How to Make Remaining Problem Plausibility-Based Let Us Consider - PowerPoint PPT Presentation

Need for Prediction Deterministic . . . Need for Statistical . . . Plausibility Approach . . . How to Make Remaining Problem Plausibility-Based Let Us Consider the . . . How to Modify . . . Forecasting More Accurate What Is the . . . Resulting . . . Kongliang Zhu 1 , Nantiworn Thianpaen 2 , Home Page and Vladik Kreinovich 3 Title Page 1 Faculty of Economics, Chiang Mai University, Thailand email 258zkl@gmail.com ◭◭ ◮◮ 2 Faculty of Economics, Chiang Mai University, Thailand, ◭ ◮ and Faculty of Management Sciences, Suratthani Rajabhat University, Thailand, nantiworn@outlook.com Page 1 of 22 3 Department of Computer Science, University of Texas at El Paso El Paso, TX 79968, USA, vladik@utep.edu Go Back Full Screen Close Quit

Need for Prediction Deterministic . . . 1. Outline Need for Statistical . . . • In recent papers, a new plausibility-based forecasting Plausibility Approach . . . method was proposed. Remaining Problem Let Us Consider the . . . • This method has been empirically successful. How to Modify . . . • One of the steps – selecting a uniform probability dis- What Is the . . . tribution for the plausibility level – is heuristic. Resulting . . . • Is this selection optimal or a modified selection would Home Page like to a more accurate forecast? Title Page • In this talk, we show that the uniform distribution does ◭◭ ◮◮ not always lead to (asymptotically) optimal estimates. ◭ ◮ • We show how to modify this step so that the resulting Page 2 of 22 estimates become asymptotically optimal. Go Back Full Screen Close Quit

Need for Prediction Deterministic . . . 2. Need for Prediction Need for Statistical . . . • One of the main objectives of science is: Plausibility Approach . . . Remaining Problem – given the available data x 1 , . . . , x n , Let Us Consider the . . . – to predict future values of different quantities y . How to Modify . . . • The usual approach to solving this problem consists of What Is the . . . two stages: Resulting . . . Home Page – first, we find a model that describes the observed data; and Title Page – then, we use this model to predict the future value ◭◭ ◮◮ of each of the quantities y . ◭ ◮ • Often, it is sufficient to have a deterministic model : Page 3 of 22 x i = f i ( p ) and y = f ( p ) for some parameters p . Go Back • We use the observed values to estimate p ; then, we use Full Screen these estimates to predict the desired future values y . Close Quit

Need for Prediction Deterministic . . . 3. Deterministic Prediction and Beyond Need for Statistical . . . • This is how, e.g., solar eclipses can be predicted for Plausibility Approach . . . centuries ahead; here: Remaining Problem Let Us Consider the . . . – parameters p includes initial locations, initial ve- How to Modify . . . locities, and masses of all the celestial bodies; What Is the . . . – observations x i are visible locations of celestial bod- Resulting . . . ies at different moments of time. Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 22 Go Back Full Screen Close Quit

Need for Prediction Deterministic . . . 4. Need for Statistical Prediction Need for Statistical . . . • In most practical problems, a fully deterministic pre- Plausibility Approach . . . diction is not possible: Remaining Problem Let Us Consider the . . . – in addition to the parameters p , How to Modify . . . – both the observed values x i and the future value y What Is the . . . are affected by parameters z j beyond our control, Resulting . . . – parameters that can be viewed as random . Home Page • Thus, we have a probabilistic model : Title Page x i = f ( p, z 1 , . . . , z m ) and y = f ( p, z 1 , . . . , z m ) , ◭◭ ◮◮ where z j are random variables. ◭ ◮ • Usually: Page 5 of 22 – we do not know the exact probability distribution Go Back for the variables z i , but Full Screen – we know a finite-parametric family of distributions that contains the actual (unknown) distribution. Close Quit

Need for Prediction Deterministic . . . 5. Need for Statistical Prediction Need for Statistical . . . • For example, we may know that the distribution is Plausibility Approach . . . Gaussian, or that it is uniform. Remaining Problem Let Us Consider the . . . • Let q denote the parameter(s) that describe this dis- How to Modify . . . tribution. What Is the . . . • In this case, both x i and y are random variables whose Resulting . . . distribution depends on all the parameters θ = ( p, q ): Home Page x i ∼ f i,θ and y ∼ f θ . Title Page • In this case, to identify the model: ◭◭ ◮◮ – we first estimate the parameters θ based on the ◭ ◮ observations x 1 , . . . , x n , and then Page 6 of 22 – we use the distribution f θ corresponding to these Go Back parameter values to predict the values y Full Screen – or, to be more precise, to predict the probability of different values of y . Close Quit

Need for Prediction Deterministic . . . 6. Need for a Confidence Interval Need for Statistical . . . • In the statistical case, we cannot predict the exact value Plausibility Approach . . . of y . Remaining Problem Let Us Consider the . . . • So, it is desirable to predict the range of possible values How to Modify . . . of y . What Is the . . . • For many distributions (e.g., for normal), it is possible Resulting . . . to have arbitrarily small and arbitrarily large values. Home Page • In such situations, there is no guaranteed range of val- Title Page ues of y . ◭◭ ◮◮ • However, we can still try to estimate a confidence in- ◭ ◮ terval , i.e., Page 7 of 22 – for a given small value α > 0, Go Back – an interval [ y α , y α ] that contains the actual value y Full Screen with confidence 1 − α . Close Quit

Need for Prediction Deterministic . . . 7. Confidence Interval (cont-d) Need for Statistical . . . • In other words, we would like to find an interval for Plausibility Approach . . . which Prob( y ∈ [ y α , y α ]) ≥ α . Remaining Problem Let Us Consider the . . . def • When we know the cdf F ( y ) = Prob( Y ≤ y ), we can How to Modify . . . F − 1 � α 1 − α � � , F − 1 � �� take [ y α , y α ] = . What Is the . . . 2 2 • In general, a statistical estimate based on a finite sam- Resulting . . . Home Page ple is only approximate. Title Page • Thus, based on a finite sample, we can predict the value of the parameters θ only approximately. ◭◭ ◮◮ • Therefore, we only have an approximate estimate of ◭ ◮ the probabilities of different values of y . Page 8 of 22 • So, instead of the actual cdf F ( y ), we only know the Go Back bounds on the cdf: F ( y ) ≤ F ( y ) ≤ F ( y ). Full Screen • We want to select [ y α , y α ] so that the probability of Close being outside this interval is guaranteed not be ≤ α . Quit

Need for Prediction Deterministic . . . 8. Confidence Interval (cont-d) Need for Statistical . . . • We can take F ( y α ) ≤ α Plausibility Approach . . . 2 for all possible F ( x ). Remaining Problem • This can be achieved for y α = ( F ) − 1 � α � . Let Us Consider the . . . 2 How to Modify . . . 1 − α • Similarly, we can take y α = ( F ) − 1 � � . What Is the . . . 2 Resulting . . . • Plausibility-based forecasting: we start by forming a Home Page likelihood function . Title Page • We assume that the probability density function corre- sponding to each observation x i has the form f i,θ ( x i ). ◭◭ ◮◮ • We assume that x i are independent, so ◭ ◮ n Page 9 of 22 � L x ( θ ) = f i,θ ( x i ) . i =1 Go Back • L x ( θ ) is normally used to find the maximum likelihood Full Screen � � estimate ˆ ˆ θ : L x θ = max L x ( θ ). Close θ Quit

Need for Prediction Deterministic . . . 9. Plausibility Approach (cont-d) Need for Statistical . . . • Instead, we use L x ( θ ) to define the plausibility function Plausibility Approach . . . Remaining Problem θ ′ L x ( θ ′ ) = L x ( θ ) L x ( θ ) pl x ( θ ) = � . Let Us Consider the . . . � sup ˆ L x θ How to Modify . . . • Based on pl x ( θ ), we define, for each ω ∈ [0 , 1], a plau- What Is the . . . sibility region Γ x ( ω ) = { θ : pl x ( θ ) ≥ ω } . Resulting . . . Home Page • We represent y as g ( θ, z ), where z is uniform on [0 , 1] and g ( θ, z ) = F − 1 Title Page θ ( z ). ◭◭ ◮◮ • We compute the belief and plausibility of each set A of possible values of θ : Bel( A ) = Prob( g (Γ x ( ω ) , z ) ⊆ A ) , ◭ ◮ Pl( A ) = Prob( g (Γ x ( ω ) , z ) ∩ A � = ∅ ) . Page 10 of 22 Go Back • Here, ω , z are uniformly distributed on [0 , 1]. Full Screen • Then, we compute F ( y ) = Bel(( −∞ , y ]), F ( y ) = Pl(( −∞ , y ]) , and the confidence intervals. Close Quit