#### Transcript StatMod - Alan Moses

Review of statistical modeling and probability theory Alan Moses ML4bio What is modeling? • Describe some observations in a simple, more compact way X = (X1,X2) What is modeling? • Describe some observations in a simple, more compact way Model: a = - Gm r2 Instead of all the observations, we only need to remember a constant ‘G’ and measure some parameters ‘m’ and ‘r’. What is statistical modeling? • Deals also with the ‘uncertainty’ in observations Deviation or Variance Expectation • Mathematics is more complicated • Also use the term ‘probabilistic’ modeling What kind of questions will we answer in this course? What’s the best linear model to explain some data? What kind of questions will we answer in this course? Are there multiple groups? What are they? What kind of questions will we answer in this course? Given new data, which group do we assign it to? 3 major areas of machine learning (that have proven useful in biology) • Regression • Clustering • Classification What’s the best linear model to explain some data? Are there multiple groups? What are they? Given new data, which group do we assign it to? Expression Level Molecular Biology example X = (L,D) Expectation Variance disease Expectation Variance disease Expression Level Expression Level Molecular Biology example “clustering” V1 E1 E2 V2 Class 2 is “enriched” for disease Expectation Variance disease Expression Level “regression” AA Aa Genotype aa Expression Level Expression Level Molecular Biology example “clustering” V1 E1 E2 V2 Class 2 is “enriched” for disease Variance disease Expression Level “regression” AA Aa Genotype aa Expression Level Expectation “clustering” V1 E1 E2 V2 Class 2 is “enriched” for disease “classification” Expression Level Expression Level Molecular Biology example Aa disease? AA Aa Genotype aa Probability theory • Probability theory quantifies uncertainty using ‘distributions’ • Distributions are the ‘models’ and they depend on constants and parameters E.g., in one dimension, the Gaussian or Normal distribution depends on two constants e and π and two parameters that have to be measured, μ and σ 2 P(X|μ,σ) = e 1 √2πσ2 –(X–μ) 2σ2 ‘X’ are the possible datapoints that could come from the distribution. In statistics jargon ‘X’ is called a random variable Probability theory • Probability theory quantifies uncertainty using ‘distributions’ • Choosing the distribution or ‘model’s the first step in a statistical model • E.g., data: mRNA expression levels, counts of sequencing reads, presence or absence of protein domains or ‘A’ ‘C’ ‘G’ and ‘T’ s • We will use different distributions to describe these different types of data. Typical data and distributions • • • • Data is categorical (yes or no, A,C,G,T) Data is a fraction (e.g., 13 out of 5212) Data is a continuous number (e.g., -6.73) Data is a ‘natural’ number (0,1,2,3,4…) • It’s also possible to do regression, clustering and classification without specifying a distribution Molecular Biology example • In this example, we might try to combine a Bernoulli for the disease data, Poisson for the genotype and Gaussian for the expression level • We also might try to classify without specifying distributions Expression Level “classification” Aa disease? AA Aa Genotype aa Molecular Biology example • genomics era means we will almost never have the expression level for just one gene or the genotype at just one locus • Each gene’s expression level can be considered another ‘dimension’ • for 1000s of genes…. Gene 1 Expression Level Gene 2 Expression Level Gene 2 Expression Level • for two genes, if each point is data for one person, we can make a graph of this type of data Gene 3 Gene 4 Gene 5 … Gene 1 Expression Level Molecular Biology example • genomics era means we will almost never have the expression level for just one gene or the genotype at just one locus Gene 2 Expression Level • We’ll usually make 2-D plots, but anything we say about 2-D can usually be generalized to n-dimensions Each “observation” , X, contains expression level for Gene 1 and Gene 2 Represent this as a vector: e.g., X = (1.3, 4.6) Or generally Gene 1 Expression Level X = (X1, X2) Molecular Biology example • genomics era means we will almost never have the expression level for just one gene or the genotype at just one locus Gene 2 Expression Level • We’ll usually make 2-D plots, but anything we say about 2-D can usually be generalized to n-dimensions Each “observation” , X, contains expression level for Gene 1 and Gene 2 Represent this as a vector: e.g., X = (1.3, 4.6) Or generally Gene 1 Expression Level X = (X1, X2) This gives a geometric interpretation to multivariate statistics Probability theory • Probability theory quantifies uncertainty using ‘distributions’ • Distributions are the ‘models’ and they depend on constants and parameters E.g., in two dimensions, the Gaussian or Normal distribution depends on two constants e and π and 5 parameters that have to be measured, μ and Σ – P(X|μ,σ) = 1 2π √|Σ| e 1 2 (X–μ)T Σ-1 (X–μ) ‘X’ are the possible datapoints that could come from the distribution. In statistics jargon ‘X’ is called a random variable What does the mean mean in 2 dimensions? What does the standard deviation mean? Bivariate Gaussian Molecular Biology example • genomics era means we will almost never have the expression level for just one gene or the genotype at just one locus Gene 2 Expression Level • We’ll usually make 2-D plots, but anything we say about 2-D can usually be generalized to n-dimensions Each “observation” , X, contains expression level for Gene 1 and Gene 2 Represent this as a vector: µ Gene 1 Expression Level X = (X1, X2) The mean is also a vector: µ = (µ1, µ2) The variance is a matrix: σ11 σ12 = Σ σ21 σ22 Σ= -4 1 0 0 1 -2 0 2 4 “spherical covariance” rmvnorm(n = 300, mean = c(1, 1), sigma = matrix(c(1, 0, 0, 1), [,1] ncol = 2))[,1] 2 Σ=σ I Σ= 1 0 -4 -2 0 4 0 2 4 “axis-aligned, diagonal covariance” rmvnorm(n = 300, mean = c(1, 1), sigma = matrix(c(1, 0, 0, 4), [,1] ncol = 2))[,1] = 300, mean = c(1, 1), sigma = matrix(c(1, 0 rmvnorm(n = 300, mean = c(1, 1), sigma = matrix(c(1, -1.9, -1.9,rmvnorm(n [,2] 1), ncol = 2))[,2] 4), ncol = 2))[,2] -4 -2 0 2 4 -4 -2 0 2 4 vnorm(n = 300, mean = c(1, 1), sigma = matrix(c(1, 0, 0, 4),rmvnorm(n [,2] = 300, mean = c(1, 1), sigma = matrix(c(1, 0 ncol = 2))[,2] ncol = 2))[,2] -4 -2 0 2 4 -4 -2 0 2 4 µ Σ= -4 1.0 0.5 0.5 1.0 -2 0 2 4 “correlated data” rmvnorm(n = 300, mean = c(1, 1), sigma = matrix(c(1, 0.5, 0.5, [,1] 1), ncol = 2))[,1] 1.0 -1.9 Σ =-1.9 4.0 -4 -2 0 2 “full covariance” 4 rmvnorm(n = 300, mean = c(1, 1), sigma = matrix(c(1, -1.9, -1.9, [,1] 4), ncol = 2))[,1] Probability theory • Probability theory quantifies uncertainty using ‘distributions’ • Distributions are the ‘models’ and they depend on constants and parameters • Once we chose a distribution, the next step is to chose the parameters • This is called “estimation” or “inference” P(X|μ,σ) = e 1 √2πσ2 2 (X–μ) – 2 2σ Expression Level Estimation We want to make a statistical model. 1.Choose a model (or probability distribution) Expectation Variance 2.Estimate its parameters • Choose the parameters so the model ‘fits the data’ 2 (X–μ) – 2 2σ 1 P(X|μ,σ) = √2πσ2 e How do we know which parameters fit the data? • There are many ways to measure how well a model fits that data • Different “Objective functions” will produce different “estimators” (E.g., MSE, ML, MAP) Laws of probability (True for all distributions) • If X1 … XN are a series of random variables (think datapoints) P(X1 , X2) is the “joint probability” and is equal to P(X1) P(X2) if X1 and X2 are independent. i=N P(X1 … XN ) =Π P(Xi) i=1 P(X1 | X2), is the “conditional probability” of event X1 given X2 Conditional probabilities are related by Bayes’ theorem: P(X1| X2) = P(X2 |X1) P(X1) P(X2) Likelihood and MLEs • Likelihood is the probability of the data (say X) given certain parameters (say θ) L = P(X|θ) • Maximum likelihood estimation says: choose θ, so that the data is most probable. L θ =0 • In practice there are many ways to maximize the likelihood. Example of ML estimation Data: Xi 5.2 9.1 8.2 7.3 7.8 P(Xi|μ=6.5, σ=1.5) 0.182737304 0.059227322 0.13996368 0.230761096 0.182737304 L = P(X|θ) = P(X1 … XN | μ, σ) i=5 = ΠP(Xi|μ=6.5, σ=1.5) = 6.39 x 10-5 i=1 L Mean, μ Example of ML estimation In practice, we almost always use the log likelihood, which becomes a very large negative number when there is a lot of data Mean, μ Log(L) Log(L) Example of ML estimation ML Estimation • In general, the likelihood is a function of multiple variables, so the derivatives with respect to all of these should be zero at a maximum • In the example of the Gaussian, we have two parameters, so that L μ =0 and L σ =0 • In general, finding MLEs means solving a set of coupled equations, which usually have to be solved numerically for complex models. MLEs for the Gaussian μML = 1 NΣ X X VML = 1 (X - μ NΣ ML) 2 X • The Gaussian is the symmetric continuous distribution that has as its “centre” a parameter given by what we consider the “average” (the expectation). • The MLE for the for variance of the Gaussian is like the squared error from the mean, but is actually a biased (but still consistent!?) estimator Other estimators • Instead of likelihood, L = P(X|θ) we can choose parameters to maximize posterior probability: P(θ|X) • Or sum of squared errors: Σ X (X – μMSE)2 – θ2 • Or a penalized likelihood: L* = P(X|θ) x e • In each case, estimation involves a mathematical optimization problem that usually has to be solved on computer • How do we choose?