Machine Learning. Module 3-4: Regression and Survival Analysis Day 2, Asst. Prof. Dr. Santitham Prom-on

Transcription

1 Machine Learning Module 3-4: Regression and Survival Analysis Day 2, Asst. Prof. Dr. Santitham Prom-on Department of Computer Engineering, Faculty of Engineering King Mongkut s University of Technology Thonburi

2 Module 3 Overview Linear regression Poisson regression Survival analysis

3 Simple linear regression Simple linear regression allows us to summarize and study relationships between two continuous (quantitative) variables: One variable, denoted x, is regarded as the predictor, explanatory, or independent variable. The other variable, denoted y, is regarded as the response, outcome, or dependent variable.

4 Types of relationship Deterministic Statistical

5 Example of statistical relationships Height and weight Alcohol consumed and blood alcohol content Spending amount and number of spending Number of exercises per week and loosing weight Credit-line and average credit usage

6 Assessing statistical linear relationship Graphical Scatter plot Statistics Correlation

7 Correlation coefficient The correlation coefficient computed from the sample data measures the strength and direction of a linear relationship between two variables. The symbol for the sample correlation coefficient is r. The symbol for the population correlation coefficient is ρ. There are several types of correlation coefficients. The one explained in this section is called the Pearson product moment correlation coefficient (PPMC)

8 Pearson product moment correlation Given n pairs of observations (x 1,y 1 ), (x 2,y 2 ),,(x n,y n ) It is natural to speak of x and y having a positive relationship if large x s are paired with large y s and small x s with small y s On the contrary, if large x s are paired with small y s and small x s with large y s, then a negative relationship between the variable is implied

9 Pearson product moment correlation Consider the quantity n s = å x -x y - y ( )( ) xy i i i= 1 Then, if the relationship is strongly positive, an x i above the mean will tend to be paired with a y i above the mean, so that and this product will also be positive whenever both x i and y i are below their means x -x y - y > ( )( ) 0 i i

10 Pearson product moment correlation To make this measure dimensionless, we divide as follow r s å ( x -x)( y - y) xy i= 1 i i = = n 2 n 2 sxx s å yy ( x ) ( ) i 1 i -x å y i 1 i - y = = A more convenient form for this equation is r = n ( å ) ( )( ) i i - å å i i n xy x y ( 2) ( ) ( 2) ( ) i - i i - i 2 2 å å å å n x x n y y

11 Correlation coefficient and scatter plot The range of the correlation coefficient is from -1 to 1. If there is a strong positive linear relationship between the variables, the value of r will be close to 1. If there is a strong negative linear relationship, the value of r will be close to -1. When there is no linear relationship between the variables or only a weak one, the value of r will be close to 0

12 Strong correlation? A frequently asked question is: what can it be said that there is a strong correlation between variables, and when is the correlation weak? A reasonable rule of thumb is to say that the correlation is weak if 0 r 0.5 strong 0.8 r 1 moderate otherwise

13 Correlation and shape

14 Python: Loading data

15 Pairwise Correlation

16 Correlation and scatter plot Negative relationship, mpg vs wt

17 Correlation and scatter plot Positive relationship, hp vs disp

18 Correlation and scatter plot No relationship, qsec vs drat

19 Scatter and trend Since we are interested in summarizing the trend between two quantitative variables, the natural question arises "what is the best fitting line?" Scatter plot shows points distribution and potentially the trend in the data Look at the figure in next pages, which lines do you think best summarizes the trend between height and weight?

20 Line equation y i denotes the observed response for experimental unit i x i denotes the predictor value for experimental unit i ŷ i is the predicted response (or fitted value) for experimental unit i The equation for best fitting line is: ŷ i = b 0 + b 1 x i

21 Regression parameters Given a value x i, how do we interpret b 0 and b 1. b 1 tells us: if the value we saw for x was one unit bigger, how much would our prediction for y changes? b 0 tells us: what would we predict for y if x = 0?

22 Residual must be normally distributed with zero mean

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24 Best line

25 Error In general, when we use ŷ i = b 0 + b 1 x i to predict the actual response y i, we make a prediction error (or residual error) of size: e i = y i ŷ i A line that fits the data "best" will be one for which the n prediction errors one for each observed data point are as small as possible in some overall sense.

26 Method of least squares Choose the b s so that the sum of the squares of the errors, e i, are minimized The error function is '! = # $%& ' = # $%& ( $ ) $ +, + & - $.

27 Ordinary least square solution Minimum of a function is the point where the slope is zero E(α) E(x) xα

28 Coefficient of determination (r 2 ) The coefficient of determination is a number that indicates the proportion of the variance in the dependent variable that is predictable from the independent variables

29 Coefficient of determination The coefficient of determination R 2 (or sometimes r 2 ) is another measure of how well the least squares equation Y = α + βx perform as a predictor of y R 2 is computed as: R 2 SS yy - SSE SS yy SSE SSE = = - = 1- SS SS SS SS yy yy yy yy R 2 measures the relative sizes of SS yy and SSE. The smaller SSE, the more reliable the predictions obtained from the model.

30 Coefficient of determination SS yy measures the deviation of the observations from their mean: yy ( ) 2 = åi i - SS y y SSE measures the deviation of observations from their predicted values ( ) 2 = åi i - SSE y Y i

31 Coefficient of determination The higher the R 2, the more useful the model R 2 takes on values between 0 and 1 Essentially, R 2 tells us how much better we can do in predicting y by using the model and computing Y than by just using the mean of y as a predictor. Note that when we use the model and compute Y the prediction depends on X because Y = α + βx. Thus, we act as if x contains information about y. If we just use the mean of y to predict y, then we are saying that x does not contribute information about y and thus our predictions of y do not depend on x.

32 Python: Separate X and Y

33 Simple linear regression No intercept

34 Coefficients Estimates: values of the coefficients Standard errors: This measures the average amount that the coefficient estimates vary from the actual average value of our response variable t-value: The coefficient t-value is a measure of how many standard deviations our coefficient estimate is far away from 0. Pr(>t): This indicates whether the probability that the impact of the parameter is due to chance.

35 Simple linear regression With intercept

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37 Residual and R 2 The Residual Standard Error is the average amount that the response (dist) will deviate from the true regression line. In multiple regression settings, the R 2 will always increase as more variables are included in the model. That s why the adjusted R 2 is the preferred measure as it adjusts for the number of variables considered.

38 F-statistics F-statistic is a good indicator of whether there is a relationship between our predictor and the response variables. The further the F-statistic is from 1 the better it is.

39 R: prediction with linear regression

40 Root mean square error

41 Multiple linear regression We move from the simple linear regression model with one predictor to the multiple linear regression model with two or more predictors. That is, we use the adjective "simple" to denote that our model has only predictor, and we use the adjective "multiple" to indicate that our model has at least two predictors

42 Multiple regression Additive model, no interaction Multiple linear regression model structure is exactly the same as the linear regression f ( x) = w 0 + w 1 x 1 + w 2 x 2 +! Mathematically, parameters are obtained by least square method

43 Multiple regression additive model

44 Interaction in multiple regression Adding interaction terms to a regression model can greatly expand understanding of the relationships among the variables in the model This occurs when two or more variables depend on one another for the outcome For example, drug and alcohol may interact and creates addition (adverse) affect Another example, credit card types and number of active month may interact (depend) and have different spending results

45 Adding interaction term

46 Multiple regression multiplicative model (interaction)

47 Survival analysis

48 Logistic regression vs time In logistic regression, we were interested in studying how risk factors were associated with presence or absence of disease. Sometimes, we are interested in how a risk factor or treatment affects time to disease or some other event. In these cases, logistic regression is not appropriate.

49 Survival analysis Survival analysis is used to analyze data in which the time until the event is of interest. The response is often referred to as a failure time, survival time, or event time.

50 Example: Time until tumor recurrence Time until a machine part fails Time until mobile phone recharge Time until the next credit card usage

51 The survival time response Usually continuous May be incompletely determined for some subjects For some subjects we may know that their survival time was at least equal to some time t. Whereas, for other subjects, we will know their exact time of event. Incompletely observed responses are censored Is always 0.

52 Analysis issue If there is no censoring, standard linear regression procedures could be used. However, these may be inadequate because Time to event is restricted to be positive and has a skewed distribution. The probability of surviving past a certain point in time may be of more interest than the expected time of event. The hazard function, used for regression in survival analysis, can lend more insight into the failure mechanism than linear regression.

53 Censoring Censoring is present when we have some information about a subject s event time, but we don t know the exact event time. For the analysis methods we will discuss to be valid, censoring mechanism must be independent of the survival mechanism.

54 Reasons censoring might occur A subject does not experience the event before the study ends A person is lost to follow-up during the study period A person withdraws from the study These are all examples of right-censoring.

55 Terminology and notation

56 Survival function The survival function gives the probability that a subject will survive past time t. As t ranges from 0 to, the survival function has the following properties It is non-increasing At time t= 0, S(t) = 1. In other words, the probability of surviving past time 0 is 1. At time t=, S(t)=S( )=0. As time goes to infinity, the survival curve goes to 0. In theory, the survival function is smooth. In practice, we observe events on a discrete time scale (days, weeks, etc.).

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59 Survival data

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65 Survival analysis data

66 Kaplan-Meier estimator

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68 Multiple groups

69 Parametric survival functions The Kaplan-Meier estimator is a very useful tool for estimating survival functions. Sometimes, we may want to make more assumptions that allow us to model the data in more detail.

70 Benefit of using parametric survival functions By specifying a parametric form for S(t), we can easily compute selected quantiles of the distribution estimate the expected failure time derive a concise equation and smooth function for estimating S(t),H(t) and h(t) estimate S(t) more precisely than KM assuming the parametric form is correct!

71 Cox proportional hazards regression model The Cox PH model is a semiparametric model makes no assumptions about the form of h(t) (nonparametric part of model) assumes parametric form for the effect of the predictors on the hazard In most situations, we are more interested in the parameter estimates than the shape of the hazard. The Cox PH model is well-suited to this goal.

72 Survival regression While the above KaplanMeierFitter is useful, they only give us an average view of the population. Often we have specific data at the individual level, either continuous or categorical, that we would like to use. For this, we turn to survival regression, specifically CoxPHFitter.

73 Python: Load survival regression data

74 Survival regression

75 Survival regression result

76 Result

77 Thank you Question?