[edit]

Life, the Universe and Machine Learning

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\newcommand{\inlineDiff}[2]{\text{d}#1/\text{d}#2} \newcommand{\inputDim}{q} \newcommand{\inputMatrix}{\mathbf{X}} \newcommand{\inputScalar}{x} \newcommand{\inputSpace}{\mathcal{X}} \newcommand{\inputVals}{\inputVector} \newcommand{\inputVector}{\mathbf{ \inputScalar}} \newcommand{\iterNum}{k} \newcommand{\kernel}{\kernelScalar} \newcommand{\kernelMatrix}{\mathbf{K}} \newcommand{\kernelScalar}{k} \newcommand{\kernelVector}{\mathbf{ \kernelScalar}} \newcommand{\kff}{\kernelScalar_{\mappingFunction \mappingFunction}} \newcommand{\kfu}{\kernelVector_{\mappingFunction \inducingScalar}} \newcommand{\kuf}{\kernelVector_{\inducingScalar \mappingFunction}} \newcommand{\kuu}{\kernelVector_{\inducingScalar \inducingScalar}} \newcommand{\lagrangeMultiplier}{\lambda} \newcommand{\lagrangeMultiplierMatrix}{\boldsymbol{ \Lambda}} \newcommand{\lagrangian}{L} \newcommand{\laplacianFactor}{\mathbf{ \MakeUppercase{\laplacianFactorScalar}}} \newcommand{\laplacianFactorScalar}{m} 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\newcommand{\naturalFrequency}{\omega} \newcommand{\neighborhood}[1]{\mathcal{N}\left( #1 \right)} \newcommand{\neilurl}{http://inverseprobability.com/} \newcommand{\noiseMatrix}{\boldsymbol{ E}} \newcommand{\noiseScalar}{\epsilon} \newcommand{\noiseVector}{\boldsymbol{ \epsilon}} \newcommand{\norm}[1]{\left\Vert #1 \right\Vert} \newcommand{\normalizedLaplacianMatrix}{\hat{\mathbf{L}}} \newcommand{\normalizedLaplacianScalar}{\hat{\ell}} \newcommand{\normalizedLaplacianVector}{\hat{\mathbf{ \ell}}} \newcommand{\numActive}{m} \newcommand{\numBasisFunc}{m} \newcommand{\numComponents}{m} \newcommand{\numComps}{K} \newcommand{\numData}{n} \newcommand{\numFeatures}{K} \newcommand{\numHidden}{h} \newcommand{\numInducing}{m} \newcommand{\numLayers}{\ell} \newcommand{\numNeighbors}{K} \newcommand{\numSequences}{s} \newcommand{\numSuccess}{s} \newcommand{\numTasks}{m} \newcommand{\numTime}{T} \newcommand{\numTrials}{S} \newcommand{\outputIndex}{j} \newcommand{\paramVector}{\boldsymbol{ \theta}} 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\newcommand{\scalarProduct}[2]{\left\langle{#1},{#2}\right\rangle} \newcommand{\sign}[1]{\text{sign}\left(#1\right)} \newcommand{\sigmoid}[1]{\sigma\left(#1\right)} \newcommand{\singularvalue}{\ell} \newcommand{\singularvalueMatrix}{\mathbf{L}} \newcommand{\singularvalueVector}{\mathbf{l}} \newcommand{\sorth}{\mathbf{u}} \newcommand{\spar}{\lambda} \newcommand{\trace}[1]{\text{tr}\left(#1\right)} \newcommand{\BasalRate}{B} \newcommand{\DampingCoefficient}{C} \newcommand{\DecayRate}{D} \newcommand{\Displacement}{X} \newcommand{\LatentForce}{F} \newcommand{\Mass}{M} \newcommand{\Sensitivity}{S} \newcommand{\basalRate}{b} \newcommand{\dampingCoefficient}{c} \newcommand{\mass}{m} \newcommand{\sensitivity}{s} \newcommand{\springScalar}{\kappa} \newcommand{\springVector}{\boldsymbol{ \kappa}} \newcommand{\springMatrix}{\boldsymbol{ \mathcal{K}}} \newcommand{\tfConcentration}{p} \newcommand{\tfDecayRate}{\delta} \newcommand{\tfMrnaConcentration}{f} \newcommand{\tfVector}{\mathbf{ \tfConcentration}} \newcommand{\velocity}{v} \newcommand{\sufficientStatsScalar}{g} \newcommand{\sufficientStatsVector}{\mathbf{ \sufficientStatsScalar}} \newcommand{\sufficientStatsMatrix}{\mathbf{G}} \newcommand{\switchScalar}{s} \newcommand{\switchVector}{\mathbf{ \switchScalar}} \newcommand{\switchMatrix}{\mathbf{S}} \newcommand{\tr}[1]{\text{tr}\left(#1\right)} \newcommand{\loneNorm}[1]{\left\Vert #1 \right\Vert_1} \newcommand{\ltwoNorm}[1]{\left\Vert #1 \right\Vert_2} \newcommand{\onenorm}[1]{\left\vert#1\right\vert_1} \newcommand{\twonorm}[1]{\left\Vert #1 \right\Vert} \newcommand{\vScalar}{v} \newcommand{\vVector}{\mathbf{v}} \newcommand{\vMatrix}{\mathbf{V}} \newcommand{\varianceDist}[2]{\text{var}_{#2}\left( #1 \right)} \newcommand{\vecb}[1]{\left(#1\right):} \newcommand{\weightScalar}{w} \newcommand{\weightVector}{\mathbf{ \weightScalar}} \newcommand{\weightMatrix}{\mathbf{W}} \newcommand{\weightedAdjacencyMatrix}{\mathbf{A}} \newcommand{\weightedAdjacencyScalar}{a} \newcommand{\weightedAdjacencyVector}{\mathbf{ \weightedAdjacencyScalar}} \newcommand{\onesVector}{\mathbf{1}} \newcommand{\zerosVector}{\mathbf{0}} $$
at St George’s Church Lecture Theatre, University of Sheffield on Sep 6, 2012 [reveal]
Neil D. Lawrence, Department of Computer Science, University of Sheffield

Abstract

What is Machine Learning? Why is it useful for us? Machine learning algorithms are the engines that are driving forward an intelligent internet. They are allowing us to uncover the causes of cancer and helping us understand the way the universe is put together. They are suggesting who your friends are on facebook, enabling driverless cars and causing flagging potentially fraudulent transactions on your credit card. To put it simply, machine learning is about understanding data. In this lecture I will try and give a sense of the challenges we face in machine learning, with a particular focus on those that have inspired my research. We will look at applications of data modelling from the early 18th century to the present, and see how they relate to modern machine learning. There will be a particular focus on dealing with uncertainty: something humans are good at, but an area where computers have typically struggled. We will emphasize the role of uncertainty in data modelling and hope to persuade the audience that correct handling of uncertainty may be one of the keys to intelligent systems.

$$\newcommand{\tk}[1]{} \newcommand{\Amatrix}{\mathbf{A}} \newcommand{\KL}[2]{\text{KL}\left( #1\,\|\,#2 \right)} \newcommand{\Kaast}{\kernelMatrix_{\mathbf{ \ast}\mathbf{ \ast}}} \newcommand{\Kastu}{\kernelMatrix_{\mathbf{ \ast} \inducingVector}} \newcommand{\Kff}{\kernelMatrix_{\mappingFunctionVector \mappingFunctionVector}} \newcommand{\Kfu}{\kernelMatrix_{\mappingFunctionVector \inducingVector}} \newcommand{\Kuast}{\kernelMatrix_{\inducingVector \bf\ast}} \newcommand{\Kuf}{\kernelMatrix_{\inducingVector \mappingFunctionVector}} \newcommand{\Kuu}{\kernelMatrix_{\inducingVector \inducingVector}} \newcommand{\Kuui}{\Kuu^{-1}} \newcommand{\Qaast}{\mathbf{Q}_{\bf \ast \ast}} \newcommand{\Qastf}{\mathbf{Q}_{\ast \mappingFunction}} \newcommand{\Qfast}{\mathbf{Q}_{\mappingFunctionVector \bf \ast}} \newcommand{\Qff}{\mathbf{Q}_{\mappingFunctionVector \mappingFunctionVector}} \newcommand{\aMatrix}{\mathbf{A}} \newcommand{\aScalar}{a} \newcommand{\aVector}{\mathbf{a}} 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\newcommand{\naturalFrequency}{\omega} \newcommand{\neighborhood}[1]{\mathcal{N}\left( #1 \right)} \newcommand{\neilurl}{http://inverseprobability.com/} \newcommand{\noiseMatrix}{\boldsymbol{ E}} \newcommand{\noiseScalar}{\epsilon} \newcommand{\noiseVector}{\boldsymbol{ \epsilon}} \newcommand{\norm}[1]{\left\Vert #1 \right\Vert} \newcommand{\normalizedLaplacianMatrix}{\hat{\mathbf{L}}} \newcommand{\normalizedLaplacianScalar}{\hat{\ell}} \newcommand{\normalizedLaplacianVector}{\hat{\mathbf{ \ell}}} \newcommand{\numActive}{m} \newcommand{\numBasisFunc}{m} \newcommand{\numComponents}{m} \newcommand{\numComps}{K} \newcommand{\numData}{n} \newcommand{\numFeatures}{K} \newcommand{\numHidden}{h} \newcommand{\numInducing}{m} \newcommand{\numLayers}{\ell} \newcommand{\numNeighbors}{K} \newcommand{\numSequences}{s} \newcommand{\numSuccess}{s} \newcommand{\numTasks}{m} \newcommand{\numTime}{T} \newcommand{\numTrials}{S} \newcommand{\outputIndex}{j} \newcommand{\paramVector}{\boldsymbol{ \theta}} 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\newcommand{\scalarProduct}[2]{\left\langle{#1},{#2}\right\rangle} \newcommand{\sign}[1]{\text{sign}\left(#1\right)} \newcommand{\sigmoid}[1]{\sigma\left(#1\right)} \newcommand{\singularvalue}{\ell} \newcommand{\singularvalueMatrix}{\mathbf{L}} \newcommand{\singularvalueVector}{\mathbf{l}} \newcommand{\sorth}{\mathbf{u}} \newcommand{\spar}{\lambda} \newcommand{\trace}[1]{\text{tr}\left(#1\right)} \newcommand{\BasalRate}{B} \newcommand{\DampingCoefficient}{C} \newcommand{\DecayRate}{D} \newcommand{\Displacement}{X} \newcommand{\LatentForce}{F} \newcommand{\Mass}{M} \newcommand{\Sensitivity}{S} \newcommand{\basalRate}{b} \newcommand{\dampingCoefficient}{c} \newcommand{\mass}{m} \newcommand{\sensitivity}{s} \newcommand{\springScalar}{\kappa} \newcommand{\springVector}{\boldsymbol{ \kappa}} \newcommand{\springMatrix}{\boldsymbol{ \mathcal{K}}} \newcommand{\tfConcentration}{p} \newcommand{\tfDecayRate}{\delta} \newcommand{\tfMrnaConcentration}{f} \newcommand{\tfVector}{\mathbf{ \tfConcentration}} \newcommand{\velocity}{v} \newcommand{\sufficientStatsScalar}{g} \newcommand{\sufficientStatsVector}{\mathbf{ \sufficientStatsScalar}} \newcommand{\sufficientStatsMatrix}{\mathbf{G}} \newcommand{\switchScalar}{s} \newcommand{\switchVector}{\mathbf{ \switchScalar}} \newcommand{\switchMatrix}{\mathbf{S}} \newcommand{\tr}[1]{\text{tr}\left(#1\right)} \newcommand{\loneNorm}[1]{\left\Vert #1 \right\Vert_1} \newcommand{\ltwoNorm}[1]{\left\Vert #1 \right\Vert_2} \newcommand{\onenorm}[1]{\left\vert#1\right\vert_1} \newcommand{\twonorm}[1]{\left\Vert #1 \right\Vert} \newcommand{\vScalar}{v} \newcommand{\vVector}{\mathbf{v}} \newcommand{\vMatrix}{\mathbf{V}} \newcommand{\varianceDist}[2]{\text{var}_{#2}\left( #1 \right)} \newcommand{\vecb}[1]{\left(#1\right):} \newcommand{\weightScalar}{w} \newcommand{\weightVector}{\mathbf{ \weightScalar}} \newcommand{\weightMatrix}{\mathbf{W}} \newcommand{\weightedAdjacencyMatrix}{\mathbf{A}} \newcommand{\weightedAdjacencyScalar}{a} \newcommand{\weightedAdjacencyVector}{\mathbf{ \weightedAdjacencyScalar}} \newcommand{\onesVector}{\mathbf{1}} \newcommand{\zerosVector}{\mathbf{0}} $$

Discovery of Ceres [edit]

Year is 1801 Planet Ceres discovered in January by Giuseppe Piazzi, an italian priest, born in Lombardy, but working in Palermo, where he founded the observatory. He was later to die in Naples.

Figure:

Figure:

Figure:

Piazzi (n.d.)

Figure:

Figure:

Gauss (n.d.),Gauss (1802)

Figure:

Figure:

Figure: The surface area of Ceres is 2,850,000 square kilometers, it’s a little bigger than Greenland, but quite a lot colder. The moone is about 27% of the width of the Earth. Ceres is 7% of the width of the Earth.

Figure: This image from http://upload.wikimedia.org/wikipedia/commons/c/c4/Planets2008.jpg

Figure: This image from http://www.popsci.com/sites/popsci.com/files/styles/large_1x_/public/dawn-two-bright-spots.jpg?itok=P5oeSRrc

Figure: Daniel Bernoulli 1700-1782 Pressure from a gas is given by velocity of gas acting on sides (but constant velocity).

Figure: Bernoulli’s simple kinetic models of gases assume that the molecules of air operate like billiard balls.

Figure: Bernoulli’s simple kinetic models of gases assume that the molecules of air operate like billiard balls.

Figure:

}{James Clerk Maxwell 1831-1879 Derived distribution of velocities of particles in an ideal gas (elastic fluid).}{gaussian-histogram}

Figure: James Clerk Maxwell (1831-1879), Ludwig Boltzmann (1844-1906) Josiah Willard Gibbs (1839-1903)

Figure: Arthur Stanley Eddington () book on the Nature of the Physical World (???)

Figure: Chandrasekhar (1910-1995)

Figure: (???)

Figure: Claude Shannon (died 2001) and Edwin Thompson Jaynes Jaynes (1965)

Shannon image source: http://www.gstatic.com/hostedimg/a23c741096cdf969_large

Figure: Bert Kappen

http://videolectures.net/aispds08_kappen_easop/

Figure: Kappen Ball

trim=0cm 20cm 0cm 20cm, clip]

[trim=0cm 25cm 0cm 15cm, clip]

Figure: Variation in the data fit term, the capacity term and the negative log likelihood for different lengthscales.

[angle=90]

Figure: Goodsell (1999)

Figure: p53. Left unbound, Right bound to DNA. Images by David S. Goodsell from http://www.rcsb.org/ (see the ‘Molecule of the Month’ feature).

ODE1 ARTIFICIAL EXAMPLE


load ../../../gpsim/matlab/demToyProblem7.mat

predt = [linspace(0, 18, 100)]';
figure(1), clf
order = [4 3 2];
lin1 = [];
colors = {blueColor, greenColor, redColor};
hold on;
for i = 1:length(order)
  lin1 = [lin1; plot(t, truey(:, order(i)-1), 'color', colors{i})];
end
set(gca, 'ylim', [0 8]);
xlim = get(gca, 'xlim');
ylim = get(gca, 'ylim')
line([xlim(1) xlim(1)], ylim, 'color', blackColor)
line(xlim, [ylim(1) ylim(1)], 'color', blackColor)

hold on
lin2 = [];
counter = 0;
presentOrder = 0;
for i = order
  presentOrder = presentOrder + 1;
  for j = 1:length(model.timesCell{i})
    counter = counter + 1;
    indices(counter, 1) = i;
    indices(counter, 2) = j;
    indices(counter, 3) = presentOrder;
  end
end
ind = [1];
tvals = cell(length(model.timesCell));
includeText = [];
for i = 1:size(indices, 1)
  offset = 0;
  for j = 1:indices(i, 1)-1
    offset = offset + length(model.timesCell{j});
  end
  ind = [ind indices(i, 2)+offset];
  tvals{indices(i, 1)} = [tvals{indices(i, 1)}; model.timesCell{indices(i, 1)}(indices(i, ...
                                                    2))];
  
  proteinKern = model.kern.comp{1};
  K = rbfKernCompute(proteinKern, 0, predt);
  counter = 0;
  for j=order
    counter = counter + 1;
    if ~isempty(tvals{j})
      K = [K; real(simXrbfKernCompute(model.kern.comp{j}, proteinKern, ...
                                      tvals{j}, predt))];
    end
  end  
  invK = pdinv(model.K(ind, ind));
  obsY = model.m(ind, 1);
  predF = K'*invK*obsY;
  varF = kernDiagCompute(proteinKern, predt) - sum(K.*(invK*K), 1)';
  
  figure(1)
  lin2 = [ plot(model.timesCell{indices(i, 1)}(indices(i, 2)), ...
               [repmat(NaN, 1, indices(i, 3)-1) model.y(ind(end) - 1)], 'x', ...
               'color', colors{indices(i, 3)})];
  set(lin1, 'lineWidth', 2);
  set(lin1, 'markersize', 10);
  set(lin2, 'lineWidth', 4);
  set(lin2, 'markersize', 10);
  %set(gca, 'fontname', 'arial', 'fontsize', 24, 'xlim', xlim, 'ylim', [0 8])
  fileName = ['toyGeneData' num2str(i)];
  printLatexPlot(fileName, directory, 0.4*textWidth);
  includeText = [includeText '###\n\n']
  includeText = [includeText '<img class="" src="' directory fileName '" width="45%" height="auto" align="left" style="background:none; border:none; box-shadow:none; display:block; margin-left:auto; margin-right:auto;vertical-align:middle">'];
   
  figure(2), clf  
  hold on
  stdVals = sqrt(varF);
  fillColor = [0.7 0.7 0.7];
  patch([predt; predt(end:-1:1)], ...
        [predF; predF(end:-1:1)] ...
        + 2*[stdVals; -stdVals(end:-1:1)], ...
            fillColor,'edgecolor',fillColor)
  % fill(, ...
  %      fillColor,'EdgeColor',fillColor)
  lin = plot(t, truef, '-', 'color', blueColor);  
  lin = [lin plot(predt, predF, '-', 'color', blackColor)];
  set(lin, 'lineWidth', 4);
  set(lin, 'markersize', 10);
  %set(gca, 'fontname', 'arial', 'fontsize', 24, 'xlim', xlim)
  ylim = [-2 4];
  xlim = get(gca, 'xlim');
  set(gca, 'ylim', ylim)
  line([xlim(1) xlim(1)], ylim, 'color', blackColor)
  line(xlim, [ylim(1) ylim(1)], 'color', blackColor)
  fileName = ['groundTruthTFData' num2str(i)];
  printLatexPlot(fileName, directory, 0.4*textWidth);
  includeText = [includeText '<img class="" src="' directory fileName '" width="45%" height="auto" align="right" style="background:none; border:none; box-shadow:none; display:block; margin-left:auto; margin-right:auto;vertical-align:middle">'];
end
    printLatexText(includeText, 'infer-tf-from-gene-text.md', directory)

include{../slides/diagrams/sysbio/infer-tf-from-gene-text.md}

Gauss, Carl Friederich. 1802. “Astronomische Untersuchungen Und Rechnungen Vornehmlich über Die Ceres Ferdinandea.”

———. n.d. “Fortgesetzte Nachrichten über Den Längst Vermutheten Neuen Haupt-Planeten Unseres Sonnen-Systems.” In, 638–49.

Goodsell, David S. 1999. “The Molecular Perspective: P53 Tumor Suppressor.” The Oncologist, Vol. 4, No. 2, 138-139, April 1999 4 (2): 138–39.

Jaynes, Edward T. 1965. “Gibbs Vs Boltzmann Entropies.” American Journal of Physics 33 (5). AAPT: 391–98. https://doi.org/10.1119/1.1971557.

Piazzi, Giuseppe. n.d. “Fortgesetzte Nachrichten über Den Längst Vermutheten Neuen Haupt-Planeten Unseres Sonnen-Systems.” In, 279–83.