domain: qfin stat math info

text:
Tuan Tran
email: anh-tuan.tran@etu.ephe.psl.eu

Nhat Nguyen
email: nhatnm@buh.edu.vn

Improving Portfolio Liquidity with Cash-Value-at-Risk for Covariance Estimations in Quantitative Trading

Keywords: covariance matrix, shrinkage estimation, portfolio liquidity

Understanding characteristics of covariance matrix is an important research topic. In quantitative trading, portfolio liquidity is a hidden dimension and important as others such as portfolio volatility. In this paper, we propose a liquidity impact measure to improve the portfolio liquidity and also a novel Cash Value at Risk to evaluate the liquidity risk from portfolio cash perspective. Experimental results on various scenarios show that our approach improve a portfolio turnover significantly and also better than others on Cash Value at Risk in almost all cases. An interesting finding is that linear shrinkage covariance estimations not only improve the covariance structure but also resolve a large partial of liquidity.

Introduction

Covariance matrix is used in several domains, e.g. econometrics or data science, therefore covariance estimation is an important research topic which try to robustify the covariance matrix, reduce estimation error and strengthen matrix structure. Particularly in quantitative trading, where the data is noisy and large, it makes the data could be seen as big data and a quantitative system with that data as a complex system. Consequently, the reliable covariance estimation plays a central role in financial economics.

In present investment practice, portfolios with minimum variance objective attach great attention since they probably have higher returns in long run. Covariance matrix used in the theoretical Minimum Variance Portfolio (MVP) is sample covariance which is a simplest estimator but has several disadvantages, such as high estimation error or illstructure when it meets the curse of dimensionality. In large dimensional covariance matrices, linear shrinkage covariance estimations which combine sample covariance matrix and other well-structured matrices show a significant improvement on portfolio volatility.

A third dimension in MVP is portfolio liquidity which is important and strongly relate to portfolio volatility. Liquidity is a complex concept, there is no definition for liquidity and also we cannot see directly the liquidity of an asset or a portfolio. Stated simply, the liquidity of an asset is an ability of trading that asset. Which means how fast to buy or sell the asset at a lowest cost, these elements are very important because it affects on investor's decisions. An asset with less expensive trading could lead to higher valuations [START_REF] Lin | Stock splits, trading continuity, and the cost of equity capital[END_REF] and high selling speed helps investors exit their position rapidly. Therefore, investors need to consider not only portfolio volatility but also portfolio liquidity.

Although there is no definition of liquidity, several studies have proposed indicators to measure the liquidity of an asset such as Sarr et al. [START_REF] Sarr | Measuring liquidity in financial markets[END_REF] described five characteristics of liquid markets:

(i) tightness: mention to low transaction cost of the market, (ii) immediacy: mention to the high speed of order execution in the market, (iii) depth: mention to the existence of limit orders in the market, (iv) breadth: mention to a small market impact of large orders, (v) resiliency: mention to a flow of new orders to correct market imbalances.

While many measures have been proposed to measure liquidity, no single measure can cover all of the above requirements. We focus on one kind of the breadth, the impact of market on portfolio performance which is easily obtained from portfolio results. While this aspect is important, the liquidity of those linear shrinkage MVPs above is still unclear, further analysis is needed to understand characteristics of them.

In this study, we propose a liquidity impact ratio and improve the portfolio liquidity by integrating our ratio into MVP optimization as a liquidity constraint. We also introduce a novel risk metric which captures an outcome of liquidity aspect in the portfolio, it uses daily ending cash of the portfolio to evaluate a Cash value at Risk (CR). Experimental results show a significant improvement on portfolio turnover and also on the CR at four alpha levels. Another contribution is those linear shrinkage estimations, by shrinking the covariance matrix it not only improve the portfolio volatility but also solve a large partial of liquidity problem in MVPs.

The rest of the paper is structured as follows. Section 2 introduces MVP optimization and covariance estimations which are used in this study, then discusses previous liquidity measures and their drawbacks. Section 3 presents our Liquidity Impact ratio and how to integrate it as a liquidity constraint in portfolio optimization. We also explain in detail the Cash Value at Risk measure for portfolio liquidity. Section 4 presents our experimental design. Finally, Section 5 concludes.

Related works

Portfolio construction and Minimum Variance Portfolio

Modern Portfolio Theory (MPT) is a mainstream theoretical approach of Portfolio Selection after Marktowitz published his essay [START_REF] Markowitz | Portfolio selection[END_REF]. To this day, researchers have analyzed various kind of characteristics and pointed out that one of the main problem of the MPT is an uncertainty of input parameters, which is expected returns and covariance matrix of the stock data [START_REF] Best | On the sensitivity of mean-variance-efficient portfolios to changes in asset means: some analytical and computational results[END_REF]. After the financial crisis of 2007-2009, active MPT portfolios showed disappointed performances and then investors looking for alternative approaches, a new passive concept of minimum variance investing is one of them and a large number of studies reported that it outperforms the active style [START_REF] Scherer | A note on the returns from minimum variance investing[END_REF]. Global Minimum Variance Portfolio (MVP) is a special case of MPT which only require covariance matrix to estimate a portfolio with lowest volatility. Given a dataset with N assets, a covariance matrix Σ of asset returns, a formula of MVP is as following:

min w w Σw s.t. w 1 = 1 (1)
where w is a weight vector. One of advantages of MVP is that the covariance matrix is less uncertain than the mean returns of asset [START_REF] Chopra | The effect of errors in means, variances, and covariances on optimal portfolio choice[END_REF], it makes the MVP is less sophisticated than the standard MPT.

Covariance matrix and Linear Shrinkage Estimations

A traditional estimator for the covariance matrix in Equation 1 is a sample covariance estimation. However, when the number of assets N greater than a number of observations, the sample covariance matrix is not well-conditioned or even singular, therefore the portfolio optimization of the MVP fail to apply. In this case, we need a well-conditioned estimator for large-dimensional covariance matrices. Recently, some linear shrinkage estimations of covariance matrices have been proposed to handle this issue. Given a target matrix F, sample covariance matrix S, a linear shrinkage estimator is constructed as:

Σ = δF + (1 -δ)S (2)
where the δ is shrinkage intensity between 0 and 1. This estimator shrinks the sample covariance matrix toward the pre-defined target matrix F which is more structured, well-conditioned. Ledoit and Wolf proposed some shrinkage estimators, such as Shrinkage towards indentity matrix [START_REF] Ledoit | A well-conditioned estimator for large-dimensional covariance matrices[END_REF], Shrinkage towards single-index model (also known as Shrinkage to market) [START_REF] Ledoit | Improved estimation of the covariance matrix of stock returns with an application to portfolio selection[END_REF] and Shrinkage towards constant correlation model [START_REF] Ledoit | Honey, i shrunk the sample covariance matrix[END_REF].

Liquidity in MVP

The MVP with any covariance estimation above try to optimize on portfolio variance. However, liquidity risk should be as important as the portfolio variance or given equal standing with other investment styles [START_REF] Ibbotson | Liquidity as an investment style[END_REF]. There is an interesting negative relationship between liquidity and asset pricing, Amihud et al. [START_REF] Amihud | Asset pricing and the bid-ask spread[END_REF] show that expected returns are an increasing function of the bid-ask spread, i.e. illiquidity cost. Other works, such as Farmer at el. [START_REF] Farmer 5 | What really causes large price changes?[END_REF] show a strong link between liquidity and volatility. Therefore, the portfolio optimization should take the liquidity into account not only in the MVP but also in the MPT.

Even though the liquidity in portfolio is important, it is often ignored by researchers and practitioners. An extensive work of Lo et al. [START_REF] Lo | It's 11 pm-do you know where your liquidity is?: The mean-variance-liquidity frontier[END_REF] presents three ways to construct three-dimensional liquidity-integrated MPT, including: liquidity filtering, liquidity constraints and mean-variance-liquidity objective function. They also generalize several liquidity measures into five styles: i) trading volume, ii) logarithm of trading volume, iii) turnover, iv) percentage bid/ask spread and v) price impact function. Those measures base on daily data or intraday data, a later case is more accurate but a first case is more easily to obtain the data and compute the measures.

A classic example of daily-data liquidity measure is Amihud's illiquidity [START_REF] Amihud | Illiquidity and stock returns ii: Cross-section and time-series effects[END_REF], denoted as ILLIQ and formulated as follows:

ILLIQ i = 1 D D t=1 |r i,t | V i,t C i,t (3)
where D is a number of days in a chosen period, r i,t , V i,t and C i,t are return value, trading volume and close price of an asset i at date t respectively. They interpret it as the market response to one unit of trading value, i.e. roughly as a price impact measure. By using the asset return, they captures only the price impact on the outcome of trading activities but other aspects are not under consideration. Although it proved to be one of the best proxy of liquidity, there is a large room to improve this measure. One of the drawbacks is that it can not handle zero-trading-volume days which is not impossible in the real market, specially with illiquidity assets in illiquidity markets. That is important information regarding illiquidity and even if we use an inverse variation of the Amihud, this issue still exists with zero-return days.

In order to compare liquidity quality of portfolios, which one is better in term of performance or risk, a suitable portfolio metric is necessary. Besides portfolio return and volatility, Lo et al. [START_REF] Lo | It's 11 pm-do you know where your liquidity is?: The mean-variance-liquidity frontier[END_REF] use Sharpe ratio and show a trajectory of liquidity-integrated portfolios which have higher levels of liquidity, but with lower expected return and higher volatility. It is interpretive for practitioners, but not enough in quantitative trading because they have to evaluate a trade-off between portfolio liquidity and portfolio performances (return and volatility). In a next section we will describe a new way to evaluate an outcome of the portfolio with liquidity integration.

Methodology -Liquidity aspect in Minimum Variance portfolio

Liquidity Impact ratio

By measuring the liquidity of an asset, we are more confident in our holding since we know the risk when we short an asset. In this paper, we define the liquidity as an impact of our trading action into asset's price. Therefore, we propose to estimate this impact by considering how much trading value affects a fluctuation of an asset. Given a standard daily OHLCV dataset with O i,t , H i,t , L i,t , C i,t , V i,t are open, high, low, close prices and trading volume of asset i at date t respectively. Let us define an intraday fluctuation of the given asset as following:

f i,t = max C i,t -O i,t C i,t , H i,t -L i,t C i,t . (4)
A first element in the formula 4 can be seen as absolute daily return, it means the O i,t could be replaced by the previous close price C i,t-1 . It will be zero if the return of the asset is zero at that day, which is not rare even with high liquidity assets. The last element could be seens as bid/ask spread in intraday data, it will only be zero if all matched prices are the same. Liquidity estimation from this spread sometimes be zero, particularly in cases of low liquidity assets. In our fluctuation estimation as combination of two approaches, they are complementary to each other. Even that, the fluctuation value can still be zero if all prices are the same, i.e. O i,t = H i,t = L i,t = C i,t , in this case we assume the fluctuation is a tick size of the asset in the market.

The liquidity impact is computed as following:

l i,t = 1 W W j=1 V i,j • C i,j f i,j , (5)
where V i,j is the volume trading of asset i at date j in a latest period with window length W . By completely using the daily stock data, our liquidity estimation could limit disadvantages of previous approaches. The liquidity impact l i of asset i can be interpreted as how much trading value could oscillate its price by 100%. For more intuition, practitioners could use an unit of 1% instead of 100%, in this case they get a sense that the price could move by 1% with l i trading value. From the definition of liquidity, the higher l i value means better liquidity asset. Base on that, given a pre-defined maximum fluctuation θ, we could estimate maximum value of our holding which will make its price fluctuate less than the threshold θ when we want to short the whole asset as soon as possible. An upper bound of the asset in our portfolio will be:

u i,t = l i,t • θ P t (6)
where P t is a portfolio value at date t. We use this upper bound equation in order to integrate the liquidity metric into the portfolio optimization.

Liquidity constrained MVP

The MVP concentrate the portfolio on few assets which have low volatility. However, those low volatility assets are not only low volatility but also low liquidity. Low liquidity makes the volatility is not fully evaluated, therefore holding a large portion of them in our portfolio is a hidden risk. Similar to the volatility, the portfolio optimization need to take into account the liquidity risk. Considering an universe of N assets, let w = (w 1 , ..., w N ) be the vector of portfolio weight, Σ be the covariance matrix of N asset returns. In this study, we only consider a long-only portfolio so that w i ≥ 0. The MVP optimization with liquidity element could be formulated as follows:

min w w Σw s.t. w 1 = 1 w i ∈ [0, min(U, u i )] ∀i = 1, N (7)
where u i is the above upper bound of asset i and U is a constant upper bound which is traditionally used in portfolio optimization.

Liquidity measurement for portfolios

In order to evaluate the liquidity qualification of an individual portfolio and also to compare the liquidity of multiple portfolios, we need a measure to quantify the liquidity of a given portfolio. In this study, we propose to use Cash Value at Risk as an indirect approach to evaluate the liquidity. The cash ratio is a result of portfolio's rebalacing activities, in which the cash from short orders will be used to pay for long orders. If the market could not fulfill the short orders (e.g. shorting low liquidity assets) then the portfolio doesn't have enough cash to pay for the long orders, in this case we allow it to borrow the cash to complete the long activities. Therefore, an ending cash of the portfolio could be negative. A high liquidity portfolio could balance the rebalacing, i.e. easily short assets to meet the cash requirement of long orders, and has positive cash ratio all the time, or at least has small nagative values in a few time. And vice versa, a low liquidity portfolio with large portion of low liquidity assets will have large negative cash ratio frequently. That is why we could use the cash ratio to evalute indirectly the portfolio's liquidity.

More details about the process, we backtesting a given portfolio through T dates with an initial portfolio value P 0 , i.e. an initial cash value C 0 . Each date t ∈ [1, T ] we receive the money from our short orders C short t and then with a previous cash C t-1 , it is used to pay for our long orders by C long t . The orders are placed in a previous trading date t -1 and their matched prices are the closing prices at the current date t. A current ending cash

C t = C t-1 + C short t -C long t , if C t-1 + C short t < C long t
then we allow cash deficits and borrow the money from the market. In a next trading date t+1, we dont short our holding assets by C t to pay off the deficit. However, we continue to trade previous unmatched long/short orders in order to meet expected portfolio weights, until all orders are complete or up to a next rebalancing date.

It is worth to mention that in this study we assume: i) a maximum volume we could long/short for an asset i at date t is up to its volume at that date V i,t , ii) a total weight w 1 always equals 100% which is respectively 100% portfolio value, and iii) we could borrow the money without any type of borrowing limits, i.e. unlimited amount, no interest and infinite period. These assumptions make this measure meaningful, to be more realistic, the first assumption should be more strict or add borrowing limits in the third assumption.

A Cash ratio at each date is an ending cash over a portfolio value, which is a total of a position value and the cash value. Let X be a distribution of Cash ratio values, the Cash Value at Risk (CR) at level α ∈ [0, 1] will be:

CR α (X) = -inf {x ∈ R : F X (x) > α} in which x t = C t P t , (8)

Empirical experiments

Data

We use daily historical data of Vietnam stock market, particularly is Ho Chi Minh City Stock Exchange -HOSE. The dataset is downloaded from their website. Backtesting period is from 2013 to the end of 2019 with 1744 trading days, prior 2013 is for computing preparation. The number of assets varies across backtesting period between N = 303 and N = 387. Shortsell are not allowed according to Vietnam trading regulation. Also, we only consider assets which have at least one year of data. Data summary for each year is in Table 1. The same procedure is repeat for every last trading day in a week, we use two-year weekly data to estimate covariance matrix by a given covariance estimation. Then optimize portfolio weights by Equation 7and rebalance the portfolio to the optimal weights. In a first trading day of next week, our orders will be filled by the close price at that day. Remaining unmatched orders continue ordering in this week in order to meet the target weights, until all of them are complete or at the week's last trading day1 .

Benchmarks

To demonstrate the advantage of our approach, we design a liquid comparison of our Liquidity Impact approach, named LIM, with three other liquidity methods, including: none is a theoretical portfolio optimization which have no liquidity perspective and only use a constraint with constant U such that w i ∈ [0, U ] ∀i = 1, N , we choose U = 10%, classic is a basic liquidity integration by using average trading volume of latest W trading days to limit the upperbound as minimum of U

and 1 /W W t=1 V i,t • C i,t P i,t
, it similars with Equation 6 with f i,t = θ for all asset i, Amihud is similar with us but we replace the fluctuation f i,t in Equation 5with the absolute return value r i,t in Equation 3, Moreover, there are several parameters in those portfolios which can affect performance. In order to show up our robustness, we use different combinations of them to test over various scenarios. In our experiments, we have four main parameters, includes: method: a method of covariance matrix estimation proposed in the literature, includes i) the sample covariance matrix (sample), ii) the Shrinkage to the identity matrix (ST IM ), iii) the Shrinkage to singleindex model (SSIM ) and iv) the Shrinkage to constant correlation matrix (SCCM )2 , N : the number of assets in current universe which we will take into account, using all available assets in the market or using only a top largest assets by market capitalization. We will test some scenarios, including N = {N, 200, 100, 50}, W : the window length in the above liquidity formulas, the liquidity metric will more stable with a larger number and a smaller number let it adapts faster to current market data. We will test a wide range for this parameter, including W = {20, 250}, θ: the threshold value in our liquidity formulas, a smaller value will strictly considers high liquidity assets and a larger value also take into account illiquidity assets. Here we use θ = {1%, 5%}.

Portfolio performance measures

In this study, we use three common portfolio performance measures: the Annual Volatility σ, the Sharpe Ratio SR and a Portfolio Turnover P T which is described below. Only out-of-sample results are reported and are denominated in units of percent. Bracket under each number is p-value level of that portfolio with our portfolio which use the proposed LIM and other parameters are the same. Moreover, we also show the Cash-Ratio-at-Risk CR α at some levels of α such as 0.05%, 0.1%, 1% or 5%.

Although we do not show the hypothesis testing results for all pairwise portfolios in our experiment, we will use them when comparing portfolio strategies in our discussion section below. Particularly for the hypothesis testing of the Sharpe ratio, we follow a bootstrapping method of Ledoit and Wolf [START_REF] Ledoit | Robust performance hypothesis testing with the sharpe ratio[END_REF] to compute robustly the p-value. 3 And we only say that the difference is significant if a p-value is smaller than 5%.

Portfolio turnover. The portfolio turnover is an indicator to measure the daily changing in the portfolio weights. In the scope of the global MVP, a lower portfolio turnover is a better portfolio which shows that the previous calculations is robust and stable. The low portfolio turnover helps to reduce not only the transaction costs but also the risk of the portfolio afterwards, for example the liquidity risks. A formal formula of the portfolio turnover is as follows:

P T = 1 T -1 T -1 t=1 N i=1 |w t+1,i -w t,i | ( 9
)
where N is a number of assets, T is a backtesting period [0, T ] and w t,i is a weight of an asset i at date t.

Analysis of results

Table 2 reports the out-of-sample portfolio performances and the corresponding level of p-value that the performance measure for that portfolio is different from that for the LIM portfolio respectively. We see that for all covariance estimation methods, our volatilities is higher than the volatilities of two basic liquidity approaches, i.e. none and classic, significantly but those are not different from the Amihud approach even at the 10% level. Similarly, our Sharpe ratios are compatible with the Amihud approach and lower than the linear shrinkage estimations while near others in sample covariance estimation. In all cases, our turnover always lower than the classic and Amihud approaches significantly. It confirms the finding in the literature that liquidity integration reduces the portfolio performance since the theoretical portfolio doesnt consider the liquidity risk.

For the Cash ratio, the LIM approach almost better than others except some cases at 1% level. There is only one portfolio that beats our cash ratios, that is ST IM portfolio with none liquidity. This exceptional portfolio has not only a lowest volatility but also a lowest portfolio turnover. 4 Moreover, in the SCCM method at 5% level, only our approach and Amihud could achieve positive cash ratios. Although an improvement of the CR at all levels for liquidity-based Amihud portfolio in sample covariance estimation are clearly visible, but in the linear shrinkage estimations at lower alpha levels (e.g. 0.05% and 0.1%) those differences with basic approaches are not clear.

On average for all alpha levels, the Cash-Ratio-at-Risk of our approach is better than the Amihud by 42.5% and by 66.08% than the theoretical none approach (without the ST IM method). Another interesting in the Table 2 is: while the Amihud liquidity in three linear shrinkage estimators improves only 4.55% on average over the sample covariance estimation, the two basic liquidity approaches in those linear shrinkage estimators (the none and the classic) improve the liquidity by 32.26% and 81.91% respectively. 5 It suggests that the linear shrinkage estimations not only shrink the covariance structure but also resolve most of the liquidity problem in the original MVP. An absolute difference between those classic liquidity in the three linear shrinkage estimations and the Amihud in the sample method is only 0.79% on average for all levels.

Table 3 reports out-of-sample performances of the SCCM portfolios on different sizes of universe. For example, by consider only top 50 assets with highest market capitalization the portfolios expect to use only 50 highest liquidity assets on the market. Similarly with the above analysis, our volatility higher than the two basic liquidity approaches but equivalent with the Amihud's volatility. However, there is no evidence to show difference of the Sharpe ratio between our LIM approach and others, except one weak case of the classic MVP on top N = 200 assets at a significance level α = 0.1. Although our turnover is lower than others in all universe sizes, its significance decreasing when the universe size is more narrow down. At higher universe sizes, the LIM 's turnover significantly lower than the classic and also the Amihud. Until at the size of 50 assets, all of portfolio turnovers are similar.

The SCCM method over all universe sizes (in the Table 3 and the last part of the Table 2) shows that our CR α values are almost the best, or be second-best in only two cases with the N = 200. Overall in those cases, our turnover is 24.55% lower than the Amihud's turnover, 84.7% and 73.65% lower than the turnover of the classic and none approaches respectively. Table 4 presents portfolio performances when we change the parameters in the liquidity formulas. Similarly with the above results, our volatility and turnover are better than the two basic approaches and undifferentiated between us and the Amihud approach. At a longer window length W = 250, our Sharpe ratio lower than the two basic but significantly higher than the Amihud. At a more risky portfolios with θ = 5%, although our Sharpe ratio is close to the Amihud but our turnover is significantly lower.

Comparing the Amihud approach between different thresholds, when the risk threshold is increased then the Cash ratios are also decreased at three over four alpha levels. On average of those differences, the threshold increased from 1% to 5% then the CR decreased 1.36%. Meanwhile, with our approach the CR slightly increased 0.06%.

Interpretations

The theoretical MVP using sample covariance matrix with no liquidity consideration in the Table 2 has the Cash Value at Risk at α = 0.05% equals -20.79%, that means there is a 0.0005 probability that our portfolio will lack of money to complete the orders by more than 20.79% of portfolio value at that day. Practitioners could explain this number as this portfolio is expected on 1 day out of 2000 days an 20.79% of portfolio value as an additional cash or more.

Another example is the MVP with the SCCM covariance estimation on top 200 assets by market cap, using W = 250, θ = 1% and our LIM approach in Table 4, at α = 0.1% the CR is -0.03%. In other words, we expects the additional cash as only 0.03% of portfolio value or more on 1 day out of 1000 days. Similarly, on 100-day periods this portfolio could keep positive cash ratio and there is only 0.05% that the cash ratio will fall by more than -0.23%.

In real world, those low percents of the additional cash are easily taken from the cash asset in the portfolio or from a cash reserve of the fund. As interpreted, a good liquidity portfolio should have the CR α less than fund's cash reserve ratio. Moreover, the CR could be seen as an estimation of portfolio's cash reserve ratio. For example at level α = 0.05%, our portfolio could reserve only 0.23% as cash and invest with 99.77% portfolio value, i.e. the total weight in our second assumption w 1 = 99.77%. Their sample portfolio above have to reserve up to 20.79% as cash and invest with only 79.21% their initial value.

Conclusion

In this study, we proposed the improved liquidity approach for MVP by taking the whole daily data into account to fully evaluate the fluctuation of assets then use it to estimate the liquidity impact LIM on the asset. And by using our novel risk metric based on cash ratio of the portfolio, which indirectly presents outcome of the liquidity, we shows that our risks on four alpha levels are almost lower than the Amihud illiquidity approach and also lower than other approaches on MVPs with four covariance estimations, including sample covariance estimation and linear shrinkage estimations. Although our results are indistinguishable from the Amihud portfolio on the two most common metrics, the volatility and the Sharpe ratio, our turnovers are almost lower than the Amihud's turnover significantly. It shows that the LIM approach robustifies the MVPs and their liquidity, i.e. reduce the risk of cash. Another contribution is about the liquidity of the linear shrinkage estimations for covariance matrices, they not only shrink the covariance structure but also strengthen the liquidity of the portfolio. Their liquidity improve-

Table 1 :

 1 Summary Statistics for HOSE dataset.

		2013 2014 2015 2016 2017 2018 2019
	Average daily return (%) 0.16 0.13 0.06 0.05 0.10 -0.02 0.02
	Standard deviation	0.08 0.03 0.04 0.03 0.03 0.03 0.03
	Average volume (10 5 )	4.53 9.03 8.12 8.88 11.84 11.19 9.66

Table 2 :

 2 Out-of-sample performance of portfolios with different covariance estimation methods and different liquidity approaches. Using all assets in the market (N = N ), use one-month data to compute the covariance matrices (W = 20) and the threshold in liquidity formulas θ = 1%.

	method liquid	σ	SR	PT CR 0.05% CR 0.1% CR 1% CR 5%
		none	8.97 (***)	68.35	7.04 (***)	-20.79 -15.47 -7.27	-3.02
	sample	classic	9.04 (***)	137.06	7.54 (***)	-11.12	-8.69 -6.02	-3.17
		Amihud 12.07 96.24	5.27 (***)	-7.33	-7.18 -1.36	-0.90
		LIM	12.17 94.92 4.23	-2.31	-2.27 -1.74	-0.44
		none	6.52 (***)	171.33 (***)	2.26 (***)	-1.56	-1.11 -0.73	-0.22
	STIM	classic	8.07 (***)	169.97 (***)	4.24 (***)	-6.68	-5.89 -3.22	-1.90
		Amihud 12.01 92.66	4.03 (***)	-4.30	-3.82 -2.29	-1.81
		LIM	12.11 90.37 3.42	-3.35	-3.27 -2.75	-1.08
		none	6.80 (***)	158.82 (*)	3.45	-4.43	-2.95 -1.56	-0.79
	SSIM	classic	8.08 (***)	175.64 (***)	5.29 (***)	-8.12	-7.79 -4.43	-2.17
		Amihud 11.77 102.25	4.66 (***)	-7.59	-7.43 -1.37	-0.75
		LIM	11.95 102.36 3.79	-2.12	-2.04 -1.53	-0.27
		none	7.49 (***)	171.80 (***)	2.64 (***)	-3.68	-3.18 -2.29	-0.83
	SCCM	classic	8.19 (***)	178.15 (**)	4.13 (***)	-7.21	-6.13 -4.34	-1.88
		Amihud 11.37 119.95	4.10 (***)	-7.53	-7.37 -1.39	0.29
		LIM	11.58 116.58 3.29	-2.44	-2.26 -0.04	0.38

Table 3 :

 3 Out-of-sample performance of SCCM portfolios on different universe sizes by taking only top N largest assets by market capitalization. Using one-month data to compute the covariance matrices (W = 20) and the threshold θ = 1%.

	N liquid	σ	SR	PT CR 0.05% CR 0.1% CR 1% CR 5%
		none	8.05 (***)	141.14	2.62 (*)	-5.58	-3.66 -1.45	-0.70
	200	classic	8.61 (***)	(*)	3.87 (**)	-7.72	-5.93 -3.83	-1.80
		Amihud 11.74 99.23	3.92 (**)	-6.18	-5.95 -1.50	0.12
		LIM	12.05 95.70 3.18	-3.92	-3.76 -1.03	-0.64
		none	9.27 (***)	89.81	2.22 (**)	-6.69	-5.09 -1.84	-0.60
	100	classic	10.05 (***)	102.53	3.32 (**)	-9.32	-9.01 -4.48	-2.08
		Amihud 13.09 86.31 3.10	-3.41	-3.00 -0.51	0.24
		LIM	13.43 84.78 2.74	-0.59	-0.59	0.10	0.29
		none	11.13 (***)	63.77 2.01	-6.91	-6.61 -3.25	-1.32
	50	classic	11.62 (***)	73.81 2.44	-8.73	-5.94 -3.15	-1.05
		Amihud 13.98 67.94 2.49	-2.62	-2.60 -0.03	0.17
		LIM	14.41 68.83 2.31	-1.41	-1.40	0.04	0.19

Table 4 :

 4 Out-of-sample performance of SCCM portfolios on top N = 200 largest market capitalization assets with different parameters, the window length W and the threshold θ.

	W θ	liquid	σ	SR	PT CR 0.05% CR 0.1% CR 1% CR 5%
		none	8.05 (***)	141.14 (**)	2.62 (***)	-5.58	-3.66 -1.45	-0.70
	250	classic	8.43 (***)	157.14 (***)	2.27 (**)	-1.43	-1.41 -0.74	-0.22
	1% Amihud 11.85	53.37 (**)	2.02	-5.28	-4.95 -0.80	0.08
	1% LIM	11.44 72.74 1.73	-0.23	-0.03	0.31	0.41
		none	8.05 (***)	141.14	2.62 (***)	-5.58	-3.66 -1.45	-0.70
	20	classic	8.61 (***)	147.06 (**)	3.87	-7.73	-5.93 -3.84	-1.80
	5% Amihud 10.22 106.96	4.77 (**)	-9.19	-5.21 -3.63	-0.93
	5% LIM	10.02 107.25 3.93	-3.75	-3.42 -1.80	-0.15

For other details of the backtesting process, see Appendix in[START_REF] Tran | K-segments under bagging approach: An experimental study on extremely imbalanced data classification[END_REF]

These portfolios are constructed by an open-source Python library, named PyPortfo-lioOpt[START_REF] Martin | Pyportfolioopt: portfolio optimization in python[END_REF]

[START_REF] Markowitz | Portfolio selection[END_REF] Thanks Ledoit and Wolf for publishing their code at https://www.econ.uzh.ch/en/ people/faculty/wolf/publications.html.

The ST IM method similars with the 1/N portfolio of DeMiguel[START_REF] Demiguel | A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms[END_REF], therefore it invests in a large number of assets but with a small weight on each asset. That is a reason why the ST IM has the exceptional performance.

Although the Amihud slightly improves between sample covariance estimation and the linear shrinkage covariance estimations, particularly in the sample covariance estimation, the Amihud could improves the liquidity by 50.08%.


Marie Fabre

Laura Beullier

Chloé Sutter

Amirezza Krebritchi

Pascale Chavet

Martin Simoneau

Michel Toupet

Jean Blouin

Laurence Mouchnino
email: laurence.mouchnino@univ-amu.fr

Cortical facilitation of somatosensory inputs using gravity-related tactile information in patients with bilateral vestibular loss

Keywords: Balance, Plantar sole afferents, bilateral vestibular loss, tactile compensation, EEG, SEP, time frequency analysis

Introduction

When standing upright, balance can be jeopardized by gravitational force if not properly controlled. This control is thought to involve feedforward processes, and to rely on the use and the quality of internal representations of body orientation and motion in space [START_REF] Borel | Vestibular syndrome: A change in internal spatial representation[END_REF][START_REF] Massion | Movement, posture and equilibrium: interaction and coordination[END_REF]Pfeiffer et al., 2014 for reviews). Observations in patients with major vestibular hypofunction (VH) suggest that the gravitational reference frame is crucial for encoding such body-related information. Investigating balance control in VH provides a unique opportunity to assess the role of this geocentric frame of reference for balance control. The vestibular system is paramount for constructing the geocentric frame of reference as it senses linear and angular body accelerations (including Earth gravitational force). The alteration of this reference frame would be responsible for the trunk deviation relative to the earth-vertical that is commonly observed when VH individuals stand on a moving surface [START_REF] Horak | Vestibulospinal control of posture[END_REF].

Similar body deviations were also reported in cats with bilateral labyrinthectomy during comparable stance perturbations [START_REF] Macpherson | Bilateral vestibular loss in cats leads to active destabilization of balance during pitch and roll rotations of the support surface[END_REF]. With impaired otoliths-based gravitational reference (see Day & Fitzpatrick, 2005 for a review), the nervous system would be able to detect body motion relative to the supporting surface (through the processing of somatosensory inputs) but would be impaired when determining how the body is moving relative to the earth-vertical (see [START_REF] Horak | Vestibulospinal control of posture[END_REF]. Other cat studies found delayed postural responses following translations of the supporting surface when somatosensory inputs were removed (with a large dose of Pyridoxine, [START_REF] Stapley | Automatic postural responses are delayed by pyridoxine-induced somatosensory loss[END_REF], but not after bilateral labyrinthectomy [START_REF] Inglis | Bilateral labyrinthectomy in the cat: effects on the postural response to translation[END_REF]. Therefore, the greater incidence of falls and the deteriorated quality of life in individuals with VH [START_REF] Herdman | Vestibular rehabilitation of patients with vestibular hypofunction or with benign paroxysmal positional vertigo[END_REF][START_REF] Ward | Prevalence and impact of bilateral vestibular hypofunction: Results from the 2008 US national health interview survey[END_REF]Zingler et al., 2008;[START_REF] Pavlou | The assessment of increased sensitivity to visual stimuli in patients with chronic dizziness Marousa[END_REF] even years after the onset of their first symptoms, cannot solely be attributed to a deficit in postural reactions. Rather, vestibular disorder complications could be linked to an impaired representation of body orientation and motion relative to the external world (i.e., earth gravity vertical).

It is well known that sensory loss or decrease in the reliability of sensory information can alter the processing of the spared sensory modalities. In such cases, intact sensory inputs are upregulated, and are given more weight by the central nervous system [START_REF] Angelaki | Time course of sensory substitution for gravity sensing in visual vertical orientation perception following complete vestibular loss[END_REF][START_REF] Bronstein | Multisensory integration in balance control[END_REF][START_REF] Lopez | Changes of visual vertical perception: A long-term sign of unilateral and bilateral vestibular loss[END_REF][START_REF] Toupet | Subjective visual vertical in idiopathic bilateral vestibular hypofunction: enhanced role of vision, neck, and body proprioception[END_REF]. Surprisingly, despite the fact that the plantar mechanoreceptors provide useful information about body position and velocity relative to the ground and to the earth-vertical (e.g., [START_REF] Carriot | Perceived body orientation in microgravity: Effects of prior experience and pressure under the feet[END_REF]Morasso & Schiepatti, 1999;Mouchnino & Blouin, 2013), it has yet to be determined whether cutaneous inputs are enhanced in absence of vestibular information. Upregulation of cutaneous inputs in VH individuals would support studies showing that the brain can selectively increase the functional gain of task-relevant sensory inputs, including those from somatosensory systems during challenging equilibrium tasks [START_REF] Duysens | Gating of sensation and evoked-potentials following foot stimulation during human gait[END_REF][START_REF] Mouchnino | Facilitation of cutaneous inputs during the planning phase of gait initiation[END_REF][START_REF] Saradjian | Cortical facilitation of proprioceptive inputs related to gravitational balance constraints during step preparation[END_REF][START_REF] Saradjian | Independent early and late sensory processes for proprioceptive integration when planning a step[END_REF].

The primary aim of the present study was to explore whether the response of the somatosensory cortex to foot cutaneous stimulation, increases in individuals with bilateral vestibular hypofunction. We tested this hypothesis by measuring the amplitude of the P1N1 cortical response, extracted from the electroencephalogram, following the electric stimulation of the plantar sole (i.e., somatosensory-evoked potential (SEP) technique). We reasoned that the amplitude of the P1N1 should be a key variable in comparing the weighting of foot cutaneous inputs between VH and healthy control individuals. Importantly, whether or not foot cutaneous inputs were to be upregulated in individuals with VH, balance-related issues are still reported in patients with vestibular hypofunction [START_REF] Herdman | Vestibular rehabilitation of patients with vestibular hypofunction or with benign paroxysmal positional vertigo[END_REF][START_REF] Ward | Prevalence and impact of bilateral vestibular hypofunction: Results from the 2008 US national health interview survey[END_REF]Zingler et al., 2008). This suggests that the sensory compensation if present, would not suffice to create a reliable gravitational reference frame.

Particularly, the mechanical action of gravity exerted on all body parts ultimately generates forces on the supporting surface depending on body configuration [START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF]. Among these forces, the shear forces that are small relative to the weight force during natural quiet standing, are readily detectable by the tactile receptors (Morasso & Schieppati, 1999;[START_REF] Knellwolf | Microneurography from the posterior tibial nerve: a novel method of recording activity from the foot in freely standing humans[END_REF]. Along these lines, our second aim was to test if increasing the differential forces between both feet would enhance the internal representation of body orientation and motion relative to a gravitational reference frame in individuals with VH. During upright standing, the pressure exerted under the loading foot increases while the pressure exerted under the unloading foot simultaneously decreases [START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF]. This loading/unloading mechanism provides gravity-related tactile information as slow adapting afferent fibers (e.g., Merkel and Ruffini) from the plantar sole increase their firing rate resulting from changes in normal force applied to the skin (Vallbo & Johansson, 1984;see Macefield, 2005 for a review). Therefore, the impulse discharge pattern alternates between the feet depending on the loading or unloading. This phenomenon is reminiscent of the effect of body tilts on vestibular neurons in healthy individuals. The neurons located in the labyrinth ipsilateral to the tilt increase their discharge rates while neurons of the contralateral labyrinth concomitantly decrease theirs [START_REF] Fernandez | Physiology of peripheral neurons innervating otolith organs of the squirrel monkey. II. Directional selectivity and force response relations[END_REF]. Shaped by this push-pull like organization of the vestibular system, vestibular signals provide key information on neural computation of gravity and body deviation [START_REF] Angelaki | Neurons compute internal models of the physical laws of motion[END_REF][START_REF] Merfeld | Neural processing of gravitoinertial cues in humans. III. Modeling tilt and translation responses[END_REF]. We tested the impact of increasing loading/unloading on plantar cutaneous information in the mediolateral (ML) axis by comparing the cortical response to plantar sole electric stimulation while a participant (individuals with VH and Controls) stood upright without or with a 1 kg mass added at each wrist.

We hypothesized that adding a weight to both wrists should enhance the detection of body sways through plantar sole mechanoreceptors due to improve sensorimotor integration (assign larger weight to proprioceptive inputs), resulting more accurate encoding of body sways in the gravitational field in the absence of vestibular information. The right posterior parietal cortex (rPPC) is a pivotal region for processing body motion relative the gravitational field (see Pfeiffer et al., 2014, for a review). Observing an increased activity of the rPPC when individuals with VH wore the added mass would support our hypothesis. The approach we used to test this hypothesis was based on the current consensus that functional processing of sensory inputs is associated with the modulation of band-specific neural oscillation power [START_REF] Haegens | Top-down controlled alpha band activity in somatosensory areas determines behavioral performance in a discrimination task[END_REF][START_REF] Lebar | On the neural basis of sensory weighting: Alpha, beta and gamma modulations during complex movements[END_REF][START_REF] Van Ede | Beyond establishing involvement: Quantifying the contribution of anticipatory α-and β-band suppression to perceptual improvement with attention[END_REF][START_REF] Zumer | Occipital Alpha Activity during Stimulus Processing Gates the Information Flow to Object-Selective Cortex[END_REF][START_REF] Pfurtscheller | Event-related EEG/MEG synchronization and desynchronization: basic principles[END_REF]. We predicted that with the added mass, this modulation would specifically target PPC alpha and beta bands power, which have been associated with the processing of tactile information [START_REF] Van Ede | Beyond establishing involvement: Quantifying the contribution of anticipatory α-and β-band suppression to perceptual improvement with attention[END_REF], notably relative to external frames of reference [START_REF] Ruzzoli | Alpha stimulation of the human parietal cortex attunes tactile perception to external space[END_REF].

Participants and methods

Participants

Ten participants (including 3 women) with bilateral idiopathic vestibular hypofunction took part in the study. Their clinical evaluations are presented in Table 1. They were aged between 44 and 71 (mean 61 years) and their body mass index ranged from 22 and 33 (mean 26, i.e., healthy weight). For all patients, the vestibular pathology started at least 5 years prior to the study. The bilateral vestibular hypofunction was confirmed by a major deficit in the four caloric vestibular tests (the speed of the nystagmus' slow phase was below 6º/s for all four caloric tests, Barany 1906). The canal function was assessed by measuring the ocular response to low frequency rotations (0.05 Hz, period 20 s), the saccular and the utricular function were tested by measuring vestibular-evoked myogenic potentials (VEMP, respectively on the sternocleidomastoid muscle and on the inferior eyelid). Most patients showed no ocular or muscular response to the rotatory and VEMP tests for utricular function, but they showed muscular response to the VEMP for saccular function. All additional pathologies (e.g., hearing loss) and/or other vestibular pathologies led to the patient's exclusion from the study. A control group (Controls) was composed of 10 self-declared healthy participants (including 3 women).

They were aged between 52 and 64 (mean 58 years) and their body mass index ranged between 22 and 31 (mean 25, healthy weight).

Task and stimulation procedure

Participants were asked to stand still in a comfortable position with their arms alongside their body and eyes open (Fig. 1A). Particular attention was paid to maintain constant the selfselected foot position before each trial to avoid potential effect of stance width on balance control [START_REF] Day | Effect of vision and stance width on human body motion when standing: implications for afferent control of lateral sway[END_REF]. Before the first session, participants were asked to self-select a comfortable side-by-side foot position and small sticker were positioned in front of each big toe. This allowed us to verify that the feet position remained similar (with an accuracy of a few mm) throughout the experiment.

Participants were asked to hold their gaze on a small cross fixed on a wall, 3m ahead at eye level. This distance was chosen to avoid vergence eye movements which can be impaired in individuals with VH leading to instability in the mediolateral direction [START_REF] Kapoula | Vergence and standing balance in subjects with idiopathic bilateral loss of vestibular function[END_REF].

We recorded cortical activity, and pelvis and head accelerations (see below) of all participants (i.e., VH and Controls) in 16 trials of 15 s. Eight trials were performed while participants wore a 1kg weight bracelet at each wrist (Weight condition) and 8 trials were performed without mass (No weight condition). To avoid potential post-effects of carrying extra mass (e.g., on the frame of reference used for perceiving earth-vertical), the No weight condition (i.e., condition closest to everyday life) was always tested first.

We chose to add 1 kg (i.e., 10 N) at each wrist to increase reaction force under each foot. Such load does not cause disequilibrium while standing upright but influences balance control (e.g., [START_REF] Aruin | The role of motor action in anticipatory postural adjustments studied with self-induced and externally triggered perturbations[END_REF][START_REF] Mouchnino | Online control of anticipated postural adjustments in step initiation: evidence from behavioral and computational approaches[END_REF][START_REF] Simoneau | Effects of predictive mechanisms on head stability during forward trunk perturbation[END_REF]. Obviously, the added weight increased the body gravitational force. Compared to adding weights at the ankle joints, body sways with weight at both wrists exacerbate the shear forces under each foot (see modeling results below). The current study was conducted in a clinical setting that was not equipped with force platforms. Consequently, we developed a biomechanical model to verify 1) that adding a weight of 20 N altered the vertical and shear reaction forces under both feet during natural body sways and, 2) that this alteration is greater when the weight is added to the wrists (i.e., 10N strapped around each wrist) than to the ankles. For the model, we simulated a double inverted pendulum model (mass = 75 kg, height = 171 cm which correspond to the mean of both groups) with stiffness (K) and damping (B) at the ankle and hip joints (Fig. 1A). The biomechanical model only swayed along the frontal plane with an amplitude of 10 mm (i.e., amplitude of center of mass displacement commonly recorded during quiet standing, e.g., [START_REF] Lafond | Comparison of three methods to estimate the center of mass during balance assessment[END_REF] with frequencies of 0.25, 0.5 or 0.75 Hz. Compared to simulations without the added weight, the outcomes of the model revealed that, as expected, adding a weight of 20 N increased the vertical ground reaction forces under each foot (Fig. 1B) regardless of the added weight location (i.e., wrists or ankles). More importantly, adding weights at the wrists increased the mediolateral shear forces compared to the control condition (i.e., no weight) while adding the weights around the ankle joints decreased these forces (Fig. 1B). Results of the model confirmed that adding weights at both wrists increased the mediolateral shear reaction forces, thereby augmenting the stimulation of the plantar sole mechanoreceptors. Because of the crossed organization of the ascending sensory pathways, analyzing cortical processes when the laterality of the stimulated foot differs between participants is tricky. In the present study, all participants self-declared being right-handed. We stimulated the left foot for all participants because it is considered as the "postural" foot in most right-handed individuals [START_REF] Coren | The lateral preference inventory for measurement of handedness, footedness, eyedness, and earedness: Norms for young adults[END_REF]. Electric stimulation of the foot was delivered via a set of 2 electrodes (5 × 9 cm Platinium Foam Electrodes) using an isolated bipolar constant current stimulator (DS5 Digitimer, Welwyn Garden City, UK). Using the same technique as in previous studies [START_REF] Fabre | Somatosensory cortical facilitation during step preparation restored by an improved body representation in obese patients[END_REF][START_REF] Mouchnino | Facilitation of cutaneous inputs during the planning phase of gait initiation[END_REF], the cathode was located under the metatarsal region and the anode underneath the heel (Fig. 2A). Because cutaneous afferents from plantar sole are distributed across the foot sole (with some regional variation, [START_REF] Strzalkowski | Cutaneous afferent innervation of the human foot sole: What can we learn from single-unit recordings[END_REF], the stimulations were deemed to activate most nervous fibres linked to mechanoreceptors. Each stimulation consisted of a single rectangular 10-ms pulse. For each participant, stimulation intensity was set 25% above the tactile detection threshold (i.e., below the motor threshold) and was innocuous. This threshold was estimated before the experimental session using a forcedchoice adaptive method [START_REF] Ehrenstein | Psychophysical methods. Modern techniques in Neuroscience research[END_REF], while the participants were standing without added mass and were gazing at the fixation cross. The stimulation intensities used for participants with VH and Controls are reported in the Results section. All stimuli were clearly detectable but did not create illusion of body motion, as mechanical stimulation of the foot can do [START_REF] Kavounoudias | The plantar sole is a 'dynamometric map' for human balance control[END_REF]. Moreover, a previous study using stimulation intensities similar to those employed here showed that such stimulations are not strong enough to evoke reflextriggered postural response that could alter normal body sways [START_REF] Mouchnino | Facilitation of cutaneous inputs during the planning phase of gait initiation[END_REF].

Fourteen stimulations were delivered in each 15 s trial, giving a total of 98 stimulations for each condition (i.e., No-weight and Weight). The inter-stimulus interval ranged between 800 and 1200 ms to restrain anticipation of the next stimulation by the participants. Because the inter-stimulus intervals were > 800 ms, the evoked response was unlikely influenced by the previous stimulation. When the inter-stimulus intervals was < 500 ms, a decrease in the evoked response was commonly observed due to the previous stimulation [START_REF] Burke | Interfering cutaneous stimulation and the muscle afferent contribution to cortical potentials[END_REF].

Electrophysiological recordings and analyses

Electroencephalographic (EEG) activity was recorded continuously from 64 Ag/AgCl surface electrodes embedded in an elastic cap (10-20 system, BioSemi ActiveTwo system: BioSemi, Netherlands). Specific to the BioSemi system, "ground" electrodes were replaced by As in previous studies [START_REF] Duysens | Gating of sensation and evoked-potentials following foot stimulation during human gait[END_REF][START_REF] Fabre | Large postural sways prevent foot tactile information from fading: neurophysiological evidence[END_REF][START_REF] Staines | Task-relevant selective modulation of somatosensory afferent paths from the lower limb[END_REF], the largest responses to foot stimulations were observed at electrode Cz (vertex). The SEPs were recorded at this electrode which overlays the sensorimotor feet regions located on the inner surface of the longitudinal fissure. These SEPs were obtained by averaging, for each participant and condition (i.e., No-weight and Weight), all synchronized epochs relative to the electrical stimulus (Fig. 2A). We searched, from 25 ms post-stimulus onward (i.e., after stimulation conditions), the latencies of P1 and N1 were respectively of 59 ± 9 ms and 97 ± 16 ms and were similar as reported in previous studies using similar methods (e.g., [START_REF] Fabre | Somatosensory cortical facilitation during step preparation restored by an improved body representation in obese patients[END_REF][START_REF] Duysens | Gating of sensation and evoked-potentials following foot stimulation during human gait[END_REF]. Amplitudes of the P1N1 were measured peak-to-peak between these positive and negative deflections (Figure 2A). Time frequency analyses were performed in EEG source space. The neural sources were estimated with dynamical Statistical Parametric Mapping (dSPM, [START_REF] Dale | Dynamic statistical parametric mapping: combining FMRI and MEG for high-resolution imaging of cortical activity[END_REF] implemented in the Brainstorm software [START_REF] Tadel | Brainstorm: a user-friendly application for MEG/EEG analysis[END_REF]. A three-shell (i.e., scalp, outer skull, and inner skull) sphere boundary element model (BEM) was used to compute the forward model on the anatomical MRI brain template from the Montreal Neurological Institute (MNI Colin27). The data was transformed into time-frequency domain using Morlet wavelet transforms. Due to the Gaussian shape of the frequency response, Morlet wavelets produce smooth-looking time-frequency plots that can be easily interpreted (see Figure 3B, where activities in distinct frequency bands can be identified). We used a 1Hz central frequency (full width at half maximum (FWHM), tc=3sec) that offered a good compromise between temporal and spectral resolutions [START_REF] Allen | Time-frequency analysis of movement-related spectral power in EEG during repetitive movements: A comparison of methods[END_REF]. We computed alpha (8-12 Hz) and beta (15-25 Hz) event related desynchronization/synchronization (ERD/ERS) using a baseline window (-500 to -50 ms). We purposely selected a 300 ms time window for the ERS/ERD computation [-100; +200 ms relative to the stimulation] because this temporal window allows to include several oscillations cycles. We then extracted the power average of alpha and beta event-related ERS/ERD from 50 to 100 ms (i.e., interval encompassing the P1N1 SEP). These computations were performed in regions of interest (ROIs) from the right and left posterior parietal cortex (PPC) (see Fig. 3A). These ROIs were defined based on the Destrieux cortical atlas [START_REF] Destrieux | Automatic parcellation of human cortical gyri and sulci using standard anatomical nomenclature[END_REF] and had similar numbers of vertices. The right PPC was chosen because it is a key region for integrating tactile information into body representation [START_REF] Tsakiris | The role of the right temporo-parietal junction in maintaining a coherent sense of one's body[END_REF][START_REF] Wang | Rhythm makes the world go round: An MEG-TMS study on the role of right TPJ theta oscillations in embodied perspective taking[END_REF]) and on the hypothesis that, for the VH participants, the added mass should enhance the internal representation of body orientation and motion relative to a gravitational reference frame. Consequently, we predicted that the patients' right PPC should show greater alpha and beta ERDs in the Weight condition than in the No-weight condition.

The left PPC, which is not considered to be involved in integrating tactile information into body representation, served as a control ROI. Due to extensive artifact in the beta band, data of 2 participants of the control group were excluded from the time-frequency analysis of the left PPC (i.e., control ROI).

Behavioral recordings and analyses

Body stability was assessed by measuring head and pelvis accelerations. Head acceleration was measured with a triaxial accelerometer (Model 4630: Measurement Specialties, USA) placed on the participants' chin. The accelerometer was securely fixed on a chin-cup. This chin-cup reduced the possibility to move the jaw. Pelvis acceleration was estimated using a triaxial accelerometer (Model ADXL335: Analog Devices, USA) located on the sacrum (i.e., an approximation of the body center of mass, [START_REF] Thirunarayan | Comparison of three methods for estimating vertical displacement of center of mass during level walking in patients[END_REF]. Signals from these sensors were recorded at a sampling rate of 1000 Hz and low-pass filtered (Butterworth 4th order, 10 Hz cutoff frequency). Head and pelvis accelerations were analyzed in both anteroposterior (AP) and mediolateral (ML) axes. We rebased, the acceleration timeseries along the AP and ML axes, with respect to their means calculated during the first 500 ms of each 15 s -trial. This procedure allowed a common baseline for all trials. Then, we computed, for each trial, the integrals of the ML and AP time-series to estimate the amount of acceleration in each direction and calculated the mean ML and AP accelerations for each condition and group. These analyses provided an estimate of head and pelvis motion in ML and AP directions for both groups (VH and Controls) and both experimental conditions (i.e., No-Weight and Weight).

Statistical Analyses

The behavioral and neurophysiological dependent variables were submitted to separate analyses of variance (ANOVA) as the data were normal distributed (Kolmogorov-Smirnov test). For SEP amplitude and latency and alpha/beta ERS/ERD, mixed-design ANOVAs were used for mean comparisons with Condition (No-weight, Weight,) as a within-subject factor and Group (VH, Controls) as a categorical predictor. For head and pelvis accelerations, we used 2 (Group) x (Condition) x 2 (Direction: AP and ML) ANOVAs with repeated measures on the last two factors. For post-hoc analyses, we used Newman-Keuls comparisons. When main effects could be solely explained by a higher order interaction, only the break-down of the interaction are reported. Significance threshold was set at p < 0.05 for all analyses.

Results

Foot stimulation perceptual threshold

The mean thresholds for perceiving foot stimulations were 6.95 ± 2.3 mA for the VH and 6.83 ± 1.3 mA for Control groups. The result of a t-test indicated that perceptual thresholds did not differ between both groups (t = -0.23; p = 0.81).

SEP response to foot tactile stimulation

Consistent with an upregulation of cutaneous inputs for the VH group, the evoked response to the foot cutaneous stimulation was larger in VH than in Control groups (Fig. 2A).

This was confirmed by the ANOVA showing that P1N1 amplitude was greater in VH than in Control groups (Figure 2B-C; F1, 18 = 5.84; p = 0.02). Adding a 2 kg mass did not change the SEP amplitude, neither as a main effect (F1, 18 = 2.19; p = 0.15) nor as an interaction between Weight x Condition (F1, 18 = 0.36; p = 0.55).

For the SEP latency, the ANOVA did not reveal a significant effect of Group (p = 0.82).

However, the main effect of Condition failed just short of the conventional 0.05 cut-off value for statistical significance (F1, 18 = 4.16; p = 0.07; Weight condition: 58 ms ± 5, No-weight condition: 61 ms ± 8). No Group by Condition interaction was observed (F1, 18 = 0.36; p = 0.55).

EEG spectral analyses

Consistent with an increased activity of the right PPC when adding weight in VH (Fig. 3B-C), the alpha power desynchronization increased for the VH group whereas for the Control group, adding weight to the wrists had no significant effect (significant Group by Condition interaction, F1, 15 = 6.01; p = 0.02). Similar results were observed for the beta band frequency without reaching the significant value for statistical significance (F1, 15 = 4.1; p = 0.062). No significant Condition modulation was observed in the left PPC for both groups, neither for alpha, nor for beta frequency bands (F1, 15 = 1.17; p = 0.29 and F1, 16 = 3.68; p = 0.072 respectively for alpha and beta).

Behavioral results

Variables related to balance control were analyzed to compare body sways between VH and Control groups, and to verify whether the added mass altered body sways. For pelvis accelerations (Fig. 4A), the ANOVA revealed a main effect of Group; pelvis accelerations being greater in the Control than in the VH groups (F1, 17 = 79; p <0.001). The pelvis acceleration of both groups was greater in the Weight compared to the Non-Weight conditions (main effect of Condition: F1, 17 = 8; p = 0.01). The analyses also revealed a significant interaction between Group and Direction (F1, 17 = 27; p <0.001). The breakdown of this interaction revealed that pelvis acceleration was greater in the AP direction than in ML for the Control group (p < 0.001), but did not differ between AP and ML directions for the VH group (p = 0.89).

Interestingly, for the head accelerations (Fig. 4B), the analyses yielded different results; the changes in head acceleration varied across groups and directions (Group by Direction interaction: F1, 17 = 19.2; p <0.001). For the Control group, contrary to what was observed for the pelvis, the head accelerations were similar along the AP and ML directions (p = 0.46) and was uninfluenced by the added mass (p = 0.51). The mean head acceleration of the Control group was not different than the threshold for vestibular detection (0.048 m.s -2 ; [START_REF] Gianna | Thresholds for detection of motion direction during passive lateral whole-body acceleration in normal participants and patients with bilateral loss of labyrinthine function[END_REF] [t-test for means against a value; t8 = 1.09, p = 0.30 and t8 = 1.03 p = 0.33 respectively for Weight (0.055 ± 0.02 m. s-2 ) and No-weight (0.052 ± 0.01 m.s -2 ) conditions]. Conversely, for VH group, the head accelerations were different between AP and ML directions; head acceleration was greater in AP direction (p < 0.001). No main effect of Group (F1, 17 = 0.34; p = 0.56) or Condition (F1, 17 = 0.01; p = 0.91) was observed.

Discussion

Due to the properties of its sensors and their afferents (e.g., otolith neurons) that cannot be turned off [START_REF] Fernandez | Physiology of peripheral neurons innervating otolith organs of the squirrel monkey. II. Directional selectivity and force response relations[END_REF]), the vestibular system is a valuable source of information used by the brain to preserve balance during bipedal stance. Being sensitive to linear and angular head accelerations, it provides key information for determining the earth gravity vertical and for estimating how the body deviates from it (Day & Fitzpatrick, 2005).

Here, we found evidence that this function could be supported, at least in part, by foot cutaneous inputs in individuals with bilateral vestibular hypofunction. Our results provide valuable insights into the mechanisms underpinning this sensory substitution.

We found that the sensorimotor cortex response to foot cutaneous stimulations was much greater in VH participants than in aged-matched healthy individuals. Although the present experimental protocol does not account for the origin of this finding, the increased VH's SEP unlikely resulted from an increase in the sensitivity of the plantar sole tactile receptors. Indeed, our results showing similar detection thresholds for foot cutaneous stimulation between VH and Controls provides evidence against this possibility. Distinct potential mechanisms could explain the larger SEP in the VH group. The multimodal cells, where vestibular and somatosensory inputs converge, could have augmented their responsiveness to cutaneous stimulations following reduced vestibular inputs. Such cells are present, for example, in the somatosensory cortex [START_REF] Zarzecki | Convergence of sensory inputs upon projection neurons of somatosensory cortex: vestibular, neck, head, and forelimb inputs[END_REF]Guldin and Grüsser, 1998) and in the thalamus [START_REF] Lopez | The thalamocortical vestibular system in animals and humans[END_REF]. Such trade-off mechanism between somatosensory and vestibular responsiveness has been observed in upright standing healthy individuals by [START_REF] Mian | Violation of the craniocentricity principle for vestibularly evoked balance responses under conditions of anisotropic stability[END_REF]. These authors showed that the gain of the vestibular-evoked balance response was lessened during finger tactile stimulation most likely enhancing the weight of somatosensory inputs as shown by [START_REF] Bolton | The impact of light fingertip touch on haptic cortical processing during a standing balance task[END_REF] in participants touching lightly an external support. Besides, the thalamus, which is the principal sensory relay station, can also send potentiated inputs to the brain through the effect of the reticular nucleus which intercepts and modulates all corticothalamic (including from prefrontal cortex) and thalamocortical communications [START_REF] Guillery | Paying attention to the thalamic reticular nucleus[END_REF][START_REF] Zikopoulos | Prefrontal projections to the thalamic reticular nucleus form a unique circuit for attentional mechanisms[END_REF]. Because cells that respond exclusively to vestibular stimulations have not yet been discovered in the brain, it is unlikely that greater SEP in participants with VH resulted from cross-modal plasticity. This phenomenon allows for instance cells which respond to visual stimulation in sighted individuals to be sensitive to cutaneous inputs in the blinds [START_REF] Bavelier | Cross-modal plasticity: where and how?[END_REF][START_REF] Sathian | Cross-modal plasticity of tactile perception in blindness[END_REF].

Our behavioral data support the suggestion that somatosensory compensation occurred in absence of vestibular cues. For the VH group, pelvis accelerations were reduced along both directions (i.e., AP and ML) while for the Control group, we observed a reduction in head accelerations in both directions (see Fig. 3). This reduced head accelerations could imply that in healthy individuals, the head served as a reference frame for controlling upright standing and locomotion. This strategy requires vestibular functioning [START_REF] Bent | When is vestibular information important during walking[END_REF][START_REF] Horak | Vestibulospinal control of posture[END_REF][START_REF] Pozzo | Head stabilization during locomotion: perturbations induced by vestibular disorders[END_REF]. Note that, along the frontal plan, the mean head acceleration of the control participants was not different than the acceleration threshold for vestibular detection [START_REF] Gianna | Thresholds for detection of motion direction during passive lateral whole-body acceleration in normal participants and patients with bilateral loss of labyrinthine function[END_REF]. Functionally, such near-vestibular threshold head acceleration puts the vestibular system in an optimal state for being stimulated in case of unexpected loss of balance.

For the participants with VH, we observed smaller pelvis accelerations. The brain, in the absence of vestibular information, likely determined body sways amplitude and direction with respect to the base of support through somatosensory information (e.g., lower limb somatosensory and plantar sole afferents signals). This change in reference frame involves a tight control of the pelvis. In particular, the body sways can be regulated within a small area by reducing the amplitude of the centre of pressure displacement (not recorded in the present study) and increasing its frequency. Such tactile exploration, by changing the pressure onto the ground with the feet, likely contributes to improve the signal-to-noise ratio of sensory cues and to determine the limit of stability [START_REF] Fabre | Large postural sways prevent foot tactile information from fading: neurophysiological evidence[END_REF][START_REF] Latash | Movement sway: changes in postural sway during voluntary shifts of the center of pressure[END_REF][START_REF] Murnaghan | Exploratory behavior during stance persists with visual feedback[END_REF].

The increase in plantar sole afferents through this strategy could be particularly important in the absence of vestibular information. Larger upregulation of cutaneous inputs could have contributed to the greater SEP observed in the participants with VH compared to controls.

Attentional processes, which are known to increase the excitability to sensory input (Eimer & Forster, 2003;[START_REF] Sambo | Sustained Spatial Attention in touch: modality-specific and multimodal mechanisms[END_REF][START_REF] Bolton | Age-related loss in attention-based modulation of tactile stimuli at early stages of somatosensory processing[END_REF] could be part of the mechanisms that contributed to this sensory upregulation.

Besides and, perhaps more surprisingly, for participants with VH, we did not observe difference in the pelvis accelerations along the AP and ML directions. In contrast, the healthy individuals exhibited larger pelvis accelerations in the AP compared to the ML directions as reported in previous studies [START_REF] Collins | Open-loop and closed-loop control of posture : A random-walk analysis of center-of-pressure trajectories[END_REF][START_REF] Lord | Lateral stability, sensorimotor function and falls in older people[END_REF][START_REF] Johnson-Hilliard | Lateral balance factors predict future falls in community-living older adults[END_REF][START_REF] Winter | Unified theory regarding A/P and M/L balance in quiet stance[END_REF]. Less accurate estimation of body sways amplitude and direction, due to the absence of vestibular information, likely increases the perception of instability leading to tight control of body sways along the frontal and sagittal planes. A tight control of body sways is commonly observed in healthy individuals when they are standing on an elevated surface [START_REF] Adkin | Postural control is scaled to level of postural threat[END_REF][START_REF] Carpenter | Surface height effects on postural control: a hypothesis for a stiffness strategy for stance[END_REF][START_REF] Lim | Postural threat influences vestibular-evoked muscular responses[END_REF] or in patients with postural phobic vertigo [START_REF] Krafczyk | Increased body sway at 3.5-8 Hz in patients with phobic postural vertigo[END_REF]. The increased risk of injury leads to tight balance control.

Our second hypothesis was that potentiating the differential pressure between both feet when standing could contribute to restoring the internal representation of the patients' body orientation and motion relative to a gravitational reference frame hosted in the right PPC [START_REF] Pérennou | Lateropulsion, pushing and verticality perception in hemisphere stroke: a causal relationship?[END_REF]Dai et al., 2021 a, b;[START_REF] Pfeiffer | The vestibular system: A spatial reference for bodily self-consciousness[END_REF]. In the present study, this potentiation was done by adding a 1 kg mass to both wrists. Our biomechanical model indicated that this procedure magnified the differential pressure between both feet, and more specifically the shear forces. The analyses of the alpha and beta power recorded in VH participants are consistent with an enhanced internal presentation of the body when standing with the added mass. We found that adding a small weight on each hanging wrist resulted in a decrease of the right PPC alpha and beta power in the VH but not in the Control groups. The powers of the alpha and beta oscillations are inversely related to cortical excitability and to the reliance on sensory processing, respectively [START_REF] Neuper | Event-related dynamics of cortical rhythms: frequencyspecific features and functional correlates[END_REF][START_REF] Romei | Resting electroencephalogram alpha-power over posterior sites indexes baseline visual cortex excitability[END_REF][START_REF] Sauseng | Spontaneous locally restricted EEG alpha activity determines cortical excitability in the motor cortex[END_REF]. For VH participants, the decreased alpha and beta power observed in the right PPC might entail integration of cutaneous inputs into the representation of body orientation relative to the gravitational field. This cortical region is known to receive thalamic projections of

  All protocols and procedures were conducted in accordance with the ethical standards set out in the Declaration of Helsinki. All participants gave their written informed consent and procedures were approved by the local ethics committee of the Centre d'Explorations Fonctionnelles Oto-neurologiques Falguière (CEFON).

Table 1 :

 1 Patients characteristics. Negative values are for the maximal slow phase velocity when nystagmus are beating in opposite side as expected. Ocular (o) and cervical (c) Vestibular Evoked Myogenic Potential (oVEMP) Tests.

Figure 1 :

 1 Figure 1: A. Biomechanical model. Model represents a double inverted pendulum with movement at the ankle and hip joints. B Upper panel depicts peak-to-peak shear forces along the mediolateral axis and the lower panel presents peak-to-peak forces along the vertical axis. Data are for the three center of mass frequencies (i.e., 0.25, 0.5 and 0.75 Hz) and for the three conditions (i.e., without weight, and with the weights at the wrists or at the ankle joints).

  Common Mode Sense active and Driven Right Leg passive electrodes. The signal was preamplified at the electrode sites and post-amplified with DC amplifiers, digitized at a sampling rate of 1024 Hz (Actiview acquisition program, BioSemi). The signals of each electrode were referenced to the mean signal of all the 64 electrodes. Three external Ag/AgCl electrodes (positioned next to the right and left external canthus and under the left eye orbital) and one from the cap (FP1) controlled for ocular movements and blinks. After artefact rejection based on visual inspection, 87 ± 2% and 83 ± 8% of the trials were included in the analyses for participants with VH and Controls, respectively.

Figure 2 :

 2 Figure 2: A. Position of the stimulation electrodes underneath the left foot (inset). Average of the somatosensory evoked Potential over electrode Cz for a participant in the VH and the Control groups for the No weight condition. The vertical dashed line depicts the onset of the electric stimulation. B. Group means amplitude of the P1N1 SEP evoked by the electric stimulation for the No Weight and Weight conditions (error bars depict standard error of the means). C. Averaged Weight and No weight conditions SEP of each participant for all Controls and VH patients. The values are ranked in ascending order for each group.

Figure 3 :

 3 Figure 3: A. Location of the regions of interest (ROIs) on the anatomical MRI Colin 27 brain template used to compute cortical activations. Similar ROIs were defined for the left and right Parietal Posterior Cortex (rPPC). B. Time-frequency power (ERS/ERD) of the signals by means of a complex Morlet's wavelet transform applied on the ROIs for each trial of each VH patients then averaged across VH participants in the Weight and No weight conditions (left panel). Cooler colors indicate ERD (eventrelated desynchronization) while warmer colors indicate ERS (event-related synchronization). Mean difference between Weight and No weight conditions (right panel). Frequency bands up to 90 Hz are illustrated to present changes in brain electrocortical activities for delta, theta, alpha, beta and gamma bands. The artifact caused by the electric stimulation is present at stimulation onset. We epoched the data from -100 to 200 ms to emphasize changes in brain electrocortical activities (ERS/ERD) near plantar sole electrical stimulation. C. Group means power for the Alpha (8-12 Hz) and Beta (15-29 Hz) frequency bands computed during [50; 100 ms] time window for the No weight and Weight conditions (error bars depict standard error of the mean).

Figure 4 :

 4 Figure 4: A. Upper panel) Pelvis acceleration time-series, along the mediolateral (ML) and antero-posterior (AP) axes, for one representative participant in the Control and VH groups for the No weight condition. Bottom panel) mean of the integration of the pelvis acceleration along the mediolateral (ML) and anteroposterior (AP) directions for each participant for the No weight and the Weight conditions (error bars depict standard error of the mean). B. Group means integration of head acceleration along the mediolateral (ML) and anteroposterior (AP) directions for the No weight and Weight conditions (error bars depict standard error of the mean)

Funding

This research supported in part by grants from Natural Sciences and Engineering Research Council of Canada to Prof. M. Simoneau (grant #04068).

(which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

through SII/parietal ventral, [START_REF] Cavada | Posterior parietal cortex in rhesus monkey: II. Evidence for segregated corticocortical networks linking sensory and limbic areas with the frontal lobe[END_REF][START_REF] Disbrow | Cortical connections of the second somatosensory area and the parietal ventral area in macaque monkeys[END_REF] pathways.

In VH participants, the contribution of the PCC to balance control could have therefore been enhanced by exteroceptive tactile afferents in the Weight condition. This would be consistent with findings suggesting that the reliance on internal representations for controlling balance increases in contexts with impaired sensory inputs [START_REF] Blouin | Vestibular signal processing in a subject with somatosensory deafferentation: The case of sitting posture[END_REF][START_REF] Fabre | Somatosensory cortical facilitation during step preparation restored by an improved body representation in obese patients[END_REF]. The importance of cutaneous inputs in building a representation of the body in space has also been suggested by observations made by [START_REF] Day | Vestibular-evoked postural responses in the absence of somatosensory information[END_REF] in a study with a deafferented patient (IW). The patient, who suffered from a rare lack of cutaneous and proprioceptive inputs (see [START_REF] Cole | Pride and a daily marathon[END_REF], experienced a "floating sensation" and a "loss of contact with the world" when receiving galvanic vestibular stimulations (GVS) while seated. This unusual effect suggests an inability to create an accurate representation of the body in space in the absence of cutaneous inputs, even when vestibular information is preserved.

In conclusion, we found that plantar cutaneous inputs are upregulated in individuals with major vestibular information deficits. Such sensory upregulation points to an increased importance of cutaneous input in balance control in these patients. The contribution of foot tactile information in VH was also suggested by their change in balance control and postural organization, which were based on reduced pelvis acceleration rather than head acceleration.

Finally, increasing loading/unloading under both feet with small weights, we were able to identify within the right PPC altered somatosensory dynamics (i.e., increased alpha-beta desynchronization). This supports our hypothesis of an increased processing of tactile information relative to an external frame of reference to enable building up an internal representation of body state.


Julien Tissot

Paul Mazoyer

Matthieu Boquet

Understanding Lidar availability in the context of wind turbine control

Conclusions

One critical-to-success factor of Lidar-assisted control is the availability of Lidar data, especially when wind turbine design and size relies on the preview of critical wind condition from the Lidar. We analyze here Lidar availability in a variety of atmospheric conditions.

RAWS:

Rotor Averaged Wind Speed is a measure of the average longitudinal wind speed over the rotor area.

(Provided at 10 distances simultaneously ahead the wind turbine, from 50 to 200m, at a 4Hz frequency).

RAWS availability: is the percentage quantity of accepted RAWS for a complete data set.

(based on embedded quality control. Provided for each measurement distance independently).

Global RAWS availability: for the sake of clarity of this analysis, we define the Global RAWS availability as a representative indicator of RAWS availability over all measurement distances.

(its status is ok if at least 50% of distances have a RAWS status ok. Alternative definitions could be used, accordingly to the control algorithms involved. The conclusions of the atmospheric sensitivity analysis are expected to remain similar).

The analysis is based on SmartEole research project which involves a Leosphere nacelle-mounted turbine control lidar deployed on a MM82 Senvion wind turbine. The entire 4 months data set collected during the field test totalizes over 360 000 000 measurements.

Global RAWS availability is analyzed as a function of different environmental conditions to characterize sensitivities :

• Sensitivity to wind speed (it is critical to assess availability for different wind turbine regimes)

• Sensitivity to atmospheric visibility (the atmospheric visibility is the most useful indicator of the aerosol load in the atmosphere (which impacts Lidar signal-to-noise). It also indicates conditions of fog, haze, clear sky. It is given in km)

• Sensitivity to precipitations (the impact of rain and snow is evaluated, defined in mm/h.)

Availability increases with wind speed Precipitations do not reduce availability

• Lower availability at lower wind speed is explained by the blocking effect of blades.

• High availability from moderate to high wind speeds can be due to the aerosols dragging effect from wind.

• A very clear sky (high visibility) decreases availability because of reduced particles load. • No fog data was registered during the field test.

• Precipitation droplets increase the Lidar signal-to-noise.

• No higher precipitation levels have been recorded.

RAWS availability is all above 95% , reaching 97% around the system focalization distance at 100m

Availability increases with wind speed → Availability is above 99% from 7m/s wind speed.

Atmospheric visibility drives availability

During light-rain events (precipitation from 1 to 3 mm/h) the availability is not lower compared to non-rainy periods

The evaluation of Lidar measurement availability enables the implementation of correct wind turbine control strategies, while understanding its limitation led to developing performant mitigation strategies and get a trustworthy evaluation of the overall benefits of Lidar-assisted control.

Wind Iris TC (Turbine Control) Lidar is designed to maximize the availability of measurements. In this study, it is particularly promising to see increasingly high Lidar availability for wind speeds above 7m/s, a wind regime for which the Lidar-Assisted Control is key for mitigating wind turbine loads and safety.

Only few data points for visibility above 25km. Confirmation of trend needed


Paul Mazoyer

Julien Tissot

Yahaoui Salma

Leosphere

Vector averaging for Windcube in complex terrain

The Windcube vertical profiler achieved its 3rd classification in 2018 and is thus fully IEC compliant [1][2]. The classification output is an evaluation of the sensitivity of a wind measurement device accuracy with regards to environmental parameters such as turbulence intensity (TI), shear or rain.

The Windcube classification shed light on the link between accuracy and TI [3]. It also allowed to better understand the influence of the averaging method called "Scalar averaging" on the accuracy. The effect of using an alternative averaging method, called "Vector averaging", was investigated on Windcube data [4] and, for high TI environment, it provided wind measurements with lower uncertainty than "Scalar averaging", mainly by removing all sensitivity to TI:

Introduction Results

Conclusions

Method and Datasets

• 3 validations of Windcube FCR using "Vector" averaged dataset have shown equivalent or better performances , i.e. within 2% to 4% of the mast measurement, than when using "Scalar average".

• Achieving better performances will require an in-depth knowledge of the cup uncertainties in complex terrain.

• Using "Vector averaging" in complex terrain will not only provide better accuracy but also a lower sensitivity to turbulence intensity and enhance repeatability of remote sensors measurements.

Meet us at A1.150

Scalar averaging

Vector averaging The "Vector averaging" method has also been shown to perform well in simple terrain [3] but there are limited materials on its application in complex terrain.

The following section presents comparative results of FCR measurements, i.e. the solution for correcting raw Windcube data for complex terrain [5], using "Vector" or "Scalar" averaging as input.

Three Windcube datasets using FCR correction were re-processed with the "Vector averaging" method applied on the raw 1 HZ data. For each campaign, the Windcube was located on a complex terrain and close to a mast allowing a precise comparison between their measurements.

We then studied and compared the deviation between the mast and the Windcube measurements using FCR with "Vector" and "Scalar" averaged data as an input. The Windcube measurements with 10 minutes availability lower than 80% were discarded, the cup measurements filtered from mast wake, icing and wind speed below 4 m/s.

The sites are presented below:

1 st site: topography moderately complex, scattered forest with trees up to 10 meters, warm climate. Maximum slope: ~36%. Average slope: ~18%.

2 nd site: mountainous terrain, no trees, high altitude (~1000 meters). Maximum slope: ~37%. Average slope: ~15%.

3 rd site: complex terrain, scattered forest with trees up to 10 meters, warm climate. Maximum slope: ~50%. Average slope: ~19%. Using FCR with "Vector" averaged data produced wind measurements with lower or equivalent uncertainties than when using it with "Scalar" averaged data. This observation was anticipated as "Vector averaging" produces slightly lower wind speeds compared to "Scalar averaging" [2]. As the Windcube FCR is generally observed to be overestimating the cup measurement in complex terrain, the "Vector averaging" was expected to have a closer match with the mast. Besides, TI is generally higher in complex terrain due to topography and the "Scalar averaging" will then be subject to over-estimation.

Results are here presented for wind speed bins of 0.5 m/s as recommended [1].

1 st site: FCR with "Vector averaging" method produces wind measurements within 2% of the cup measurement. The mean deviation is 0.08 m/s. 2 nd site: FCR with "Vector averaging" method produces wind measurements within 2% of the cup measurement at high wind speed. There is no significant difference with "Scalar averaging". This could be due to a low TI onsite. The mean deviation is 0.15 m/s.

3 rd site: FCR with "Vector averaging" method produces wind measurements within 4% of the cup measurement. Deviations observed are equivalent, the "Vector averaging" underestimates the cup measurements while "Scalar averaging" overestimates, likely due to the very high complexity of the terrain (see Figure 2-c) to high TI conditions onsite. The mean deviation is -0.3 m/s.

  61400-12-1:ed2 -Wind energy generation systems -Part 12-1: Power performance measurements of electricity producing wind turbines; 2017 2. Summary of Classification of Remote Sensing Device; Franke K., Albers A.; 2018 3. Observed reduction of sensitivities by vector averaging; Albers A., Franke K.; 2018; Workshop on Vector Averaging Versus Scalar Averaging, Monostatic RSDs 4. Influence of Environmental Parameters on Windcube Accuracy final; Mazoyer P., Tissot J., Rebeyrat F.; 2018; Workshop on Vector Averaging Versus Scalar Averaging, Monostatic RSDs 5. Windcube FCR measurements; 2017; Leosphere

Figure

  Figure 1a-b : Windcube deviation to cup for vector and scalar averaging in simple terrain (red line) compared to the calibration uncertainty (black line). From [1].

  Figure 2a-b-c: Elevation profiles in the mean wind direction. Graphs from Google Earth®.

Figure

  Figure 3a-b-c: Deviation between cup and Windcube FCR using either « Vector» or « Scalar » averaging