Predictions Are Important Traditional Statistics . . . Predictive Approach Measurement . . . How Better Are Predictive Robust Interval . . . Models: Analysis on the Analysis of the Problem How Accurate Are . . . Practically Important Comparison of Two . . . How More Accurate? Example of Robust Interval Home Page Uncertainty Title Page ◭◭ ◮◮ Vladik Kreinovich 1 , Hung T. Nguyen 2 , 3 , ◭ ◮ Songsak Sriboonchitta 3 , and Olga Kosheleva 1 1 University of Texas at El Paso, El Paso, Texas 79968, USA Page 1 of 22 olgak@utep.edu, vladik@utep.edu 2 Department of Mathematics, New Mexico State University Go Back Las Cuces, New Mexico 88003, USA, hunguyen@nmsu.edu Full Screen 3 Faculty of Economics, Chiang Mai University Chiang Mai 50200 Thailand, songsakecon@gmail.com Close Quit
Predictions Are Important Traditional Statistics . . . 1. Predictions Are Important Predictive Approach • One of the main applications of science and engineering Measurement . . . is to predict what will happen in the future. Robust Interval . . . Analysis of the Problem • In science, we are most interesting in predicting what How Accurate Are . . . will happen “by itself”. Comparison of Two . . . • Examples: where the Moon will be a year from now? How More Accurate? Home Page • In engineering, we are more interested in what will hap- pen if we apply a certain control strategy. Title Page • Example: where a spaceship will be if we apply a cer- ◭◭ ◮◮ tain trajectory correction? ◭ ◮ • In both science and engineering, prediction is one of Page 2 of 22 the main objectives. Go Back Full Screen Close Quit
Predictions Are Important Traditional Statistics . . . 2. Traditional Statistics Approach to Prediction: Predictive Approach Estimate then Predict Measurement . . . • In the traditional statistical approach, we first fix a Robust Interval . . . statistical model with unknown parameters. Analysis of the Problem How Accurate Are . . . • For example, we can assume that the dependence of y Comparison of Two . . . on x 1 , . . . , x n is linear: How More Accurate? n � Home Page y = a 0 + a i · x i + ε, ε ∼ N (0 , σ ) . Title Page i =1 • In this case, the parameters are a 0 , a 1 , . . . , a n , and σ . ◭◭ ◮◮ ◭ ◮ • Then, we use the observations to confirm this model and estimate the values of these parameters. Page 3 of 22 • After that, we use the model with the estimated pa- Go Back rameters to make the corresponding predictions. Full Screen Close Quit
Predictions Are Important Traditional Statistics . . . 3. Traditional Statistical Approach to Prediction: Predictive Approach Advantages And Limitations Measurement . . . • In the traditional approach: Robust Interval . . . Analysis of the Problem – when we perform estimations, How Accurate Are . . . – we do not take into account what exactly charac- Comparison of Two . . . teristic we plan to predict. How More Accurate? • Advantage of this approach: a computationally inten- Home Page sive parameter estimation part is performed only once. Title Page • In the past, when computations were much slower than ◭◭ ◮◮ now, this was a big advantage. ◭ ◮ • With this advantages, come a potential limitation: Page 4 of 22 – hopefully, by tailoring parameter estimation to a Go Back specific prediction problem, Full Screen – we may be able to make more accurate predictions. Close Quit
Predictions Are Important Traditional Statistics . . . 4. Predictive Approach Predictive Approach • In the past, because of the computer limitations, we Measurement . . . had to save on computations. Robust Interval . . . Analysis of the Problem • Thus, the traditional approach was, in most cases, all How Accurate Are . . . we could afford. Comparison of Two . . . • However, now computers have become much faster. How More Accurate? Home Page • As a result, it has become possible to perform intensive computations in a short period of time. Title Page • So, we can directly solve the prediction problem. ◭◭ ◮◮ • In other words: ◭ ◮ Page 5 of 22 – on the intermediate step of estimating the param- eters, Go Back – we can take into account what exactly quantities Full Screen we need to predict. Close Quit
Predictions Are Important Traditional Statistics . . . 5. What We Do in This Talk Predictive Approach • There are many examples of successful use of the pre- Measurement . . . dictive approach. Robust Interval . . . Analysis of the Problem • However, most of these examples remain anecdotal. How Accurate Are . . . • In this talk: Comparison of Two . . . – on a practically important simple example of robust How More Accurate? Home Page interval uncertainty, – we prove a general result showing that predictive Title Page models indeed lead to more accurate predictions. ◭◭ ◮◮ • Moreover, we provide a numerical measure of accuracy ◭ ◮ improvement. Page 6 of 22 Go Back Full Screen Close Quit
Predictions Are Important Traditional Statistics . . . 6. Measurement Uncertainty: Reminder Predictive Approach • Data processing starts with values that come from Measurement . . . measurements. Robust Interval . . . Analysis of the Problem • Measurement are not 100% accurate: How Accurate Are . . . – the measurement result � x is, in general, different Comparison of Two . . . from How More Accurate? – the actual (unknown) value x of the corresponding Home Page quantity. Title Page • In other words, in general, we have a non-zero mea- ◭◭ ◮◮ def surement error ∆ x = � x − x . ◭ ◮ • In some situations, we know the probability distribu- Page 7 of 22 tion of the measurement error. Go Back • For example, we often that the ∆ x is normally dis- Full Screen tributed, with 0 mean and known st. dev. σ . Close Quit
Predictions Are Important Traditional Statistics . . . 7. Robust Interval Uncertainty Predictive Approach • However, often, the only information that we have Measurement . . . about ∆ x is the upper bound ∆: | ∆ x | ≤ ∆. Robust Interval . . . Analysis of the Problem • This bound is provided by the manufacturer of the How Accurate Are . . . measuring instrument. Comparison of Two . . . • In other words: How More Accurate? – we only know that the probability distribution of Home Page the measurement error ∆ x is located on [ − ∆ , ∆], Title Page – but we do not have any other information about ◭◭ ◮◮ the probability distribution. ◭ ◮ • Such interval uncertainty is a particular case of the Page 8 of 22 general robust statistics . Go Back • Why cannot we always get this additional information? Full Screen • To get information about ∆ x = � x − x , we need to have information about the actual value x . Close Quit
Predictions Are Important Traditional Statistics . . . 8. Robust Interval Uncertainty (cont-d) Predictive Approach • In many practical situations, this is possible; namely: Measurement . . . Robust Interval . . . – in addition to our measuring instrument (MI), Analysis of the Problem – we often also have a much more accurate (“stan- How Accurate Are . . . dard”) MI, Comparison of Two . . . – so much more accurate that the corresponding mea- How More Accurate? surement error can be safely ignored, Home Page – and thus, the results of using the standard MI can Title Page be taken as the actual values. ◭◭ ◮◮ • We can then find the prob. distribution for ∆ x if we ◭ ◮ measure quantities by both our MI and standard MI. Page 9 of 22 • In many situations, however, our MI is already state- Go Back of-the-art, no more-accurate standard MI is possible. Full Screen Close Quit
Predictions Are Important Traditional Statistics . . . 9. Robust Interval Uncertainty (cont-d) Predictive Approach • For example, in fundamental science, we use state-of- Measurement . . . the-art measuring instruments. Robust Interval . . . Analysis of the Problem • For a billion-dollar project like space telescope or par- How Accurate Are . . . ticle super-collider, the best MI are used. Comparison of Two . . . • Another frequent case when we have to use ∆ is the How More Accurate? case of routine manufacturing. Home Page • In this case: Title Page – theoretically, we can calibrate every sensor, but ◭◭ ◮◮ – sensors are cheap and calibrating them costs a lot ◭ ◮ – since it means using expensive standard MIs. Page 10 of 22 • In view of the practical importance, in this talk, we Go Back consider the case of robust interval uncertainty. Full Screen Close Quit
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