Sketch-map builds on the notion that in many cases the high-dimensional data - that one is trying to represent with few parameters does actually contain - some genuinely high-dimensional features, which would interfere with the - dimensionality reduction procedure. -

-

The histogram of pairwise distances

-

- A first step is therefore to analyze the data in order to understand how - the samples are distributed in high-dimensional space. A cursory - examination of two-dimensional slices of the data set can be useful to - achieve a superficial understanding of the structure of the data. - Such a plot for the 12-alanine data shows that samples - are gathered in seemingly Gaussian blobs, connected by a spiders web - of less-populated transition states.
- To better guide the choice of parameters one should perform a more quantitative - analysis. To this aim, we can use the dimdist - utility to generate a histogram of the pairwise - distances of samples, i.e. the probability distribution for two points picked - from the sample at random to be at a given distance. -

- -
-
- Running dimdist [+-] -
-
-

- The dimdist utility can be used to compute the distribution - of pairwise distances within the high-dimensional data set. For this example, you should type -

-
- dimdist -D 24 -P highd-dataset -pi 6.283185 -maxd 15 -nbin 300 -wbin 0.05 -lowmem > highd-histo -
-

- Here the -D options specifies the dimensionality, - -P the location of the data file, -pi - indicates that in computing distances the coordinates have to be taken to have the - specified periodicity, and the other options control the range, bins and smoothening of - the histogram. See --MANUAL-- for a more detailed description of the options which may - apply to a different kind of data.
- Note that the computational expense of this analysis scales with the square of the number of - data points, so it can be quite time-consuming. It there are more than a few tens of thousands - points, it is often sufficient to use a random subset of the samples. -

-
-
- - - Histogram of pairwise distances in D=24 -

- The resulting histogram may be at first surprising. In most cases, as for the 12-ala - data, one would observe a very low probability of finding two points very close together. - This is a consequence of the high dimensionality: - taking two points from a high-D-dimensional Gaussian distribution it will be unlikely for - them to be at distances smaller than several standard deviations! As a rule of thumb, if - on all the d-dimensional slices of the original data the samples seems to have a Gaussian - distribution with standard deviation s, the distribution of distances in the full, - D-dimensional description will be peaked at s*sqrt(D/d). In the example, the 2-d plot - indicates a standard deviation of about 0.5, and the distribution shows a shoulder which - corresponds to these Gaussian correlations at about R=2.
- A second observation is that the overall distribution is likely to be strongly peaked at - rather large distances. In fact, in this case - it can be shown that the long-range part of the distribution is very similar to a uniform - distribution of samples, even if clearly the samples are not uniform at all! - This is again a consequence of the quirks of working in high dimension: the "intrinsic" volume - element grows so rapidly with distance that far-away points weight a lot in the histogram, - even though they carry very little information on the structure of the data set. -

-

Proximity in high dimension

-

- The idea behind sketch-map is to transform the distances in such a way that the (not-so-relevant) - information contained in the short-range and long-range point-point distances is not considered, - and the dimensionality reduction algorithm can focus on reproducing the intermediate distances, - which carry information on the relative position of different stable conformations rather - than on Gaussian fluctuations or on far-away relations which are dominated by the topology - of the high-dimensional space rahter than on the specific features of the problem. -

- n(r) and a superimposed s(R) -

- This is achieved by means of a sigmoid function which is applied to distances,
-\[ - s_{\sigma,a,b}(r) = 1 - (1+ (2^{a/b}-1)(r/\sigma)^a)^{-b/a} -\] - which is zero for x→0, one for x→∞ and 1/2 for x=σ. The a and be parameters - define the rate at which s goes to zero and to one, respectively. The critical parameter to select - is the switching distance σ: points which are closer than σ will be considered "close" - and points which are farther apart will be considered "far". σ should be placed somewhere - between the range of distances characteristic of Gaussian fluctuations (0-3 in this example) - and the range dominated by the intrinsic high dimensionality (8 and larger in this example). - This leaves a range of σ to be chosen in the 4-7 range. We choose here σ=6, but - values not too far from this yield comparable results.
- Once σ is chosen, one should select a and b. While the method is not very sensitive to these - values, one should make sure that the step around R=σ is not too sharp, and that at the same - time distances within the Gaussian basins are mapped to zero. b should be chosen in such a way that - s(x) approaches 1 by the time n(R) has become very small. If one observes that the final map projects - some outliers very far away from anything else, it may help to reduce the value of b in high dimension, - so that s(x) does not approach 1 too rapidly. -

- -
-
- Further examples of histogram and effective sigmoid functions [+-] -
-
-

- -- PUT HERE HISTOGRAM/CHOSEN FUNCTION PLOTS FOR THE LIGAND BINDING, THE 3D EXAMPLE, AND FURTHER - DISCUSS THE RATIONALE -- -

-
-
- - - -

Proximity in low dimension

-

- One of the key-features of sketch-map is that also in low dimensions the similarity of points - is not evaluated using Euclidean distances. Another sigmoid function is applied, so that the - close→close/far→far mapping can be enforced. The σ parameter in low dimension is - irrelevant, since using a value different from the one chosen in high dimension simply corresponds - to a rigid scaling of the low-dimensional map. The high-dimension σ should always be used.
- The a and b parameters do not need to be the same. If they are chosen to be equal to their - high-dimension counterparts, sketch-map would ultimately try to match distances, since distances - are transformed in low and high dimension in the same way. This may be desirable when the mapping - is not too complex, e.g. when one knows already that high-dimensional points lie very close to a - hyperplane.
- When this is not the case, it is useful to choose different values: the advantage is that one can - better accommodate the counter-intuitive features of high-dimensional spaces in the low-dimensional - projection. For instance, if D is the original dimension and d the dimension of the projection, - one can "equalize" the volume elements in the low and high-dimensional cases by choosing - ad/d≈aD/D. For the same reason, it is usually a good idea to choose - bd to be smaller than bD. As a rule of thumb, one or two are good values - to start experimenting for both ad and bd.
- In this example, we take ad=1 and bd=2. -

- -
diff --git a/pages/plumed_tuts/gnuplot.html b/pages/plumed_tuts/gnuplot.html deleted file mode 100644 index 13169025..00000000 --- a/pages/plumed_tuts/gnuplot.html +++ /dev/null @@ -1,47 +0,0 @@ -
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z#xH&5ep0Fr78e%4Ms_ADh#u&!L9;VYkr()#SzAshah1J-gjWImY*Ftd*8U&|n|Zn~ zSQEx;TO#GWca~W4E3;|n9S|>bWFpE+0h3`ul&ejd~js)E=heo7^gnuSpz}Bz48Z4Ik~8=MrZZJBV;G_ zt#(=_YN6qP3Y+b$Z-}esgfyPgK0_1dey) z?g8~n^<8b}CgJSLpx?boJshQUVF}2YWOa=xtg^Cnha#F%M6zODMjY&`JR%)%3l?kZ zFA0$qLk>et+9UX7^t1wBb5bWabevMQ7pqxSuudnr-7#6U`@IxEH?NrUb(&@V^Aw|- z&@eN3iry#qVeUA_et&C7JHlKCafl-ilTqs?a#Xb5xtL(Gz!Bq-V$VBhp(q9mY*lrY z4Sx3wokQS2K{DxJ4~=w{-{aAn{NVsyY;`_9`=d&;45nJzj^CySGmL;3^>LT?v&yA9 z!P!~#&vCIF@Y1`eAR0+_qB2+0 z&hImVnTn!C(O%wh#K>cT^1$t;2BqGsggK4!ZO?s1xPc(t>lJxu9)EECksPKtYLW{- zsXSf~{d5l4Ueowd&qtb0xR{r$TEGgF ziRU=!mgdB~K4#rQBnu>Ha`*VA1$4$3>RGiZ9nRkf35LD*D!j@|>`q5mZi8B1Ogp-#yvkQmkoeMch zBIs%E^7_X-3~_v<#6`XE+ - - 100 minmax landmarks picked from 12ala data. Circles area is proportional to weight. -

Selecting landmark points with PLUMED

-

- The sketch-map calculation is expensive and its expense increases sharply with the number of points whose projections you are - endeavoring to find simultaneously. Moreover, the iterative minimization has more chances of getting stuck in a poor local minimum - when the number of variables to be optimized grows larger. It is therefore not feasible to fit projections for all the points in the trajectory - at once. Hence, before running the sketch-map algorithm we must select a set of representative, landmark, points from the trajectory. - There are following strategies we use to do this selection: -

- -
- - -

- All of these strategies are implemented in the SelectRandomFrames, FarthestPointSampling, SelectWithStride code and StagedSampling. -

- The input file for the following routines can be generated using the script provided [here] - - diff --git a/pages/plumed_tuts/projecting.html b/pages/plumed_tuts/projecting.html deleted file mode 100644 index 40b9cc8f..00000000 --- a/pages/plumed_tuts/projecting.html +++ /dev/null @@ -1,50 +0,0 @@ -
- -

Out-of-sample embedding

- -

- After the low-d embedding of the landmark points has been - worked out, one can use it to define a map which allows to - project additional points. In other terms, a set of sketch-map - collective coordinates can be computed for any configuration - of the system. This final step is delegated to the - dimred utility. -

- -
-
- running dimproj [+-] -
-
-

- We need first to prepare two files with the landmarks, one containing - the high-dimensional coordinates (plus weights) and another containing the sketch-mapped - ones, without headers or extra columns. Then we run -

-
- dimproj -D 24 -d 2 -P highd-landmarks -p lowd-landmarks -pi 6.283185 -w - -grid 15.0,21,201 -fun-hd 6,12,12 -fun-ld 6,1,2 -cgmin 3 < highd-new > lowd-new -
-

- Here most of the options correspond to those in the sketch-map step, - given the close analogy between the sketch-map stress function and the - one which is used for the out-of-sample embedding. After having performed - a grid search based on the parameters provided, a few conjugate gradient - steps are performed (-cgmin). -

-
-
- - - -

- The image below shows the embedding of the circa 50000 configurations of - the trajectory, colored as usual based on just one of the backbone dihedral. - Despite the fact that the landmarks are a very coarse sampling of the - configuration space, the out-of-sample sketch-map embedding separates nicely - between different structures, and shows a remarkably rich picture of the - many metastable states accessible to this simple model polypeptide. -

- Linear MDS embedding of 12-ala data, colored based on one of the dihedrals - -
diff --git a/pages/plumed_tuts/results.html b/pages/plumed_tuts/results.html deleted file mode 100644 index 4d225d4f..00000000 --- a/pages/plumed_tuts/results.html +++ /dev/null @@ -1,52 +0,0 @@ -
- - Sketch-map embedding of the 12-ala colored according to the different dihedrals -

Analyzing the results

-

- Unless you are working for a model system for which you know already which kind of embedding - to expect, it is not straightforward to judge the success of the fitting procedure. There are - many ways to post-process the projected points to get some qualitative -- and quantitative -- - assessment of the outcome of the fit.
- A first possibility is to color the projected points according to different coordinates of the - corresponding high-dimensional samples: if the fitting was successful one would expect points - of similar color to be clustered together, rather than mixed up with points of different color. -

-

- Another, more quantitative, analysis can be performed using the dimdist - utility. With the appropriate parameters, dimdist generates a D-d plot, - i.e. an histogram which measures the probability that, for any high-dimensional pairwise distance D, - the distance between the corresponding projections will be d. A completely faithful embedding would - preserve the distances, so the D-d histogram should be a series of delta function along the D=d diagonal. - Having performed sketch-map, instead, one expects would already be content when the majority of the - density is concentrated in the D<σ,d<σ and D>σ,d>σ regions. -

- -
-
- computing D-d plots [+-] -
-
- D-d plots can be generated using the dimdist utility. - Note that both the high-dim and low-dim data set must be provided, in files containing - one point per line, without headers. Optionally, the high-dimension file may contain - weights (-w). -
- dimdist -D 24 -d 2 -w -P highd_landmarks_noheader -p lowd_projection -pi 6.283185 -lowmem - -gnuplot -maxd 15 50 -nbin 150 250 -wbin 0.2 > ddplot.histo -
- Dd-plot and the corresponding embedding for different mixing parameters -

- Mixed metric MDS/sketch-map embedding, with mixing parameter going from pure MDS to pure sketch-map. - The embedding (coloured according to one of the dihedrals) is shown side by side with the corresponding - D-d plot. Note how metric MDS tends to have high density in the D>σ,d<σ. This means that - "far" points are mapped to "close" points in low-dimension: this is detrimental if the mapping is to - be used to classify configurations, to compute free energies or perform enhanced sampling. - Sketch-map, on the other hand, manages to keep most of the density in the "good" regions: even though - the structure of the data set is deformed, most of the topological relations that matter - have been preserved. -

-
-
- - -
diff --git a/pages/plumed_tuts/sketch-map.html b/pages/plumed_tuts/sketch-map.html deleted file mode 100644 index d397fdaa..00000000 --- a/pages/plumed_tuts/sketch-map.html +++ /dev/null @@ -1,194 +0,0 @@ -
- -

Running sketch-map

- -

- Having determined some reasonable parameters which define - "proximity" for the data set we want to project, we are now - ready to run sketch-map. The minimization procedure can be quite - time-consuming, in particular if one is using many landmarks - and/or very high dimensionality. -

- -
-
- running Sketchmap [+-] -
-
-

- Following code uses all dataset to run the sketch-map optimization which is decsribed in detail below. -

-
- d: READ FILE=input.in VALUES=c.*
- SKETCHMAP ...
- ARG=d.*
- NLOW_DIM=2
- USE_ALL_DATA
- HIGH_DIM_FUNCTION={SMAP R_0=1 A=3 B=2}
- LOW_DIM_FUNCTION={SMAP R_0=1 A=1 B=2}
- OUTPUT_FILE=smap-out
- EMBEDDING_OFILE=smap-embed
- ... SKETCHMAP
- -
- The parameters required and the functionality is detailed below: -
    -
  • The NLOW_DIM allows user to specify the dimension to which the projection is done
  • -
  • The USE_ALL_DATA allows user to perform sketchmap optimization on the complete dataset. This can be replaced with Landmark Selection to do optimization on only the selected landmark points.
  • -
  • The SMAP is specified to use SMACOF algorithm for sketch-map optimization.
  • -
  • The R_0, A and B specify the sigma value and the parameters of sigmoid function respectively.
  • -
- -
-
- - - -
-
- -

A detailed description of the process

- - In order to better understand the inner workings of the minimization, we will - discuss in some length how to manually perform the steps which are automated - by the sketch-map script. - - Linear MDS embedding of 12-ala data, colored based on one of the dihedrals - -

Getting a starting map: running linear MDS

-

- Sketch-map owes its flexibility to the fact of using an iterative minimization - of a strongly non-linear fitness function. The downside is that it is exposed - to the risk of getting stuck in a poor local minimum, which doesn't preserve - as well as possible the proximity relations found in the high-dimensional data. - In order to generate a reasonable starting configuration for the low-dimensional - projections, we perform linear MDS by running dimred - without any iterative minimization parameters. -

-

- Images show the two-dimensional embedding at different stages of - the optimization (here after the initial linear MDS projection). - Dots have areas proportional to the weight of the corresponding landmark, and - are colored according to the value of one of the dihedrals - (so points of different colors should be projected far apart). -

-
-
- running linear MDS [+-] -
-
-

- You can get a linear MDS (PCA if the distance is Euclidean with no periodicity) - by just running dimred without iterative minimization options -

-
- grep -v \# highd-landmarks | dimred -D 24 -pi 6.283185 -d 2 -w -center > lowd-mds -
-

- Note that dimred expects on standard input the sequence of the - landmarks given as -

-
- CLASSICAL_MDS ...
- ATOMS=1-256
- METRIC=OPTIMAL-FAST
- USE_ALL_DATA
- NLOW_DIM=2
- OUTPUT_FILE=rmsd-embed
- ... CLASSICAL_MDS
- -
-

- In this case we specify the -w option, so the weights must be present - as the last entry on each line. Note that for MDS weights are ignored, but we keep them in the file - as will be used for later steps. Comments lines which may be present in the landmark file should be - deleted or filtered out as we are doing here.
- The output will contain a just a list of the low-dimensional projections. If options are specified - which make the output more verbose, a header or some additional columns may be present as well. -

-
-
- - - Linear MDS embedding of 12-ala data, colored based on one of the dihedrals -

Iterative metric MDS and pointwise-global optimization

-

- Conventional MDS is not really trying to match pairwise distances in high dimension to low-dimension, - minimizing --write here chi2_id--. It is trying to do so with the constraint of the embedding - being a linear projection. This additional assumption makes the problem solvable with spectral - methods, but means that the projected distances won't match the original ones as well as they could. - On the other hand, iterative minimization of the stress function is prone to getting stuck in local - minima. To partially circumvent this problem, dimred allows one to - perform a pointwise-global optimization. This means that one the program tries to find the - global minimum of the stress for one point at a time, keeping the others fixed, and performing - local conjugate-gradient steps whenever one projection is moved from its initial location. - This global optimization is performed by grid search: the stress is computed on a coarse grid as - a function of the coordinates of the projected point. Then an interpolation is built on a finer - grid, and one sees if there exists a node in the fine grid with lower stress than the current one. -

-
-
- running metric MDS (with pointwise global optimization) [+-] -
-
-

- We start from the embedding obtained by linear MDS and perform a metric MDS optimization - with pointwise-global steps by launching -

-
- grep -v \# highd-landmarks | dimred -D 24 -pi 6.283185 -d 2 -w -center -init lowd-mds -grid 10.0,21,201 -preopt 50 -gopt 3 > lowd-gmds-metric -
-

- Here -grid gw,g1,g2 specifies the extent of the grid (which runs from -gw to gw in all directions), - the number of points in the coarse grid g1 and the number of points in the fine grid g2. -preopt - indicates the number of conjugate-gradient steps to be performed before starting the pointwise-global procedure - and -gopt the number of CG steps after each successful global move. - -init specify a file with the initial embedding positions, which should just contain the - d-dimensional embedding coordinates, one point per line. -

-
-
- - -

- While the resulting embedding is still far from perfect, one can see how points with similar dihedrals start - clustering together, even though there is still considerable overlap and regions where all the values are - mixed up. -

- - Simulated annealing optimization of sketch-map: varying from metric MDS to sketchmap as imig goes to zero -

Finally - sketch-map

-

- In simpler cases, it is now possible to just run again dimred, starting from the - metric MDS embedding, and perform sketch-map with the chosen parameters. In complex cases such as this one, where - the sketch-map embedding is very different (and much more informative!) than the metric MDS one, doing - sketch-map optimization directly may lead to a poor optimization -- just because the very bad initial configuration. - In this case, it can prove very effective to perform a simulated annealing procedure in which - the stress function of a metric MDS and that of sketch-map with the chosen parameters are mixed together, - and successive optimizations are performed with a decreasing fraction of the metric MDS stress. -

- -
-
- getting an optimized sketch-map embedding (simulated annealing) [+-] -
-
-

- In order to perform sketch-map, we must specify the sketch-map function parameters - σ, a, b in both high and low dimension, with the options -fun-hd sigma,a,b - and -fun-ld sigma,a,b: -

-
- grep --color=auto -v \# highd-landmarks | dimred -D 24 -pi 6.283185 -d 2 -w -center -init lowd-gmds-metric -grid 15.0,21,201 -fun-hd 6,12,12 -fun-ld 6,1,2 -imix 0.1 -preopt 50 -gopt 3 > lowd-gmds-smap_0.1 -
-

- -imix frac specifies that a fraction frac of metric MDS stress will be - mixed with (1-frac) sketch-map stress. Ideally, one should proceed iteratively, running with - say frac=0.1, and using the outcome embedding as the starting configuration from an optimization - with frac=0.01 and so on until the embedding does not change anymore. -

-
-
- - -
diff --git a/pages/plumed_tuts/summary.html b/pages/plumed_tuts/summary.html deleted file mode 100644 index 0e4f699b..00000000 --- a/pages/plumed_tuts/summary.html +++ /dev/null @@ -1,41 +0,0 @@ -
- Dodecaalanine in a alpha-helix configuration -

A sketch-map tutorial

-
- This tutorials aims at being a "walk-through" to help you apply sketch map to your - high-dimensional data. We will use as an example a quite challenging case, namely - the mapping in two dimensions of a set of configurations of the model polypeptide dodeca-alanine - which are represented by the 24 backbone dihedral angles.
- Before you start, you should download and compile the sketch-map tools. - You can then either experiment with the data file for - dodeca-alanine, or prepare your own data set in a file formatted with one sample per line, - e.g -
- x1(1) x2(1) .... xD(1)
- x1(2) x2(2) .... xD(2)
- .... -
-
-

- As with most of non-linear dimensionality reduction techniques, sketch-map works - by finding a set of low-dimensional points which reproduce the pair-wise similarity - between the samples of the high-dimensional data set. Whereas other NLDR methods - try to match some sort of distance between the points, sketch-map focuses on - reproducing proximity, i.e. making sure that points that are "close" in high dimension - will be "close" in low dimension, and those which are "far" in high dimension will - be "far" in high dimension. Finding a viable definition of proximity requires - some preliminary data analysis, which allows one to determine optiimal parameters - for running sketch-map. -

-

- The tutorial covers all the steps necessary to obtain a satisfactory low-dimensional - embedding, including: -

-
    -
  • data analysis and selection of sketch-map parameters
  • -
  • selection of landmark points
  • -
  • the actual sketch-map optimization
  • -
  • out-of-sample embedding of the remaining configurations
  • -
- -
- -

Gnuplot wisdom

- -

- The figures in this tutorial were produced using Mathematica. - However, very similar results can be obtained with a recent version - of gnuplot.
- Here, a few tips are listed which can be helpful to get the most out - of gnuplot in order to obtain insightful and visually compelling - representations of the low-dimensional embedding. -

- -

- A first suggestion is to use lt palette to - color the points according to the value of individual high-dimensional - descriptors: this can help verifying that the proximity information - has been embedded properly - with the caveat that if uninteresting CVs - had been included in the high-D set, there may be very little correlation - between their values and the position in the embedding. -

-
- gnuplot> set palette functions (2*cos(gray*pi+pi/3)**4+cos(gray*pi+pi*0.66)**2)/3, 0.5*cos(gray*pi+pi*0.55)**8, (cos(gray*pi-pi/6)**4*2+cos(gray*pi+pi*0.66)**8)/3;
- gnuplot> set cbrange [-pi:pi]
- gnuplot> plot [-100:100][-100:100] "< paste lowd-proj highd-dataset" u 1:2:15 w p pt 7 ps 0.1 lt pal
-
- sketchmap embedding of 12-ala data, colored based on one of the dihedrals - -

- Note that in cases such as this where the high-dimensional points have periodic nature it is important - to define a palette which is periodic as well. Also, note how the different data set can be combined - from within gnuplot by using the "< command" construct. -

- -

- When one is representing just the embedding of the landmark points, - and these have a variable weight factor, it is useful to represent such - weight to avoid being distracted by the location of landmarks of negligible - weight. To do so, one can use ps variable, and read - in the weights by a using x:y:size:color clause. -

-
- gnuplot> p [-100:100][-100:100] "<paste lowd-gmds-smap_0 highd-landmarks" u 1:2:(sqrt($27)*0.2):16 w p pt 7 lt pal ps var -
- sketchmap embedding of 12-ala landmarks, colored based on one of the dihedrals - -
+ +

Gnuplot wisdom

+ +

+ The figures in this tutorial were produced using Mathematica. + However, very similar results can be obtained with a recent version + of gnuplot.
+ Here, a few tips are listed which can be helpful to get the most out + of gnuplot in order to obtain insightful and visually compelling + representations of the low-dimensional embedding. +

+ +

+ A first suggestion is to use lt palette to + color the points according to the value of individual high-dimensional + descriptors: this can help verifying that the proximity information + has been embedded properly - with the caveat that if uninteresting CVs + had been included in the high-D set, there may be very little correlation + between their values and the position in the embedding. +

+
+ gnuplot> set palette functions (2*cos(gray*pi+pi/3)**4+cos(gray*pi+pi*0.66)**2)/3, 0.5*cos(gray*pi+pi*0.55)**8, (cos(gray*pi-pi/6)**4*2+cos(gray*pi+pi*0.66)**8)/3;
+ gnuplot> set cbrange [-pi:pi]
+ gnuplot> plot [-100:100][-100:100] "< paste lowd-proj highd-dataset" u 1:2:15 w p pt 7 ps 0.1 lt pal
+
+ sketchmap embedding of 12-ala data, colored based on one of the dihedrals + +

+ Note that in cases such as this where the high-dimensional points have periodic nature it is important + to define a palette which is periodic as well. Also, note how the different data set can be combined + from within gnuplot by using the "< command" construct. +

+ +

+ When one is representing just the embedding of the landmark points, + and these have a variable weight factor, it is useful to represent such + weight to avoid being distracted by the location of landmarks of negligible + weight. To do so, one can use ps variable, and read + in the weights by a using x:y:size:color clause. +

+
+ gnuplot> p [-100:100][-100:100] "<paste lowd-gmds-smap_0 highd-landmarks" u 1:2:(sqrt($27)*0.2):16 w p pt 7 lt pal ps var +
+ sketchmap embedding of 12-ala landmarks, colored based on one of the dihedrals + +