Jan Palczewski

Interpreting Random Forest Classification Models Using a Feature Contribution Method

Model interpretation is one of the key aspects of the model evaluation process. The explanation of the relationship between model variables and outputs is relatively easy for statistical models, such as linear regressions, thanks to the availability of model parameters and their statistical significance . For “black box” models, such as random forest, this information is hidden inside the model structure. This work presents an approach for computing feature contributions for random forest classification models. It allows for the determination of the influence of each variable on the model prediction for an individual instance. By analysing feature contributions for a training dataset, the most significant variables can be determined and their typical contribution towards predictions made for individual classes, i.e., class-specific feature contribution “patterns”, are discovered. These patterns represent a standard behaviour of the model and allow for an additional assessment of the model reliability for new data. Interpretation of feature contributions for two UCI benchmark datasets shows the potential of the proposed methodology. The robustness of results is demonstrated through an extensive analysis of feature contributions calculated for a large number of generated random forest models.

Finite Horizon Optimal Stopping of Time-Discontinuous Functionals with Applications to Impulse Control with Delay

We study finite horizon optimal stopping problems for continuous-time Feller-Markov processes. The functional depends on time, state, and external parameters and may exhibit discontinuities with respect to the time variable. Both left- and right-hand discontinuities are considered. We investigate the dependence of the value function on the parameters, on the initial state of the process, and on the stopping horizon. We construct

From Discrete to Continuous Time Evolutionary Finance Models

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Dynamic portfolio optimization with transaction costs and state-dependent drift

-optimal stopping times and provide conditions under which an optimal stopping time exists. We demonstrate how to approximate this optimal stopping time by solutions to discrete-time problems. Our results are applied to the study of impulse control problems with finite time horizon, decision lag, and execution delay.

Interpreting random forest models using a feature contribution method

This paper aims to open a new avenue for research in continuous-time financial market models with endogenous prices and heterogenous investors. To this end we introduce a discrete-time evolutionary stock market model that accommodates time periods of arbitrary length. The dynamics is time-consistent and allows the comparison of paths with different frequency of trade. The main result in this paper is the derivation of the limit model as the length of the time period tends to zero. The resulting model in continuous time generalizes the workhorse model of mathematical finance by introducing asset prices that are driven by the market interaction of investors following self-financing trading strategies. Our approach also offers a numerical scheme for the simulation of the continuous-time model that satisfies constraints such as market clearing at every time step. An illustration is provided.

Theoretical and empirical estimates of mean–variance portfolio sensitivity

The problem of dynamic portfolio choice with transaction costs is often addressed by constructing a Markov Chain approximation of the continuous time price processes. Using this approximation, we present an efficient numerical method to determine optimal portfolio strategies under time- and state-dependent drift and proportional transaction costs. This scenario arises when investors have behavioral biases or the actual drift is unknown and needs to be estimated. Our numerical method solves dynamic optimal portfolio problems with an exponential utility function for time-horizons of up to 40 years. It is applied to measure the value of information and the loss from transaction costs using the indifference principle.

Real Option Valuation for Reserve Capacity

Model interpretation is one of the key aspects of the model evaluation process. The explanation of the relationship between model variables and outputs is easy for statistical models, such as linear regressions, thanks to the availability of model parameters and their statistical significance. For “black box” models, such as random forest, this information is hidden inside the model structure. This work presents an approach for computing feature contributions for random forest classification models. It allows for the determination of the influence of each variable on the model prediction for an individual instance. Interpretation of feature contributions for two UCI benchmark datasets shows the potential of the proposed methodology. The robustness of results is demonstrated through an extensive analysis of feature contributions calculated for a large number of generated random forest models.

Stopping of functionals with discontinuity at the boundary of an open set

This paper studies properties of an estimator of mean–variance portfolio weights in a market model with multiple risky assets and a riskless asset. Theoretical formulas for the mean square error are derived in the case when asset excess returns are multivariate normally distributed and serially independent. The sensitivity of the portfolio estimator to errors arising from the estimation of the covariance matrix and the mean vector is quantified. It turns out that the relative contribution of the covariance matrix error depends mainly on the Sharpe ratio of the market portfolio and the sampling frequency of historical data. Theoretical studies are complemented by an investigation of the distribution of portfolio estimator for empirical datasets. An appropriately crafted bootstrapping method is employed to compute the empirical mean square error. Empirical and theoretical estimates are in good agreement, with the empirical values being, in general, higher.

Comparison of the Predictive Performance and Interpretability of Random Forest and Linear Models on Benchmark Data Sets

Motivated by the potential use of electricity storage to smooth fluctuations in supply and demand, we study the problem of writing American-type call options when the holder’s exercise strategy is of threshold type (so that the time of exercise is known, but random). The writer must provide physical cover by buying and storing the asset before selling the option. We optimise the writer’s strategy for a single option and for an infinite sequence of options, these two strategies being different. The latter is motivated by the lifetime valuation of an energy storage unit when used as reserve capacity in a power system. Our stochastic process is a Brownian motion representing the real-time system imbalance, and which we rescale to represent an imbalance price. The single option leads to an optimal stopping problem in which the principle of smooth fit may be violated and the stopping region may be disconnected. The lifetime analysis uses techniques and results for the single option to construct a certain fixed point characterising the value function.

American Call Options for Power System Balancing

We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set . The stopping horizon is either random, equal to the first exit from the set , or fixed (finite or infinite). The payoff function is continuous with a possible jump at the boundary of . Using a generalization of the penalty method, we derive a numerical algorithm for approximation of the value function for general Feller-Markov processes and show existence of optimal or [epsilon]-optimal stopping times.