Interaction terms are often misinterpreted in the empirical economics literature by assuming that the coefficient of interest represents unconditional marginal changes. I present the correct way to estimate conditional marginal changes in a series of **non**-**linear** **models** including (ordered) logit/probit regressions, censored and truncated regressions. The **linear** regression model is used as the benchmark case.

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Engle (1982) presented ARCH **models** and generalized as GARCH (Bollerslev, 1986; Taylor, 1986). These **models** are usually used in various climatic researches, especially in climatic time series investigation. Before predicating ARCH **models** for the 40 years of rainfall data series extracted from IRIMO, ARCH **models** were expanded as the first pattern to identify the variability **models** of the monthly and annual rainfall series. Firstly, the rainfalls series were confirmed for the conditional mean equation to validate the condition of an appropriate ARCH family model. Secondly, the conditional variance was analyzed to identify the ARCH model that best explains the resulted rainfalls series variability. Thirdly, the conditional error distribution was evaluated to identify the reliability model that best explains the predicted rainfalls series. The use of ARCH **models** is to assess or predict the nonlinearity of the precipitation series for comparing the **linear** and nonlinear **models** of the predict precipitation series. In this study, the six **non**-**linear** **models** of ARCH family are used to predict the rainfall series. The autoregressive conditional heteroskedasticity model is studying the effect of the conditional variance series with no reflection on the mean which may change, and is affected by some variables of the model. In this paper, an ARCH model which improves a heteroskedasticity amount into the conditional mean equation was used to indicate the influence of variability on mean forecast and calculate the mean and variability of rainfall series. The conditional variance can be illustrated as follows

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This paper describes our work with the data of the WMT’12 Confidence Estimation shared task. Our contribution is twofold: i) we first present an analysis of the provided data that will stress the difficulty of the task and motivate the choice of our approach; ii) we show how using **non**-**linear** **models**, namely ran- dom forests, with a simple and limited features set succeed in modeling the complex decisions require to assess translation quality and achieve the second best results of the shared task.

The chemical structure is represented by numerical entities called molecular descriptors, which are used to describe different characteristics of a certain structure to obtain yield information about the activity being studied [9]. We can also develop a kind of computer program with network topology called artificial neural networks (ANN) that after adequate training learns to predict target proteins for a given drug. It means, ANNs are network-like software that may use as inputs of topological indices and/or physico chemical parameters calculated in the previous steps to predict which molecular structures or network-like structures, present a desire property or not [10,11]. The ANN are known as a good method in expressing highly **non**-**linear** relationship between the input and output variables, hence, greater interests were attracted in applying them to the pattern classification of complex compounds. A genetic algorithm maintains a population of candidate solutions for the problem at hand, and makes it evolve by iteratively applying a set of stochastic operators. GAs are stochastic optimization methods that provide powerful means to perform directed random searches in a large problem space as encountered in chemometrics and drug design [12,13].

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The future forecasts of rice production in Pakistan were calculated based on the best fitted model in **Linear** and **Non**-**Linear** **models**. Among all the **models** cubic was found to be best fitted model for rice production in Pakistan as it has exhibited highest Theil’s U-Statistic (model accuracy), highest R 2 and Adjusted R 2 having least MAPE

Table 1 shows the parameters of the bank queuing system in this research study. The , , , e f in the bank decreases as the number of lines increases. This values shows that in any banking services, in term of “First Come, First Serve”, based on the principle of fairness and technicality, a line is better than more lines in the banking structure. Also, to avoid unnecessary delay of customers in the bank for a particular transaction purpose or waiting lines based on the collected data; some mathematical **models** were formulated based on regression analysis that can determine the expected number or waiting time of customers in the queue while been attended to in the bank. The model was based on **linear** and **non**-**linear** **models**. The reason is to check which of the model will be recommended best for the bank. The findings show that the **non**-**linear** model will be recommended to the bank because of it high effect of coefficient of determination ( ] 0 ! and the p-values of the calculated F-statistic that is lesser than the test of significant at 5% level as shown in Table 2. The value of the optimal servers (0.03660%) shows a significant effect because there is practically no existence of queue in the bank because a customer arriving to the bank will not have to wait more before been attended to.

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Fig 1 shows the architecture of our ensemble learning model. The core framework of our ensemble **models** includes the two stages. The ensemble model is an improved version of stacking (Wolpert, 1992; Zhou, 2012). After data processing and feature engineering, the features are sent to the stage1. We test various kinds of regression **models**. Finally, we find the four regression **models** can achieve satisfying performance on these features. (Table 6) In stage1, including **Linear** Regression, Huber Regression, Gradient Boost Decision Trees and XGBoost. The former two **models** are **linear** **models** and the latter two are **non**-**linear**. The output of the two different **models** will be gotten by **linear** and **non**-**linear** algorithms based on raw features, which guarantees the diverse representation of the raw features. With the output of stage1 serves as the input, stage2 also has both **linear** and **non**-**linear** **models**, including Huber Regression and XGBoost, which are covered by Ensemble block in the figure1.According to the characteristics of data, we carefully tune these **models** and find some tricks (such as ‘emoji’ expression) to achieve better performance than raw data. The tuning work will be discussed in following single model sections.

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Abstract. Reproducing Kernel Hilbert Space (RKHS) is a common used tool in statistics and machine learning to generalize from **linear** **models** to **non**-**linear** **models**. In this paper we will try to understand the basic theoretical results in studying RKHS: to construct a RKHS starting from a given kernel function. This view is highly related to the kernel methods for regression and classification in the area of machine learning.

One of the most remarkable application of probability theory is to depict the real life phenomenon through probability model or distribution. Statistical **models** describe a phenomenon in the form of mathematical equations. In the literature (Hogg & Crag (1970), Johnson & Kotz (1970), Lawless (1982) etc.) we come across different types of **models** e.g., **Linear** **models**, **Non**-**linear** **models**, Generalized **linear** **models**, Generalized addition **models**, Analysis of variance **models**, Stochastic **models**, Empirical **models**, Simulation **models**, Diagnostic **models**, Operation research **models**, Catalytic **models**, Deterministic **models**, etc. Out of large number of methods and tools developed so far for analyzing data (on the life sciences etc.), the statistical **models** are the latest innovations.

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Abstract. Model Output Statistics (MOS) refers to a method of post-processing the direct outputs of numerical weather prediction (NWP) **models** in order to reduce the biases in- troduced by a coarse horizontal resolution. This technique is especially useful in orographically complex regions, where large differences can be found between the NWP elevation model and the true orography. This study carries out a com- parison of **linear** and **non**-**linear** MOS methods, aimed at the prediction of minimum temperatures in a fruit-growing re- gion of the Italian Alps, based on the output of two dif- ferent NWPs (ECMWF T511–L60 and LAMI-3). Temper- ature, of course, is a particularly important NWP output; among other roles it drives the local frost forecast, which is of great interest to agriculture. The mechanisms of cold air drainage, a distinctive aspect of mountain environments, are often unsatisfactorily captured by global circulation mod- els. The simplest post-processing technique applied in this work was a correction for the mean bias, assessed at individ- ual model grid points. We also implemented a multivariate **linear** regression on the output at the grid points surround- ing the target area, and two **non**-**linear** **models** based on ma- chine learning techniques: Neural Networks and Random Forest. We compare the performance of all these techniques on four different NWP data sets. Downscaling the temper- atures clearly improved the temperature forecasts with re- spect to the raw NWP output, and also with respect to the basic mean bias correction. Multivariate methods generally yielded better results, but the advantage of using **non**-**linear** algorithms was small if not negligible. RF, the best perform- ing method, was implemented on ECMWF prognostic output at 06:00 UTC over the 9 grid points surrounding the target area. Mean absolute errors in the prediction of 2 m temper- ature at 06:00 UTC were approximately 1.2 ◦ C, close to the natural variability inside the area itself.

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One of the approaches, that is often used in control engineering to check the observability of **non**-**linear** **models** is to use **linear** observability theory applied piece-wise in time. In other words, it is assumed that exactly the same reduced-order ASM1, as described in Section (2), is composed of a **linear** model at each sampling point. Under this assumption, the Kalman rank test ([C CA…CA n-1 ]) for observability of **linear** systems has been successfully applied piece-wise in time (at each sampling point, during 10 days of simulation), assuming up to three measurements. Results are presented in Figures (2) and (3) by the black bars (piece-wise: 2 meas. / 3 meas.) and in each case, these last ones represent more than three thousand operating points. In other words, the Kalman rank test is performed at each sampling time during 10 days of simulation. In the first case (dry influent), two measurements (S O and S NO ) are necessary to achieve

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As discussed in Hills and Smith (1992) the parameterizations used in the prior and in the likelihood will affect the accuracy and efficiency of Markov chain sampling methods. When sampling each component separately using Gibbs sampling it is desirable to have low correlations between the parameters as this generally improves mixing/convergence. For a simple example, Hills and Smith showed th a t really high correlations greatly slow down the convergence of the Gibbs sampler. By reducing this correlation, even only to 0.8, convergence was much faster with even bad starting values quickly forgotten. Using a suitable parameterization is one way to reduce the correlations. For example, when fitting straight lines it is common to centre the age values by subtracting the mean age from each value. Doing this means th at the intercept is the weight at the mean age rather than at age/tim e zero. This parameter may be less correlated with the slope parameter. For **non**-**linear** **models** the situation is less clear cut. Ross (1990) gives some suggestions on stable parameterizations. By stable he means th at the parameters should represent contrasting features of the d ata and so be less likely to be correlated. When using Metropolis-Hastings algorithms an alternative to reparameterization is to use a well chosen proposal distribution which has the same correlation structure as the target distribution. Unfortunately, this is not always straightforward.

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We know that periodic oscillations can arise in standard **non**-**linear** **models**, but before the advent of chaos to explain the aperiodicity often observed in actual economic variables it was necessary to rely on an unexplained exogenous random variable. Chaos, on the contrary, gives us an endogenous explanation of erraticity. As Goodwin (1991, p. 425) aptly put it, “Poincare generalized an equilibrium point to an equilibrium motion; a chaotic attractor generalizes the motion to a bounded equilibrium region towards which all motions tend, or within which all motions remain; the conception of equilibrium is more or less lost since all degrees of a periodic, or erratic fluctuations can occur within the region. The special relevance of this to economics is that it offers not one but two types of explanation of the pervasive irregularity of economic time-series - an endogenous one in addition to the conventional exogenous shock” (emphasis added).

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(1968). In general The main disadvantage of the VAR model is, that it has too many extra estimated parameters which are usually insignificant , as a re- sult out of sample forecasting is poor as noted by Simkins (1995) although in sample forecasting is good. Performance of VAR **models** were even worst at level when variables involved are **non** stationary. Engle and Granger (1987) comes up with a solution called cointegration and Error Correction **models**. These **models** have good out sample forecasting on the long horizon. Some modified and improved versions of the method are also available. Besides these some **non** **linear** **models** like regime switching **models**, Artificial Neural Network **models**, GARCH **models** and ARFIMA **models** are also used for forecasting purpose.

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is gradually becoming a more realistic representation of data generation processes. In finance, for instance, stock returns tend to be more correlated when there is low volatility than when volatility is high. A similar behavior has been observed in exchange rate mechanisms where the exchange rate may be constrained to lie within a pre-defined target zone [4]. To accommodate this kind of dynamic behavior using time series data, regime-switching **models** (RSM) have been introduced ([15] & [16]; [7]). Threshold autoregressive (TAR) model begins to be regularly appears in the agricultural economics literature as a model that is popularly used [18], and extensively discussed in Tong [17].

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objectives are quickly to screen out a large number of alternatives and to reduce the search space at each hierarchical step. The methodology can be used to design blended solvent for extracting phytochemicals from any herb where the scope and size of the case study depend on the solvent database available and availability of **models**. This methodology is described and highlighted for binary solvent mixtures, but can easily be extended to multicomponent mixtures. For future work, this systematic methodology needs to be verified for extraction of different phytochemicals from various herbs as case studies.

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For example, when Appel and Haken proved the four-color theorem in 1976, they used a computer to enumerate 1,936 special cases that were, in some sense, lemmas of their proof. At the time, many mathematicians did not consider the theorem truly proved. Now computer-assisted proofs are common and generally (but not universally) accepted. Conversely, a substantial body of economic analysis is based on a model of human behavior called “Economic man,” or, with tongue in cheek, Homo economicus. Research based on this model was highly-regarded for several decades, especially if it involved mathematical virtuosity. More recently, this model is treated with more skepticism, and **models** that include imperfect information and bounded rationality are hot topics.

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Whilst the Bayes factor has a nice, simple theoretical underpinning, computation of the marginal likelihood of the highly dimensional time varying **models** proposed in Section 4.2 presents a **non**-trivial computational concern. Early attempts to compute the likelihood of such highly dimensional **models** replied on estimation techniques which utilize the conditional likelihood (e.g., Koop et al. [2009]). More recently, authors have begun to show that this approach can be extremely inaccurate. For in- stance, Chan and Grant [2015] show that the marginal likelihood estimates computed using the (modified) harmonic mean as in Gelfand and Dey [1994] can have a sub- stantial finite sample bias and can thus lead to inaccurate model selection. Frühwirth- Schnatter [1994a] provide a similar inference for Chib’s marginal likelihood method (Chib [1995]). To overcome these issues we follow a series of work on the efficient estimation of marginal likelihood functions for TVP-VARs which are not based on the conditional likelihood as those previously mentioned, but instead, the integrated likelihood (i.e. the marginal density of the data unconditional on the time-varying coefficients and log-volatilities). More precisely, estimation of the marginal likeli- hood for the CVAR and TVP-VAR follows the work of Chan and Grant [2014], whilst estimation of the VAR-SV and TVP-VAR-SV **models** follow the methods outlined in Chan and Eisenstat [2015a] and Chan and Eisenstat [2015b]. We refer the reader to those papers for precise details.

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In Chapter 4, we consider both **linear** and nonlinear variables with measure- ment errors. An estimation procedure and asymptotic theory for the case where the **linear** variables are measured with measurement errors are given in Section 4.1. The common estimator given in (1.2.2) is modified by applying the so-called “correction for attenuation”, and hence deletes the inconsistence caused by measurement error. The modified estimator is still asymptotically normal as (1.2.2) but with a more complicated form of the asymptotic variance. Section 4.2 discusses the case where the nonlinear variables are measured with measurement errors. Our conclusion shows that asymptotic normality heavily depends on the distribution of the measurement error when T is measured with error. Examples and numerical discussions are presented to support the theoretical results.

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